Update

79 files changed, 1905 insertions, 703 deletions

diff --git a/.cspell.yaml b/.cspell.yaml
index 896a692..eff8b31 100644
--- a/.cspell.yaml
+++ b/.cspell.yaml
@@ -5,11 +5,19 @@ dictionaryDefinitions:
   - name: names
     addWords: true
     path: "./.cspell/my-cspell-dicts/names.txt"
+  - name: math
+    addWords: true
+    path: "./.cspell/my-cspell-dicts/math.txt"
+  - name: liquid
+    addWords: true
+    path: "./.cspell/liquid.txt"
 ignorePaths:
   - "./.cspell/"
 dictionaries:
   - latex
   - names
+  - math
+  - liquid
 words:
   - abs
   - gfm
@@ -21,9 +29,14 @@ words:
   - thms
   - jxir
   - srv
+  - bigg
+  - bigl
+  - bigr
+  - lparen
+  - rparen
   - mathbb
+  - mathscr
   - enspace
   - callouts
-  - injective
   - Closedness
   - vcs
diff --git a/.cspell/liquid.txt b/.cspell/liquid.txt
new file mode 100644
index 0000000..155a881
--- /dev/null
+++ b/.cspell/liquid.txt
@@ -0,0 +1,8 @@
+endaxiom
+endcorollary
+enddefinition
+endexample
+endlemma
+endproof
+endproposition
+endtheorem
diff --git a/.cspell/my-cspell-dicts b/.cspell/my-cspell-dicts
-Subproject 5f3c438665f20f7fa0ad17c35b87946426a0cc4
+Subproject 1e177028a4a8a45a9927479811411ee2beb5c59
diff --git a/_config.yaml b/_config.yaml
index 0bcb235..ae73c4c 100644
--- a/_config.yaml
+++ b/_config.yaml
@@ -22,13 +22,15 @@ search:
   tokenizer_separator: /[\s\-\u2013/]+/
 
 #color_scheme: dark
+#incremental: true
 
 kramdown:
   input: GFMKatex
-  parse_block_html: true
+    #  parse_block_html: true
   math_engine: katex
   math_engine_opts:
     {
+      output: "html",
       macros:
         {
           "\\NN": "\\mathbb{N}",
@@ -36,12 +38,19 @@ kramdown:
           "\\QQ": "\\mathbb{Q}",
           "\\RR": "\\mathbb{R}",
           "\\CC": "\\mathbb{C}",
+          "\\HH": "\\mathbb{H}",
           "\\KK": "\\mathbb{K}",
+          "\\FF": "\\mathbb{F}",
           "\\hilb": "\\mathcal{#1}",
           "\\abs": "\\lvert #1 \\rvert",
           "\\norm": "\\lVert #1 \\rVert",
           "\\braces": "\\lbrace #1 \\rbrace",
+          "\\angles": "\\langle #1 \\rangle",
           "\\innerp": "\\langle #1, #2 \\rangle",
+          "\\diam": "\\operatorname{diam}(#1)",
+          "\\dist": "\\operatorname{dist}(#1)",
+          "\\Re": "\\operatorname{Re}",
+          "\\Im": "\\operatorname{Im}",
           "\\dom": "D(#1)",
           "\\ran": "R(#1)",
           "\\graph": "G(#1)",
@@ -49,11 +58,12 @@ kramdown:
           "\\cspec": "\\sigma_c(#1)",
           "\\rspec": "\\sigma_r(#1)",
           "\\Res": "\\operatorname{Res}",
+          "\\co": "\\operatorname{co}",
+          "\\bal": "\\operatorname{bal}",
+          "\\aco": "\\operatorname{aco}",
         },
     }
 
-qed: '<span style="float:right;">$\square\enspace$</span>'
-
 callouts:
   definition:
     title: Definition
diff --git a/_plugins/enunciation.rb b/_plugins/enunciation.rb
index 75c777a..3301e0d 100644
--- a/_plugins/enunciation.rb
+++ b/_plugins/enunciation.rb
@@ -4,29 +4,36 @@ module Jekyll
     def initialize(tag_name, arg, parse_context)
       super
       @arg = arg.strip
+      @slugtemplate = Liquid::Template.parse("{{ string | slugify }}")
     end
 
     def render(context)
       text = super
       # If the enunciation ends with a KaTeX displayed equation,
       # then we remove the bottom margin.
-      if text[-3,3] == "$$\n"
+      if text[-5,5] == "  $$\n"
+        text += '  {: .katex-display .mb-0 }'
+      elsif text[-3,3] == "$$\n"
         text += '{: .katex-display .mb-0 }'
       end
       # Check if a description was given.
       if @arg.length < 1
         "{: .#{block_name} }\n #{text.gsub!(/^/,'> ')}"
       else
-        "{: .#{block_name}-title }\n> #{block_name.capitalize} (#{@arg})\n>\n#{text.gsub!(/^/,'> ')}"
+        if @arg[0] == "*"
+          title = @arg.delete_prefix("* ")
+        else
+          title = "#{block_name.capitalize} (#{@arg})"
+        end
+        slug = @slugtemplate.render( 'string' => title )
+        "{: .#{block_name}-title }\n> #{title}\n> {: \##{slug} }\n>\n#{text.gsub!(/^/,'> ')}"
       end
     end
 
   end
 end
 
-Liquid::Template.register_tag('definition', Jekyll::EnunciationTagBlock)
-Liquid::Template.register_tag('theorem', Jekyll::EnunciationTagBlock)
-Liquid::Template.register_tag('proposition', Jekyll::EnunciationTagBlock)
-Liquid::Template.register_tag('lemma', Jekyll::EnunciationTagBlock)
-Liquid::Template.register_tag('corollary', Jekyll::EnunciationTagBlock)
-Liquid::Template.register_tag('axiom', Jekyll::EnunciationTagBlock)
+# TODO: avoid reading configuration twice
+Jekyll.configuration({})['callouts'].each do |name, value|
+  Liquid::Template.register_tag(name, Jekyll::EnunciationTagBlock)
+end
diff --git a/_plugins/example.rb b/_plugins/example.rb
new file mode 100644
index 0000000..e149f23
--- /dev/null
+++ b/_plugins/example.rb
@@ -0,0 +1,12 @@
+module Jekyll
+  class ExampleTagBlock < Liquid::Block
+
+    def render(context)
+      text = super
+      "<span style=\"text-transform: uppercase; font-weight: bold; font-size: .75em;\">Example</span> &nbsp; #{text}"
+    end
+
+  end
+end
+
+Liquid::Template.register_tag('example', Jekyll::ExampleTagBlock)
diff --git a/_plugins/proof.rb b/_plugins/proof.rb
index ab824e5..415b37f 100644
--- a/_plugins/proof.rb
+++ b/_plugins/proof.rb
@@ -3,7 +3,7 @@ module Jekyll
 
     def render(context)
       text = super
-      "<span style=\"text-transform: uppercase; font-weight: bold; font-size: .75em;\">Proof</span> #{text} <span style=\"float:right;\">$\\square\\enspace$</span>"
+      "<span style=\"text-transform: uppercase; font-weight: bold; font-size: .75em;\">Proof</span> &nbsp; #{text} <span style=\"float:right;\">$\\square\\enspace$</span>"
     end
 
   end
diff --git a/_sass/custom/custom.scss b/_sass/custom/custom.scss
index 43a40c8..9b22a37 100644
--- a/_sass/custom/custom.scss
+++ b/_sass/custom/custom.scss
@@ -1,3 +1,4 @@
-blockquote.theorem {
-  border-left: none;
+ul {
+  margin-top: 0;
+  margin-bottom: 0;
 }
diff --git a/_scripts/new.lua b/_scripts/new.lua
index e57100d..308bf2c 100755
--- a/_scripts/new.lua
+++ b/_scripts/new.lua
@@ -65,16 +65,12 @@ if arg[1] == "page" then
 		end
 		io.write([[
 nav_order: 1
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .theorem-title }
-> {{ page.title }}
-> {: #{{ page.title | slugify }} }
->
-> ...
+{% theorem %}
+{% endtheorem %}
 
 {% proof %}
 {% endproof %}
@@ -100,7 +96,6 @@ elseif arg[1] == "dir" then
 nav_order: 1
 has_children: true
 has_toc: false
-# cspell:words
 ---
 
 # {{ page.title }}
diff --git a/pages/complex-analysis/index.md b/pages/complex-analysis/index.md
index d07109e..60c8ed2 100644
--- a/pages/complex-analysis/index.md
+++ b/pages/complex-analysis/index.md
@@ -1,8 +1,7 @@
 ---
 title: Complex Analysis
-nav_order: 2
+nav_order: 3
 has_children: true
-# cspell:words
 ---
 
 # {{ page.title }}
diff --git a/pages/complex-analysis/one-complex-variable/basics.md b/pages/complex-analysis/one-complex-variable/basics.md
index b30d18c..bbbbd30 100644
--- a/pages/complex-analysis/one-complex-variable/basics.md
+++ b/pages/complex-analysis/one-complex-variable/basics.md
@@ -3,21 +3,19 @@ title: Basics
 parent: One Complex Variable
 grand_parent: Complex Analysis
 nav_order: 1
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .theorem }
-> {: #holomorphic-function-is-constant-if-derivative-vanishes }
->
-> If the derivative of a holomorphic function vanishes
-> throughout a connected open subset of the complex plane,
-> then it must be constant on that set.
->
-> More generally, if the derivative of a holomorphic function vanishes
-> throughout an open subset of the complex plane,
-> then it must be constant on any connected component of that set.
+{% theorem %}
+If the derivative of a holomorphic function vanishes
+throughout a connected open subset of the complex plane,
+then it must be constant on that set.
+
+More generally, if the derivative of a holomorphic function vanishes
+throughout an open subset of the complex plane,
+then it must be constant on any connected component of that set.
+{% endtheorem %}
 
 {% proof %}
 {% endproof %}
diff --git a/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md b/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md
index ccdd0ea..3cf81f7 100644
--- a/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md
+++ b/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md
@@ -3,39 +3,33 @@ title: Cauchy's Integral Formula
 parent: One Complex Variable
 grand_parent: Complex Analysis
 nav_order: 3
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .theorem-title }
-> {{ page.title }}
->
-> Let $f : G \to \CC$ be a function holomorphic in an open subset $G \subset \CC$.
-> Let $\gamma$ be a contour in $G$ such that the interior of $\gamma$ is contained in $G$.
-> Then for any point $a$ in the interior of $\gamma$,
->
-> $$
-> f(a) = \frac{1}{2 \pi i} \int_{\gamma} \frac{f(z)}{z-a} \, dz.
-> $$
-> {: .katex-display .mb-0 }
+{% theorem * Cauchy's Integral Formula %}
+Let $f : G \to \CC$ be a function holomorphic in an open subset $G \subset \CC$.
+Let $\gamma$ be a contour in $G$ such that the interior of $\gamma$ is contained in $G$.
+Then for any point $a$ in the interior of $\gamma$,
+
+$$
+f(a) = \frac{1}{2 \pi i} \int_{\gamma} \frac{f(z)}{z-a} \, dz.
+$$
+{% endtheorem %}
 
 {% proof %}
 {% endproof %}
 
-{: .theorem-title }
-> {{ page.title }} (Generalization)
-> {: #cauchys-integral-formula-generalized }
->
-> Let $f : G \to \CC$ be a function holomorphic in an open subset $G \subset \CC$.
-> Then the $n$th derivative $f^{(n)}$ exists for every $n \in \NN$.
-> If $\gamma$ is a contour in $G$ such that the interior of $\gamma$ is contained in $G$,
-> then for any point $a$ in the interior of $\gamma$,
->
-> $$
-> f^{(n)} (a) = \frac{n!}{2 \pi i} \int_{\gamma} \frac{f(z)}{(z-a)^{n+1}} \, dz.
-> $$
-> {: .katex-display .mb-0 }
+{% theorem * Cauchy's Integral Formula (Generalization) %}
+Let $f : G \to \CC$ be a function holomorphic in an open subset $G \subset \CC$.
+Then the $n$th derivative $f^{(n)}$ exists for every $n \in \NN$.
+If $\gamma$ is a contour in $G$ such that the interior of $\gamma$ is contained in $G$,
+then for any point $a$ in the interior of $\gamma$,
+
+$$
+f^{(n)} (a) = \frac{n!}{2 \pi i} \int_{\gamma} \frac{f(z)}{(z-a)^{n+1}} \, dz.
+$$
+{% endtheorem %}
 
 {% proof %}
 {% endproof %}
@@ -50,20 +44,17 @@ and is often used to compute the integral.
 
 ## Many Consequences
 
-{: .theorem-title }
-> Cauchy's Estimate 
-> {: #cauchys-estimate }
->
-> Let $f$ be holomorphic on an open set containing the disc with center $a$ and radius $r>0$.
-> Then
->
-> $$
-> \norm{f^{(n)}(a)} \le \frac{n!}{r^n} \sup_{\abs{z-a} = r} \norm{f(z)} \qquad \forall n \in \NN.
-> $$
-> {: .katex-display .mb-0 }
+{% theorem * Cauchy's Estimate %}
+Let $f$ be holomorphic on an open set containing the disc with center $a$ and radius $r>0$.
+Then
+
+$$
+\norm{f^{(n)}(a)} \le \frac{n!}{r^n} \sup_{\abs{z-a} = r} \norm{f(z)} \qquad \forall n \in \NN.
+$$
+{% endtheorem %}
 
 {% proof %}
-From [{{ page.title }}](#cauchys-integral-formula-generalized)
+From [{{ page.title }}](#cauchy-s-integral-formula-generalization)
 for the circular contour around $a$ with radius $r$ we obtain
 
 $$
@@ -82,16 +73,14 @@ and the right hand side of the inequality reduces to the desired expression.
 
 Recall that an *entire* function is a holomorphic function that is defined everywhere in the complex plane.
 
-{: .theorem-title }
-> Liouville's Theorem
-> {: #liouvilles-theorem }
->
-> Every bounded entire function is constant.
+{% theorem * Liouville's Theorem %}
+Every bounded entire function is constant.
+{% endtheorem %}
 
 {% proof %}
 Consider an entire function $f$ and assume that $\norm{f(z)} \le M$ for all $z \in \CC$ and some $M > 0$.
 Since $f$ is holomorphic on the whole plane, we may make
-[Cauchy's Estimate](#cauchys-estimate)
+[Cauchy's Estimate](#cauchy-s-estimate)
 for all disks centered at any point $a \in \CC$ and with any radius $r>0$.
 For the first derivative, we have $\norm{f'(a)} \le M/r$, which tends to $0$ for $r \to \infty$.
 Hence $f' = 0$ in the whole plane. This
diff --git a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
index 15412bc..2445b8b 100644
--- a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
+++ b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
@@ -3,37 +3,34 @@ title: Cauchy's Theorem
 parent: One Complex Variable
 grand_parent: Complex Analysis
 nav_order: 2
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .theorem-title }
-> {{ page.title }} (Homotopy Version)
->
-> Let $G$ be a connected open subset of the complex plane.
-> Let $f : G \to \CC$ be a holomorphic function.
-> If $\gamma_0$, $\gamma_1$ are homotopic closed curves in $G$, then
->
-> $$
-> \int_{\gamma_0} \! f(z) \, dz =
-> \int_{\gamma_1} \! f(z) \, dz 
-> $$
->
-> If $\gamma$ is a null-homotopic closed curve in $G$, then
->
-> $$
-> \int_{\gamma} f(z) \, dz = 0
-> $$
+{% theorem Cauchy's Theorem (Homotopy Version) %}
+Let $G$ be a connected open subset of the complex plane.
+Let $f : G \to \CC$ be a holomorphic function.
+If $\gamma_0$, $\gamma_1$ are homotopic closed curves in $G$, then
+
+$$
+\int_{\gamma_0} \! f(z) \, dz =
+\int_{\gamma_1} \! f(z) \, dz 
+$$
+
+If $\gamma$ is a null-homotopic closed curve in $G$, then
+
+$$
+\int_{\gamma} f(z) \, dz = 0
+$$
+{% endtheorem %}
 
 {% proof %}
 {% endproof %}
 
 {{ page.title }} has a converse:
 
-{: .theorem-title }
-> Morera's Theorem
->
-> Let $G \subset \CC$ be open and let $f : G \to \CC$ be a continuous function.
-> If $\int_{\gamma} f(z) \, dz = 0$ for every contour $\gamma$ contained in $G$,
-> then $f$ is holomorphic in $G$.
+{% theorem * Morera's Theorem %}
+Let $G \subset \CC$ be open and let $f : G \to \CC$ be a continuous function.
+If $\int_{\gamma} f(z) \, dz = 0$ for every contour $\gamma$ contained in $G$,
+then $f$ is holomorphic in $G$.
+{% endtheorem %}
diff --git a/pages/complex-analysis/one-complex-variable/index.md b/pages/complex-analysis/one-complex-variable/index.md
index 4942ff8..5830a81 100644
--- a/pages/complex-analysis/one-complex-variable/index.md
+++ b/pages/complex-analysis/one-complex-variable/index.md
@@ -3,7 +3,6 @@ title: One Complex Variable
 parent: Complex Analysis
 nav_order: 1
 has_children: true
-# cspell:words
 ---
 
 # {{ page.title }}
diff --git a/pages/complex-analysis/one-complex-variable/power-series.md b/pages/complex-analysis/one-complex-variable/power-series.md
index 0147f31..31793ab 100644
--- a/pages/complex-analysis/one-complex-variable/power-series.md
+++ b/pages/complex-analysis/one-complex-variable/power-series.md
@@ -3,59 +3,57 @@ title: Power Series
 parent: One Complex Variable
 grand_parent: Complex Analysis
 nav_order: 1
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .definition-title }
-> Definition ({{ page.title }})
->
-> Let $X$ be a complex Banach space.
-> A *power series* (with values in $X$) is an infinite series of the form
->
-> 
-> $$
-> \sum_{n=0}^{\infty} x_n (z - a)^n = x_0 + x_1 (z-a) + x_2 (z-a)^2 + \cdots,
-> $$
->
-> where $x_n \in X$ is the *$n$th coefficient*,
-> $z$ is a complex variable and
-> $a$ is the *center* of the series.
-
-{: .lemma }
-> Suppose the Banach space valued power series $\sum_{n=0}^{\infty} x_n (z - a)^n$ converges for $z = a + w$.
-> Then it converges absolutely for all $z$ with $\abs{z-a} < \abs{w}$.
+{% definition Power Series %}
+Let $X$ be a complex Banach space.
+A *power series* (with values in $X$) is an infinite series of the form
+
+$$
+\sum_{n=0}^{\infty} x_n (z - a)^n = x_0 + x_1 (z-a) + x_2 (z-a)^2 + \cdots,
+$$
+
+where $x_n \in X$ is the *$n$th coefficient*,
+$z$ is a complex variable and
+$a$ is the *center* of the series.
+{% enddefinition %}
+
+{% lemma %}
+Suppose the Banach space valued power series $\sum_{n=0}^{\infty} x_n (z - a)^n$ converges for $z = a + w$.
+Then it converges absolutely for all $z$ with $\abs{z-a} < \abs{w}$.
+{% endlemma %}
 
 {% proof %}
 TODO
 {% endproof %}
 
-{: .theorem }
-> Suppose $\sum_{n=0}^{\infty} x_n (z - a)^n$ is a Banach space valued power series.
-> Then either
->
-> - the series converges only for $z=a$ (formally $R=0$), or
-> - there exists a number $0<R<\infty$ such that
->   the series converges absolutely whenever $\abs{z-a} < R$ 
->   and diverges whenever $\abs{z-a} > R$, or
-> - the series converges absolutely for all $z \in \CC$ (formally $R=\infty$).
->
-> The number $R \in [0,\infty]$ is called the *radius of convergence* of the power series.
+{% theorem %}
+Suppose $\sum_{n=0}^{\infty} x_n (z - a)^n$ is a Banach space valued power series.
+Then either
+
+- the series converges only for $z=a$ (formally $R=0$), or
+- there exists a number $0<R<\infty$ such that
+  the series converges absolutely whenever $\abs{z-a} < R$ 
+  and diverges whenever $\abs{z-a} > R$, or
+- the series converges absolutely for all $z \in \CC$ (formally $R=\infty$).
+
+The number $R \in [0,\infty]$ is called the *radius of convergence* of the power series.
+{% endtheorem %}
 
 {% proof %}
 TODO
 {% endproof %}
 
-{: .theorem-title }
-> Cauchy–Hadamard Formula
->
-> Let $\sum_{n=0}^{\infty} x_n (z - a)^n$ be a Banach space valued power series
-> with radius of convergence $R$. Then
-> 
-> $$
-> \frac{1}{R} = \limsup_{n \to \infty} \norm{x_n}^{1/n}.
-> $$
+{% theorem * Cauchy–Hadamard Formula %}
+Let $\sum_{n=0}^{\infty} x_n (z - a)^n$ be a Banach space valued power series
+with radius of convergence $R$. Then
+
+$$
+\frac{1}{R} = \limsup_{n \to \infty} \norm{x_n}^{1/n}.
+$$
+{% endtheorem %}
 
 {% proof %}
 TODO
diff --git a/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md b/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md
index b49cdf4..a2fa53d 100644
--- a/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md
+++ b/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md
@@ -3,16 +3,13 @@ title: The Calculus of Residues
 parent: One Complex Variable
 grand_parent: Complex Analysis
 nav_order: 4
-# cspell:words
-#published: false
 ---
 
 # {{ page.title }}
 
-{: .definition-title }
-> Definition (Residue)
->
-> TODO
+{% definition Residue %}
+TODO
+{% enddefinition %}
 
 Calculation of Residues
 
@@ -32,24 +29,17 @@ $$
 \Res(f,c) = \frac{g(c)}{h'(c)}
 $$
 
+{% theorem * Residue Theorem (Basic Version) %}
+Let $f$ be a function meromorphic in an open subset $G \subset \CC$.
+Let $\gamma$ be a contour in $G$ such that
+the interior of $\gamma$ is contained in $G$
+and contains finitely many poles $c_1, \ldots, c_n$ of $f$.
+Then
 
-
-
-{: .theorem-title }
-> Residue Theorem (Basic Version)
-> {: #residue-theorem-basic-version }
->
-> Let $f$ be a function meromorphic in an open subset $G \subset \CC$.
-> Let $\gamma$ be a contour in $G$ such that
-> the interior of $\gamma$ is contained in $G$
-> and contains finitely many poles $c_1, \ldots, c_n$ of $f$.
-> Then
->
-> 
-> $$
-> \int_{\gamma} f(z) \, dz = 2 \pi i \sum_{k=1}^n \Res(f,c_k)
-> $$
-> {: .katex-display .mb-0 }
+$$
+\int_{\gamma} f(z) \, dz = 2 \pi i \sum_{k=1}^n \Res(f,c_k)
+$$
+{% endtheorem %}
 
 {% proof %}
 {% endproof %}
diff --git a/pages/complex-analysis/several-complex-variables/edge-of-the-wedge.md b/pages/complex-analysis/several-complex-variables/edge-of-the-wedge.md
index 5adc3f6..4e7666c 100644
--- a/pages/complex-analysis/several-complex-variables/edge-of-the-wedge.md
+++ b/pages/complex-analysis/several-complex-variables/edge-of-the-wedge.md
@@ -3,16 +3,12 @@ title: Edge of the Wedge
 parent: Several Complex Variables
 grand_parent: Complex Analysis
 nav_order: 1
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .theorem-title }
-> {{ page.title }}
-> {: #{{ page.title | slugify }} }
->
-> ...
+{% theorem %}
+{% endtheorem %}
 
 {% proof %}
 {% endproof %}
diff --git a/pages/complex-analysis/several-complex-variables/index.md b/pages/complex-analysis/several-complex-variables/index.md
index 49763d5..803eea4 100644
--- a/pages/complex-analysis/several-complex-variables/index.md
+++ b/pages/complex-analysis/several-complex-variables/index.md
@@ -3,7 +3,6 @@ title: Several Complex Variables
 parent: Complex Analysis
 nav_order: 2
 has_children: true
-# cspell:words
 ---
 
 # {{ page.title }}
diff --git a/pages/complex-analysis/weak-and-strong-analyticity.md b/pages/complex-analysis/weak-and-strong-analyticity.md
index 7db1dbf..c7ffb85 100644
--- a/pages/complex-analysis/weak-and-strong-analyticity.md
+++ b/pages/complex-analysis/weak-and-strong-analyticity.md
@@ -3,16 +3,9 @@ title: Weak and Strong Analyticity
 parent: Complex Analysis
 nav_order: 3
 published: false
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .definition-title }
-> {{ page.title }}
-> {: #{{ page.title | slugify }} }
->
-> ...
-
 {% proof %}
 {% endproof %}
diff --git a/pages/distribution-theory/definitions.md b/pages/distribution-theory/definitions.md
index a800e03..405eff2 100644
--- a/pages/distribution-theory/definitions.md
+++ b/pages/distribution-theory/definitions.md
@@ -2,7 +2,6 @@
 title: Definitions
 parent: Distribution Theory
 nav_order: 10
-# cspell:words
 published: false
 ---
 
diff --git a/pages/distribution-theory/index.md b/pages/distribution-theory/index.md
index b4b50a8..3055c8f 100644
--- a/pages/distribution-theory/index.md
+++ b/pages/distribution-theory/index.md
@@ -1,6 +1,6 @@
 ---
 title: Distribution Theory
-nav_order: 3
+nav_order: 5
 has_children: true
 has_toc: false
 published: true
@@ -10,17 +10,16 @@ published: true
 
 As usual, let $\mathcal{S}$ denote the space of Schwartz test functions on $\RR^n$.
 
-{: .definition-title }
-> Definition (Operator Valued Distribution)
-> 
-> Let $\hilb{H}$ be a Hilbert space.
-> An *operator valued tempered distribution* $\Phi$ (on $\RR^n$)
-> is a mapping that associates to each test function $f \in \mathcal{S}$
-> an unbounded linear operator $\Phi(f)$ in $\hilb{H}$ such that
-> {: .mb-0 }
-> 
-> {: .my-0 }
-> - there is a dense linear subspace $\mathcal{D}$ of $\hilb{H}$ that
-> is contained in the domain of all the $\Phi(f)$ 
-> - for every fixed pair of vectors $\phi, \psi \in \hilb{D}$
-> the mapping $f \mapsto \innerp{\phi}{\Phi(f) \psi}$ is a tempered distribution.
+{% definition Operator Valued Distribution %}
+Let $\hilb{H}$ be a Hilbert space.
+An *operator valued tempered distribution* $\Phi$ (on $\RR^n$)
+is a mapping that associates to each test function $f \in \mathcal{S}$
+an unbounded linear operator $\Phi(f)$ in $\hilb{H}$ such that
+{: .mb-0 }
+
+{: .my-0 }
+- there is a dense linear subspace $\mathcal{D}$ of $\hilb{H}$ that
+is contained in the domain of all the $\Phi(f)$ 
+- for every fixed pair of vectors $\phi, \psi \in \hilb{D}$
+the mapping $f \mapsto \innerp{\phi}{\Phi(f) \psi}$ is a tempered distribution.
+{% enddefinition %}
diff --git a/pages/distribution-theory/sobolev-theory.md b/pages/distribution-theory/sobolev-theory.md
index 931731f..d7a91e2 100644
--- a/pages/distribution-theory/sobolev-theory.md
+++ b/pages/distribution-theory/sobolev-theory.md
@@ -2,7 +2,6 @@
 title: Sobolev Theory
 parent: Distribution Theory
 nav_order: 10
-# cspell:words
 published: false
 ---
 
diff --git a/pages/functional-analysis-basics/banach-alaoglu-theorem.md b/pages/functional-analysis-basics/banach-alaoglu-theorem.md
index 59e4a92..91906cd 100644
--- a/pages/functional-analysis-basics/banach-alaoglu-theorem.md
+++ b/pages/functional-analysis-basics/banach-alaoglu-theorem.md
@@ -2,18 +2,21 @@
 title: Banach–Alaoglu Theorem
 parent: Functional Analysis Basics
 nav_order: 3
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .theorem-title }
-> {{ page.title }}
-> {: #{{ page.title | slugify }} }
->
-> The closed unit ball in the dual of a normed space is weak\* compact.
+{% theorem * Banach–Alaoglu Theorem %}
+The closed unit ball in the dual of a normed space is weak\* compact.
+{% endtheorem %}
 
 {% proof %}
 {% endproof %}
 
-## Generalization: Alaoglu–Bourbaki
+The {{ page.title }} is a special case of the following result:
+
+{% theorem * Alaoglu–Bourbaki Theorem %}
+The polar of a neighborhood of zero in a locally convex space is weak\* compact.
+{% endtheorem %}
+
+See [Alaoglu–Bourbaki Theorem]({% link pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md %}) for more information.
diff --git a/pages/functional-analysis-basics/compact-operators.md b/pages/functional-analysis-basics/compact-operators.md
index b114c24..92e94ba 100644
--- a/pages/functional-analysis-basics/compact-operators.md
+++ b/pages/functional-analysis-basics/compact-operators.md
@@ -3,42 +3,37 @@ title: Compact Operators
 parent: Functional Analysis Basics
 nav_order: 4
 published: false
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .definition-title }
-> Definition (Compact Linear Operator)
-> {: #compact-operator }
->
-> A linear operator $T : X \to Y$,
-> where $X$ and $Y$ are normed spaces,
-> is said to be a  *compact linear operator*,
-> if for every bounded subset $M \subset X$
-> the image $TM$ is relatively compact in $Y$.
+{% definition Compact Linear Operator %}
+A linear operator $T : X \to Y$,
+where $X$ and $Y$ are normed spaces,
+is said to be a *compact linear operator*,
+if for every bounded subset $M \subset X$
+the image $TM$ is relatively compact in $Y$.
+{% enddefinition %}
 
-{: .proposition-title }
-> Proposition (Characterisation of Compactness)
->
-> Let $X$ and $Y$ be normed spaces.
-> A linear operator $T : X \to Y$ is compact if and only if
-> for every bounded sequence $(x_n)$ in $X$
-> the image sequence $(Tx_n)$ in $Y$ has a convergent subseqence.
+{% proposition Characterization of Compactness %}
+Let $X$ and $Y$ be normed spaces.
+A linear operator $T : X \to Y$ is compact if and only if
+for every bounded sequence $(x_n)$ in $X$
+the image sequence $(Tx_n)$ in $Y$ has a convergent subsequence.
+{% endproposition %}
 
-{: .proposition-title }
-> Every compact linear operator is bounded.
+{% proposition %}
+Every compact linear operator is bounded.
+{% endproposition %}
 
-{: .proposition-title }
-> Proposition (Compactness of Zero and Identity)
->
-> The zero operator on any normed space is compact.
-> The indentity operator on a normed space $X$ is compact if and only if $X$ has finite dimension.
+{% proposition Compactness of Zero and Identity %}
+The zero operator on any normed space is compact.
+The identity operator on a normed space $X$ is compact if and only if $X$ has finite dimension.
+{% endproposition %}
 
-{: .proposition-title }
-> Proposition (The Space of Compact Linear Operators)
->
-> The set $C(X,Y)$ of compact linear operator from a normed space $X$ into a normed space $Y$
-> form a linear subspace of the space $B(X,Y)$ of bounded linear operators from $X$ into $Y$.
-> If $Y$ is a Banach space, then $C(X,Y)$ is a closed linear subspace of the Banach space
-> $B(X,Y)$ and hence itself a Banach space.
+{% proposition The Space of Compact Linear Operators %}
+The set $C(X,Y)$ of compact linear operator from a normed space $X$ into a normed space $Y$
+form a linear subspace of the space $B(X,Y)$ of bounded linear operators from $X$ into $Y$.
+If $Y$ is a Banach space, then $C(X,Y)$ is a closed linear subspace of the Banach space
+$B(X,Y)$ and hence itself a Banach space.
+{% endproposition %}
diff --git a/pages/functional-analysis-basics/hilbert-spaces.md b/pages/functional-analysis-basics/hilbert-spaces.md
new file mode 100644
index 0000000..b3ef52b
--- /dev/null
+++ b/pages/functional-analysis-basics/hilbert-spaces.md
@@ -0,0 +1,10 @@
+---
+title: Hilbert Spaces
+parent: Functional Analysis Basics
+nav_order: 1
+---
+
+# {{ page.title }}
+
+{% proof %}
+{% endproof %}
diff --git a/pages/functional-analysis-basics/reflexive-spaces.md b/pages/functional-analysis-basics/reflexive-spaces.md
index dee0e55..781fb1f 100644
--- a/pages/functional-analysis-basics/reflexive-spaces.md
+++ b/pages/functional-analysis-basics/reflexive-spaces.md
@@ -2,59 +2,59 @@
 title: Reflexive Spaces
 parent: Functional Analysis Basics
 nav_order: 2
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .definition-title }
-> Definition (Canonical Embedding)
->
-> Let $X$ be a normed space.
-> The mapping
->
-> $$
-> C : X \longrightarrow X'', \quad x \mapsto g_x,
-> $$
->
-> where the functional $g_x$ on $X'$ is defined by
->
-> $$
-> g_x(f) = f(x) \quad \text{for $f \in X'$,} 
-> $$
->
-> is called the *canonical embedding* of $X$ into its bidual $X''$.
-
-{: .lemma }
-> The canonical embedding $C : X \to X''$ of a normed space into its bidual
-> is well-defined and an embedding of normed spaces.
+{% definition Canonical Embedding %}
+Let $X$ be a normed space.
+The mapping
+
+$$
+C : X \longrightarrow X'', \quad x \mapsto g_x,
+$$
+
+where the functional $g_x$ on $X'$ is defined by
+
+$$
+g_x(f) = f(x) \quad \text{for $f \in X'$,} 
+$$
+
+is called the *canonical embedding* of $X$ into its bidual $X''$.
+{% enddefinition %}
+
+{% lemma %}
+The canonical embedding $C : X \to X''$ of a normed space into its bidual
+is well-defined and an embedding of normed spaces.
+{% endlemma %}
 
 {% proof %}
 {% endproof %}
 
 In particular, $C$ is isometric, hence injective.
 
-{: .definition-title }
-> Definition (Reflexivity)
->
-> A normed space is said to be *reflexive*
-> if the canonical embedding into its bidual
-> is surjective.
+{% definition Reflexivity %}
+A normed space is said to be *reflexive*
+if the canonical embedding into its bidual
+is surjective.
+{% enddefinition %}
 
 If a normed space $X$ is reflexive,
 then $X$ is isomorphic with $X''$, its bidual.
 James gives a counterexample for the converse statement.
 
-{: .theorem }
-> If a normed space is reflexive,
-> then it is complete; hence a Banach space.
+{% theorem %}
+If a normed space is reflexive,
+then it is complete; hence a Banach space.
+{% endtheorem %}
 
 {% proof %}
 {% endproof %}
 
-{: .theorem }
-> If a normed space $X$ is reflexive,
-> then the weak and weak$^*$ topologies on $X'$ agree.
+{% theorem %}
+If a normed space $X$ is reflexive,
+then the weak and weak$^*$ topologies on $X'$ agree.
+{% endtheorem %}
 
 {% proof %}
 By definition, the weak and weak$^*$ topologies on $X'$
@@ -65,9 +65,10 @@ Since $X$ is reflexive, those sets are equal.
 
 The converse is true as well. Proof: TODO
 
-{: .theorem }
-> If a normed space $X$ is reflexive,
-> then its dual $X'$ is reflexive.
+{% theorem %}
+If a normed space $X$ is reflexive,
+then its dual $X'$ is reflexive.
+{% endtheorem %}
 
 {% proof %}
 Since $X$ is reflexive,
@@ -115,9 +116,10 @@ This shows that $D$ is surjective, hence $X'$ is reflexive.
 In fact, we have shown more: $D = (C')^{-1}$.
 {% endproof %}
 
-{: .theorem }
-> Every finite-dimensional normed space is reflexive.
->
+{% theorem %}
+Every finite-dimensional normed space is reflexive.
+{% endtheorem %}
 
-{: .theorem }
-> Every Hilbert space is reflexive.
+{% theorem %}
+Every Hilbert space is reflexive.
+{% endtheorem %}
diff --git a/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md
index f8b8254..f6a9783 100644
--- a/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md
+++ b/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md
@@ -3,29 +3,40 @@ title: Closed Graph Theorem
 parent: The Fundamental Four
 grand_parent: Functional Analysis Basics
 nav_order: 4
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .theorem-title }
-> {{ page.title }}
-> {: #{{ page.title | slugify }} }
->
-> An (everywhere-defined) linear operator between Banach spaces is bounded
-> iff its graph is closed.
+{% theorem * Closed Graph Theorem %}
+An (everywhere-defined) linear operator between Banach spaces is bounded
+iff its graph is closed.
+{% endtheorem %}
 
 We prove a slightly more general version:
 
-{: .theorem-title }
-> {{ page.title }}
-> {: #{{ page.title | slugify }}-variant }
->
-> Let $X$ and $Y$ be Banach spaces
-> and $T : \dom{T} \to Y$ a linear operator
-> with domain $\dom{T}$ closed in $X$.
-> Then $T$ is bounded if and only if
-> its graph $\graph{T}$ is closed.
+{% theorem * Closed Graph Theorem (Variant) %}
+Let $X$ and $Y$ be Banach spaces
+and $T : \dom{T} \to Y$ a linear operator
+with domain $\dom{T}$ closed in $X$.
+Then $T$ is bounded if and only if
+its graph $\graph{T}$ is closed.
+{% endtheorem %}
 
 {% proof %}
+Let us assume first that $T$ is bounded.
+Let $(x_n,Tx_n)_n$ be a sequence in $\graph{T}$ that converges to some element $(x,y) \in X \times Y$.
+This means that $x_n \to x$ and $Tx_n \to y$ for $n \to \infty$.
+The continuity of $T$ implies $Tx_n \to Tx$.
+Since a convergent series in a Hausdorff space has a unique limit,
+it follows that $Tx = y$; hence $(x,y)$ lies in $\graph{T}$.
+This shows that $\graph{T}$ is closed.
+
+Conversely, suppose that $\graph{T}$ is a closed subspace of $X \times Y$.
+Note that $X \times Y$ is a Banach space with norm $\norm{(x,y)} = \norm{x} + \norm{y}$.
+Therefore $\graph{T}$ is itself as Banach space in the restricted norm $\norm{(x,Tx)} = \norm{x} + \norm{Tx}$.
+The canonical projections $\pi_X : \graph{T} \to X$ and $\pi_Y : \graph{T} \to Y$ are bounded.
+Clearly, $\pi_X$ is bijective, so its inverse $\pi_X^{-1} : X \to \graph{T}$ is a bounded operator by the
+[Bounded Inverse Theorem](/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.html#bounded-inverse-theorem).
+Consequently the composition, $\pi_Y \circ \pi_X^{-1} : X \to Y$ is bounded.
+To complete the proof, observe that $\pi_Y \circ \pi_X^{-1} = T$.
 {% endproof %}
diff --git a/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md
index 9d21d41..18cf64a 100644
--- a/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md
+++ b/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md
@@ -6,6 +6,7 @@ nav_order: 1
 ---
 
 # {{ page.title }}
+{: .no_toc }
 
 In fact, there are multiple theorems and corollaries
 which bear the name Hahn–Banach.
@@ -13,111 +14,112 @@ All have in common that
 they guarantee the existence of linear functionals
 with various additional properties.
 
-{: .definition-title }
-> Definition (Sublinear Functional)
->
-> A functional $p$ on a real vector space $X$
-> is called *sublinear* if it is
-> {: .mb-0 }
->
-> {: .mt-0 .mb-0 }
-> - *positive-homogenous*, that is
->   {: .mt-0 .mb-0 }
->
->   $$
->   p(\alpha x) = \alpha \, p(x) \qquad \forall \alpha \ge 0, \ \forall x \in X,
->   $$
->
-> - and satisfies the *triangle inequality*
->   {: .mt-0 .mb-0 }
->
->   $$
->   p(x+y) \le p(x) + p(y) \qquad \forall x,y \in X.
->   $$
->   {: .katex-display .mb-0 }
+<details open markdown="block">
+  <summary>
+    Table of contents
+  </summary>
+  {: .text-delta }
+- TOC
+{:toc}
+</details>
+
+{% definition Sublinear Functional %}
+A functional $p$ on a real vector space $X$
+is called *sublinear* if it is
+{: .mb-0 }
+
+{: .mt-0 .mb-0 }
+- *positively homogenous*, that is
+  {: .mt-0 .mb-0 }
+
+  $$
+  p(\alpha x) = \alpha \, p(x) \qquad \forall \alpha \ge 0, \ \forall x \in X,
+  $$
+
+- and *subadditive*, that is
+  {: .mt-0 .mb-0 }
+
+$$
+p(x+y) \le p(x) + p(y) \qquad \forall x,y \in X.
+$$
+{% enddefinition %}
 
 If $p$ is a sublinear functional,
 then $p(0)=0$ and $p(-x) \ge -p(x)$ for all $x$.
 
 Every norm on a real vector space is a sublinear functional.
 
-{: .theorem-title }
-> {{ page.title }} (Basic Version)
->
-> Let $p$ be a sublinear functional on a real vector space $X$.
-> Then there exists a linear functional $f$ on $X$ satisfying
-> $f(x) \le p(x)$ for all $x \in X$.
+{% theorem * Hahn–Banach Theorem (Basic Version) %}
+Let $p$ be a sublinear functional on a real vector space $X$.
+Then there exists a linear functional $f$ on $X$ satisfying
+$f(x) \le p(x)$ for all $x \in X$.
+{% endtheorem %}
 
 ## Extension Theorems
 
-{: .theorem-title }
-> {{ page.title }} (Extension, Real Vector Spaces)
->
-> Let $p$ be a sublinear functional on a real vector space $X$.
-> Let $f$ be a linear functional
-> which is defined on a linear subspace $Z$ of $X$
-> and satisfies
->
-> $$
-> f(x) \le p(x) \qquad \forall x \in Z.
-> $$
->
-> Then $f$ has a linear extension $\tilde{f}$ to $X$ such that
->
-> $$
-> \tilde{f}(x) \le p(x) \qquad \forall x \in X.
-> $$
+{% theorem * Hahn–Banach Theorem (Extension, Real Vector Spaces) %}
+Let $p$ be a sublinear functional on a real vector space $X$.
+Let $f$ be a linear functional
+which is defined on a linear subspace $Z$ of $X$
+and satisfies
+
+$$
+f(x) \le p(x) \qquad \forall x \in Z.
+$$
+
+Then $f$ has a linear extension $\tilde{f}$ to $X$ such that
+
+$$
+\tilde{f}(x) \le p(x) \qquad \forall x \in X.
+$$
+{% endtheorem %}
 
 {% proof %}
 {% endproof %}
 
-{: .definition-title }
-> Definition (Semi-Norm)
->
-> We call a real-valued functional $p$ on a real or complex vector space $X$
-> a *semi-norm* if it is
-> {: .mb-0 }
->
-> {: .mt-0 .mb-0 }
-> - *absolutely homogenous*, that is
->   {: .mt-0 .mb-0 }
->
->   $$
->   p(\alpha x) = \abs{\alpha} \, p(x) \qquad \forall \alpha \in \KK \ \forall x \in X,
->   $$
-> - and satisfies the *triangle inequality*
->   {: .mt-0 .mb-0 }
->
->   $$
->   p(x+y) \le p(x) + p(y) \qquad \forall x,y \in X.
->   $$
->   {: .katex-display .mb-0 }
-
-{: .theorem-title }
-> {{ page.title }} (Extension, Real and Complex Vector Spaces)
->
-> Let $p$ be a semi-norm on a real or complex vector space $X$.
-> Let $f$ be a linear functional
-> which is defined on a linear subspace $Z$ of $X$
-> and satisfies
->
-> $$
-> \abs{f(x)} \le p(x) \qquad \forall x \in Z.
-> $$
->
-> Then $f$ has a linear extension $\tilde{f}$ to $X$ such that
->
-> $$
-> \abs{\tilde{f}(x)} \le p(x) \qquad \forall x \in X.
-> $$
-
-{: .theorem-title }
-> {{ page.title }} (Extension, Normed Spaces)
->
-> Let $X$ be a real or complex normed space
-> and let $f$ be a bounded linear functional
-> defined on a linear subspace $Z$ of $X$.
-> Then $f$ has a bounded linear extension $\tilde{f}$ to $X$ such that $\norm{\tilde{f}} = \norm{f}$.
+{% definition Semi-Norm %}
+We call a real-valued functional $p$ on a real or complex vector space $X$
+a *semi-norm* if it is
+{: .mb-0 }
+
+{: .mt-0 .mb-0 }
+- *absolutely homogenous*, that is
+  {: .mt-0 .mb-0 }
+
+  $$
+  p(\alpha x) = \abs{\alpha} \, p(x) \qquad \forall \alpha \in \KK \ \forall x \in X,
+  $$
+- and satisfies the *triangle inequality*
+  {: .mt-0 .mb-0 }
+
+  $$
+  p(x+y) \le p(x) + p(y) \qquad \forall x,y \in X.
+  $$
+{% enddefinition %}
+
+{% theorem * Hahn–Banach Theorem (Extension, Real and Complex Vector Spaces) %}
+Let $p$ be a semi-norm on a real or complex vector space $X$.
+Let $f$ be a linear functional
+which is defined on a linear subspace $Z$ of $X$
+and satisfies
+
+$$
+\abs{f(x)} \le p(x) \qquad \forall x \in Z.
+$$
+
+Then $f$ has a linear extension $\tilde{f}$ to $X$ such that
+
+$$
+\abs{\tilde{f}(x)} \le p(x) \qquad \forall x \in X.
+$$
+{% endtheorem %}
+
+{% theorem * Hahn–Banach Theorem (Extension, Normed Spaces) %}
+Let $X$ be a real or complex normed space
+and let $f$ be a bounded linear functional
+defined on a linear subspace $Z$ of $X$.
+Then $f$ has a bounded linear extension $\tilde{f}$ to $X$ such that $\norm{\tilde{f}} = \norm{f}$.
+{% endtheorem %}
 
 {% proof %}
 We apply the preceding theorem with $p(x) = \norm{f} \norm{x}$
@@ -131,17 +133,66 @@ Corollaries
 
 Important consequence: canonical embedding into bidual
 
+{% theorem * Hahn–Banach Theorem (Existence of Functionals) %}
+Let $X$ be a real or complex normed space
+and let $x$ be a nonzero element of $X$.
+Then there exists a bounded linear functional $f$ on $X$
+with $f(x) = \norm{x}$ and $\norm{f} = 1$.
+{% endtheorem %}
+
+{% proof %}
+On the linear subspace $\KK x \subset X$ spanned by $x$
+we define a functional $f_0$ by $f_0(\alpha x) = \alpha \norm{x}$ for $\alpha \in \KK$.
+It is easy to check that $f_0$ is linear and bounded with norm $\norm{f_0} = 1$.
+By the Hahn–Banach Extension Theorem for Normed Spaces,
+there exists a bounded linear functional $f$ on $X$ extending $f_0$ with identical norm.
+Hence we have $f(x) = f_0(x) = \norm{x}$ and $\norm{f} = \norm{f_0} = 1$.
+{% endproof %}
+
+Recall that for a normed space $X$ we denote its (topological) dual space by $X'$.
+
+{% corollary %}
+For every element $x$ of a real or complex normed space $X$ one has
+
+$$
+\norm{x} = \sup_{f \in X' \setminus \braces{0}} \frac{\abs{f(x)}}{\norm{f}} 
+$$
+
+and the supremum is attained.
+{% endcorollary %}
+
+{% corollary %}
+The elements of a real or complex normed space $X$
+are separated by the elements of its dual $X'$.
+{% endcorollary %}
+
 ## Separation Theorems
 
-{: .theorem-title }
-> {{ page.title }} (Separation, Point and Closed Subspace)
->
-> Suppose $Z$ is a closed subspace
-> of a normed space $X$ and $x$ lies in $X \setminus Z$.
-> Then there exists a bounded linear functional on $X$
-> which vanishes on $Z$ but has a nonzero value at $x$.
-
-{: .theorem-title }
-> {{ page.title }} (Separation, Convex Sets)
->
-> TODO
+{% theorem * Hahn–Banach Theorem (Separation, Point and Closed Subspace) %}
+Suppose $Z$ is a closed subspace of a normed space $X$
+and $x$ lies in $X \setminus Z$.
+Then there exists a bounded linear functional $f$ on $X$
+vanishing on $Z$ and with nonzero value $f(x) = \dist{x,Z}$.
+{% endtheorem %}
+
+{% proof %}
+Since $Z$ is a closed subspace of $X$,
+the quotient vector space $X/Z$ becomes a normed space
+with the quotient norm given by
+
+$$
+\norm{y + Z} = \dist{y,Z} = \inf_{z \in Z} \norm{y-z} \quad \forall y \in X.
+$$
+
+Moreover, the canonical mapping $\pi : X \to X/Z$, $y \mapsto y+Z$, is bounded.
+Given a $x \in X$ that does not lie in $Z$, the null space of $\pi$,
+we see that $\pi(x)$ is a nonzero element of $X/Z$.
+By Hahn–Banach, there exists a bounded linear functional $g$ on $X/Z$
+with $g(\pi(x)) = \norm{x} = \dist{x,Z} \ne 0$.
+Now the composition $f = g \circ \pi$ is a bounded functional on $X$
+with the desired properties.
+{% endproof %}
+
+{% theorem * Hahn–Banach Theorem (Separation, Convex Sets) %}
+TODO
+{% endtheorem %}
diff --git a/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md
index 53da008..b191bb2 100644
--- a/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md
+++ b/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md
@@ -3,7 +3,6 @@ title: Open Mapping Theorem
 parent: The Fundamental Four
 grand_parent: Functional Analysis Basics
 nav_order: 3
-# cspell:words surjective bijective
 ---
 
 # {{ page.title }}
@@ -13,12 +12,10 @@ where $X$ and $Y$ are topological spaces,
 is called *open* if the image under $T$ of each open set of $X$
 is open in $Y$.
 
-{: .theorem-title }
-> {{ page.title }}
-> {: #{{ page.title | slugify }} }
->
-> A bounded linear operator between Banach spaces is open
-> if and only if it is surjective.
+{% theorem * Open Mapping Theorem %}
+A bounded linear operator between Banach spaces is open
+if and only if it is surjective.
+{% endtheorem %}
 
 {% proof %}
 Let $X$ and $Y$ be Banach spaces
@@ -91,19 +88,22 @@ Conversely, suppose that $T$ is open. TODO
 
 ---
 
-XXX injective
 For a bijective mapping between topological spaces, to say that it is open,
 is equivalent to saying that its inverse is continuous.
 The inverse of a bijective linear map between normed spaces is automatically linear
 and thus continuous if and only if it is bounded.
 As a corollary to the {{ page.title }} we obtain the following:
 
-{: .corollary-title }
-> Bounded Inverse Theorem
-> {: #bounded-inverse-theorem }
->
-> If a bounded linear operator between Banach spaces is bijective,
-> then its inverse is bounded.
-XXX relax to injective
+{% corollary * Bounded Inverse Theorem %}
+If a bounded linear operator between Banach spaces is bijective,
+then its inverse is bounded.
+{% endcorollary %}
 
 Also known as *Inverse Mapping Theorem*.
+
+{% corollary %}
+Let $T: X \to Y$ be a bounded linear operator between Banach spaces
+and suppose that $T$ is injective, so that the inverse $T^{-1} : R(T) \to X$
+is defined on the range of $T$.
+The linear operator $T^{-1}$ is bounded if and only if $R(T)$ is closed in $X$.
+{% endcorollary %}
diff --git a/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md
index 13460da..47ddd3f 100644
--- a/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md
+++ b/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md
@@ -3,27 +3,35 @@ title: Uniform Boundedness Theorem
 parent: The Fundamental Four
 grand_parent: Functional Analysis Basics
 nav_order: 2
-description: >
-  The
-# spellchecker:words preimages pointwise
 ---
 
 # {{ page.title }}
 
 Also known as *Uniform Boundedness Principle* and *Banach–Steinhaus Theorem*.
 
-{: .theorem-title }
-> {{ page.title }}
-> {: #{{ page.title | slugify }} }
->
-> If $\mathcal{T}$ is a set of bounded linear operators
-> from a Banach space $X$ into a normed space $Y$ such that
-> $\braces{\norm{Tx} : T \in \mathcal{T}}$
-> is a bounded set for every $x \in X$, then
-> $\braces{\norm{T} : T \in \mathcal{T}}$
-> is a bounded set.
+{% definition Pointwise and Uniform Boundedness %}
+Let $X$, $Y$ be normed spaces.
+We say that a collection $\mathcal{T}$ of bounded linear operators
+from $X$ to $Y$ is
+{: .mb-0 }
+- *pointwise bounded* if the set $\braces{\norm{Tx} : T \in \mathcal{T}}$ is bounded for every $x \in X$,
+- *uniformly bounded* if the set $\braces{\norm{T} : T \in \mathcal{T}}$ is bounded.
+{% enddefinition %}
+
+Clearly, every uniformly bounded collection of operators is pointwise bounded since $\norm{Tx} \le \norm{T} \norm{x}$.
+The converse is true, if $X$ is complete:
+
+{% theorem * Uniform Boundedness Theorem %}
+If a collection of bounded linear operators
+from a Banach space into a normed space
+is pointwise bounded,
+then it is uniformly bounded.
+{% endtheorem %}
 
 {% proof %}
+Suppose $X$ is a Banach space, $Y$ is a normed space
+and $\mathcal{T}$ is a pointwise bounded collection
+of bounded linear operators from $X$ to $Y$.
 For each $n \in \NN$ the set
 
 $$
@@ -38,9 +46,8 @@ the set $\braces{\norm{Tx} : T \in \mathcal{T}}$ is bounded by assumption.
 This means that there exists a $n \in \NN$
 such that $\norm{Tx} \le n$ for all $T \in \mathcal{T}$.
 In other words, $x \in A_n$.
-Thus we have show that $\bigcup A_n = X$.
-XXX Apart from the trivial case $X = \emptyset$,
-the union $\bigcup A_n$ has nonempty interior.
+Thus, we have shown that $\bigcup A_n = X$.
+In particular, $\bigcup A_n$ has nonempty interior.
 Now, utilizing the completeness of $X$, the
 [Baire Category Theorem]({% link pages/general-topology/baire-spaces.md %})
 implies that there exists a $m \in \NN$ such that $A_m$ has nonempty interior.
@@ -74,3 +81,8 @@ $$
 $$
 
 If $X$ is not complete, this may be false.
+
+TODO:
+- strong operator convergence
+- Kreyszig 4.9-5
+- Haase 15.6
diff --git a/pages/general-topology/baire-spaces.md b/pages/general-topology/baire-spaces.md
index 6bd7d9f..1332984 100644
--- a/pages/general-topology/baire-spaces.md
+++ b/pages/general-topology/baire-spaces.md
@@ -1,25 +1,24 @@
 ---
 title: Baire Spaces
 parent: General Topology
-nav_order: 1
+nav_order: 7
 description: >
   A Baire space is a topological space with the property that the intersection
   of countably many dense open subsets is still dense. One version of the Baire
   Category Theorem states that complete metric spaces are Baire spaces. We give
   a self-contained proof of Baire's Category Theorem by contradiction.
-# spellchecker:words
 ---
 
 # {{ page.title }}
 
-{: .definition }
-> A topological space is said to be a *Baire space*,
-> if any of the following equivalent conditions holds:
-> {: .mb-0 }
->
-> - The intersection of countably many dense open subsets is still dense.
-> - The union of countably many closed subsets with empty interior has empty interior.
-> {: .mt-0 .mb-0 }
+{% definition Baire Space %}
+A topological space is said to be a *Baire space*,
+if any of the following equivalent conditions holds:
+{: .mb-0 }
+
+- The intersection of countably many dense open subsets is still dense.
+- The union of countably many closed subsets with empty interior has empty interior.
+{% enddefinition %}
 
 Note that
 a set is dense in a topological space
@@ -33,11 +32,9 @@ of which there are several.
 Here we give one
 that is commonly used in functional analysis.
 
-{: .theorem-title }
-> Baire Category Theorem #1
-> {: #baire-category-theorem }
->
-> Every complete metric space is a Baire space.
+{% theorem * Baire Category Theorem #1 %}
+Every complete metric space is a Baire space.
+{% endtheorem %}
 
 {% proof %}
 Let $X$ be a metric space
@@ -76,8 +73,6 @@ On the other hand, $x \in \overline{B_1} \subset X \setminus U$,
 in contradiction to the preceding statement.
 {% endproof %}
 
-{: .theorem-title }
-> Baire Category Theorem #2
-> {: #baire-category-theorem }
->
-> Every compact Hausdorff space is a Baire space.
+{% theorem * Baire Category Theorem #2 %}
+Every locally compact Hausdorff space is a Baire space.
+{% endtheorem %}
diff --git a/pages/general-topology/baire-spaces.md.txt b/pages/general-topology/baire-spaces.md.txt
deleted file mode 100644
index eabe792..0000000
--- a/pages/general-topology/baire-spaces.md.txt
+++ /dev/null
@@ -1,73 +0,0 @@
----
-title: Baire Spaces
-parent: General Topology
-nav_order: 1
-description: >
-  A Baire space is a topological space with the property that the intersection
-  of countably many dense open subsets is still dense. One version of the Baire
-  Category Theorem states that complete metric spaces are Baire spaces. We give
-  a self-contained proof of Baire's Category Theorem by contradiction.
-# spellchecker:words
----
-
-# 
-
-
-A topological space is said to be a *Baire space*,
-if any of the following equivalent conditions holds:
->
-- The intersection of countably many dense open subsets is still dense.
-- The union of countably many closed subsets with empty interior has empty interior.
-
-
-Note that
-a set is dense in a topological space
-if and only if
-its complement has empty interior.
-
-Any sufficient condition
-for a topological space to be a Baire space
-constitutes a *Baire Category Theorem*,
-of which there are several.
-Here we give one
-that is commonly used in functional analysis.
-
-
-Baire Category Theorem
-
->
-Complete metric spaces are Baire spaces.
-
-**Proof:**
-Let C-C-C be a metric space
-with complete metric D-D-D.
-Suppose that F-F-F is not a Baire space.
-Then there is a countable collection G-G-G of dense open subsets of B-B-B
-such that the intersection C-C-C is not dense in D-D-D.
-
-In a metric space, any nonempty open set contains an open ball.
-It is also true, that any nonempty open set contains a closed ball,
-since F-F-F if G-G-G.
-
-We construct a sequence B-B-B of open balls C-C-C satisfying
-
-V-V-V
-as follows: By hypothesis,
-the interior of D-D-D is not empty (otherwise F-F-F would be dense in G-G-G),
-so we may choose an open ball B-B-B with C-C-C
-such that D-D-D.
-Given F-F-F,
-the set G-G-G is nonempty, because B-B-B is dense in C-C-C,
-and it is open, because D-D-D and F-F-F are open.
-This allows us to choose an open ball G-G-G as desired.
-
-Note that by construction B-B-B for C-C-C,
-thus D-D-D.
-Therefore, the sequence F-F-F is Cauchy
-and has a limit point G-G-G by completeness.
-In the limit B-B-B, we obtain C-C-C (strictness is lost),
-hence D-D-D for all F-F-F.
-This shows that G-G-G for all B-B-B, that is C-C-C.
-On the other hand, D-D-D,
-in contradiction to the preceding statement.
-
diff --git a/pages/general-topology/compactness/basics.md b/pages/general-topology/compactness/basics.md
index a1dded7..c7249ff 100644
--- a/pages/general-topology/compactness/basics.md
+++ b/pages/general-topology/compactness/basics.md
@@ -4,40 +4,43 @@ parent: Compactness
 grand_parent: General Topology
 nav_order: 1
 published: false
-# cspell:words
 ---
 
 # {{ page.title }} of Compact Spaces
 
 *Compact space* is short for compact topological space.
 
-{: .definition }
-> Suppose $X$ is a topological space.
-> A *covering* of $X$ is a collection $\mathcal{A}$
-> of subsets of $X$ such that
-> $\bigcup \mathcal{A} = X$.
-> A covering $\mathcal{A}$ of $X$ is called *open*
-> if each member of the collection $\mathcal{A}$
-> is open in $X$.
-> A covering $\mathcal{A}$ is called *finite*
-> the collection $\mathcal{A}$ is finite.
-> A *subcovering* of a covering $\mathcal{A}$ of $X$
-> is a subcollection $\mathcal{B}$ of $\mathcal{A}$
-> such that $\mathcal{B}$ is a covering of $X$.
-
-{: .definition }
-> A topological space $X$ is called *compact*
-> if every open covering of $X$
-> has a finite subcovering.
-
-{: .theorem }
-> Every closed subspace of a compact space is compact.
+{% definition %}
+Suppose $X$ is a topological space.
+A *covering* of $X$ is a collection $\mathcal{A}$
+of subsets of $X$ such that
+$\bigcup \mathcal{A} = X$.
+A covering $\mathcal{A}$ of $X$ is called *open*
+if each member of the collection $\mathcal{A}$
+is open in $X$.
+A covering $\mathcal{A}$ is called *finite*
+the collection $\mathcal{A}$ is finite.
+A *subcovering* of a covering $\mathcal{A}$ of $X$
+is a subcollection $\mathcal{B}$ of $\mathcal{A}$
+such that $\mathcal{B}$ is a covering of $X$.
+{% enddefinition %}
+
+{% definition %}
+A topological space $X$ is called *compact*
+if every open covering of $X$
+has a finite subcovering.
+{% enddefinition %}
+
+{% theorem %}
+Every closed subspace of a compact space is compact.
+{% endtheorem %}
 
 {% proof %}
 {% endproof %}
 
-{: .theorem }
-> Every compact subspace of a Hausdorff space is closed.
+{% theorem %}
+Every compact subspace of a Hausdorff space is closed.
+{% endtheorem %}
 
 {% proof %}
 {% endproof %}
diff --git a/pages/general-topology/compactness/index.md b/pages/general-topology/compactness/index.md
index 60c29a0..37e9b4d 100644
--- a/pages/general-topology/compactness/index.md
+++ b/pages/general-topology/compactness/index.md
@@ -1,9 +1,46 @@
 ---
 title: Compactness
 parent: General Topology
-nav_order: 1
+nav_order: 5
 has_children: true
-# cspell:words
+has_toc: false
 ---
 
 # {{ page.title }}
+
+## Compactness in Terms of Closed Sets
+
+{% theorem %}
+A topological space $X$ is compact
+if and only if it has the following property:
+- Given any collection $\mathcal{C}$ of closed subsets of $X$,
+  if every finite subcollection of $\mathcal{C}$ has nonempty intersection,
+  then $\mathcal{C}$ has nonempty intersection.
+{% endtheorem %}
+
+{% proof %}
+By definition, a topological space $X$ is compact
+if and only if it has the following property:
+- Given any collection $\mathcal{O}$ of open subsets of $X$,
+  if $\mathcal{O}$ covers $X$,
+  then there exists a finite subcollection of $\mathcal{O}$ that covers $X$.
+
+If $\mathcal{A}$ is a collection of subsets of $X$,
+let $\mathcal{A}^c = \braces{ X \setminus A : A \in \mathcal{A}}$ denote the collection of the complements of its members.
+Clearly, $\mathcal{B}$ is a subcollection of $\mathcal{A}$
+if and only if $\mathcal{B}^c$ is a subcollection of $\mathcal{A}^c$.
+Moreover, note that $\mathcal{B}$ covers $X$ if and only if
+$\mathcal{B}^c$ has empty intersection.
+Taking the contrapositive, we reformulate above property:
+- Given any collection $\mathcal{O}$ of open subsets of $X$,
+  if every finite subcollection of $\mathcal{O}^c$ has nonempty intersection,
+  then $\mathcal{O}^c$ has nonempty intersection.
+
+To complete the proof, observe that a collection $\mathcal{A}$ consists of open subsets of $X$
+if and only if $\mathcal{A}^c$ consists of closed subsets of $X$.
+{% endproof %}
+
+{% definition Finite Intersection Property%}
+TODO
+{% enddefinition %}
+
diff --git a/pages/general-topology/compactness/tychonoff-product-theorem.md b/pages/general-topology/compactness/tychonoff-product-theorem.md
index 2ae78e4..ddf3800 100644
--- a/pages/general-topology/compactness/tychonoff-product-theorem.md
+++ b/pages/general-topology/compactness/tychonoff-product-theorem.md
@@ -3,17 +3,17 @@ title: Tychonoff Product Theorem
 parent: Compactness
 grand_parent: General Topology
 nav_order: 2
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .theorem-title }
-> {{ page.title }}
-> {: #{{ page.title | slugify }} }
->
-> The product of (an arbitrary family of) compact spaces is compact.
+{% theorem * Tychonoff Product Theorem %}
+The product of (an arbitrary family of) compact spaces is compact.
+{% endtheorem %}
 
 {% proof %}
 TODO
 {% endproof %}
+
+Important Application:
+[Banach–Alaoglu Theorem](/pages/functional-analysis-basics/banach-alaoglu-theorem.html).
diff --git a/pages/general-topology/connectedness.md b/pages/general-topology/connectedness.md
new file mode 100644
index 0000000..b422775
--- /dev/null
+++ b/pages/general-topology/connectedness.md
@@ -0,0 +1,13 @@
+---
+title: Connectedness
+parent: General Topology
+nav_order: 5
+---
+
+# {{ page.title }}
+
+{% theorem %}
+{% endtheorem %}
+
+{% proof %}
+{% endproof %}
diff --git a/pages/general-topology/continuity-and-convergence.md b/pages/general-topology/continuity-and-convergence.md
new file mode 100644
index 0000000..7ae4534
--- /dev/null
+++ b/pages/general-topology/continuity-and-convergence.md
@@ -0,0 +1,24 @@
+---
+title: Continuity & Convergence
+parent: General Topology
+nav_order: 2
+---
+
+# {{ page.title }}
+
+{% definition Continuity %}
+A mapping $f: X \to Y$ between topological spaces $X$ and $Y$ is called *continuous*,
+if for each open subset $V$ of $Y$ the inverse image $f^{-1}(V)$ is an open subset of $X$.
+{% enddefinition %}
+
+Slogan: continuous $=$ The inverse image of every open subset is open.
+
+{% definition Homeomorphism %}
+Suppose $X$ and $Y$ are topological spaces.
+A mapping $f: X \to Y$ is said to be a *homeomorphism*,
+if $f$ is bijective and both $f$ and the inverse mapping $f^{-1} : Y \to X$ are continuous.
+{% enddefinition %}
+
+## Nets
+
+## Filters
diff --git a/pages/general-topology/jordan-curve-theorem.md b/pages/general-topology/jordan-curve-theorem.md
index 9da141e..e043fdc 100644
--- a/pages/general-topology/jordan-curve-theorem.md
+++ b/pages/general-topology/jordan-curve-theorem.md
@@ -3,16 +3,12 @@ title: Jordan Curve Theorem
 parent: General Topology
 nav_order: 1
 published: false
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .theorem-title }
-> {{ page.title }}
-> {: #{{ page.title | slugify }} }
->
-> ...
+{% theorem %}
+{% endtheorem %}
 
 {% proof %}
 {% endproof %}
diff --git a/pages/general-topology/metric-spaces/index.md b/pages/general-topology/metric-spaces/index.md
new file mode 100644
index 0000000..c0dc45a
--- /dev/null
+++ b/pages/general-topology/metric-spaces/index.md
@@ -0,0 +1,215 @@
+---
+title: Metric Spaces
+parent: General Topology
+nav_order: 8
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
+
+{% definition Metric, Metric Space %}
+A *metric* (or *distance function*) on a set $X$ is
+a mapping $d : X \times X \to \RR$ with the properties \
+**(M1)** $\ \forall x,y \in X : d(x,y) = 0 \iff x=y \quad$ *(point separation)* \
+**(M2)** $\ \forall x,y \in X : d(x,y) = d(y,x) \quad$ *(symmetry)* \
+**(M3)** $\ \forall x,y,z \in X : d(x,z) \le d(x,y) + d(y,x) \quad$ *(triangle inequality)* \
+We say that $d(x,y)$ is the *distance* between $x$ and $y$. \
+A *metric space* is a pair $(X,d)$ consisting of a set $X$
+and a metric $d$ on $X$.
+{% enddefinition %}
+
+Setting $x=z$ in **(M3)** and applying **(M1)** & **(M2)**
+gives us $0 = d(x,x) \le 2 d(x,y)$, hence $d(x,y) \ge 0$.
+This *nonnegativity* of the metric is often part of the definition.
+
+Relaxing **(M1)** to the condition $\forall x \in X : d(x,x) = 0$
+leads to the notion of a *semi-metric*
+and that of a *semi-metric space*.
+Nonnegativity still follows as shown above.
+
+*Pseudo-metric* is usually a synonym for *semi-metric*.
+
+*Quasi-metric* refers to dropping **(M2)**.
+
+An *ultrametric* satisfies in place of **(M3)** the stronger condition
+$d(x,z) \le \max \braces{d(x,y),d(y,z)}$.
+
+{% definition Metric Subspace %}
+A *metric subspace* of a metric space $(X,d)$ is a pair $(S,d_S)$
+where $S$ is a subset of $X$ and
+$d_S$ is the restriction of $d$ to $S \times S$.
+{% enddefinition %}
+
+Clearly, a metric subspace of a metric space is itself a metric space.
+
+{% proposition %}
+Let $(X,d)$ be a (semi-)metric space.
+- For all $x,y,z \in X$ we have the *inverse triangle inequality*
+  
+  $$
+  \abs{d(x,y) - d(y,z)} \le d(x,z).
+  $$
+
+- For all $v,w,x,y \in X$ we have the *quadrilateral inequality*
+  
+  $$
+  \abs{d(v,w) - d(x,y)} \le d(v,x) + d(w,y)
+  $$
+{% endproposition %}
+
+The proofs are straightforward.
+
+TODO
+- isometry
+- metric induced by a norm
+- metric product
+
+{% definition Diameter %}
+The *diameter* of a subset $S$ of a metric space $(X,d)$ is the number
+
+$$
+\diam{S} = \sup \braces{d(x,y) : x,y \in S} \in \braces{-\infty} \cup [0,\infty].
+$$
+{% enddefinition %}
+
+Note that $\diam{S} = -\infty$ iff $S = \varnothing$,
+and $\diam{S} = 0$ iff $S$ is a singleton set.
+
+{% definition Distance %}
+Suppose $(X,d)$ is a metric space.
+The *distance* from a point $x \in X$ to a subset $S \subset X$ is
+
+$$
+\dist{x,S} = \inf \braces{d(x,y) : y \in S} \in [0,\infty].
+$$
+{% enddefinition %}
+
+Note that $\dist{x,S} = \infty$ iff $S = \varnothing$.
+
+{% definition Convergence, Limit, Divergence %}
+Let $(X,d)$ be a metric space.
+A sequence $(x_n)_{n \in \NN}$ in $X$ is said to *converge to a point $x \in X$*, if
+
+$$
+\forall \epsilon > 0 \ \ \exists N \in \NN \ \ \forall n \ge N : d(x,x_n) < \epsilon.
+$$
+
+In this case, we call $x$ a *limit (point)* of the sequence.
+Symbolically this is expressed by
+
+$$
+\lim_{n \to \infty} x_n = x
+$$
+
+or by saying that $x_n \to x$ as $n \to \infty$.
+
+We call a sequence in $X$ *convergent*
+if it converges to some point of $X$
+and *divergent* otherwise.
+{% enddefinition %}
+
+For a semi-metric space the definition remains the same.
+However, the notation $\lim x_n = x$ can be misleading,
+because there might be more than one limit point.
+
+{% proposition %}
+A sequence in a metric space has at most one limit.
+{% endproposition %}
+
+In other words: The limit of a convergent sequence in a metric space is unique.
+
+{% proof %}
+Let $(x_n)$ be a convergent sequence in a metric space $(X,d)$ with limit point $x$.
+If $x'$ is another limit point of $(x_n)$,
+then $d(x,x') \le d(x,x_n) + d(x_n,x')$ for all $n \in \NN$ by **(M3)**.
+Given $\epsilon >0$, there exist natural numbers $N$ and $N'$ such that
+$d(x,x_n) < \epsilon$ for all $n \ge N$ and
+$d(x,x_n) < \epsilon$ for all $n \ge N'$.
+Both hold, if $n$ is large enough ($\ge \max \braces{N,N'}$ to be precise).
+It follows that $d(x,x') < 2 \epsilon$.
+Since $\epsilon$ was arbitrary, $d(x,x') = 0$.
+Therefore, $x=x'$ by **(M1)**.
+{% endproof %}
+
+{% corollary %}
+A semi-metric space $X$ is a metric space if and only if
+every sequence in $X$ has at most one limit.
+{% endcorollary %}
+
+{% definition %}
+Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces.
+A mapping $f: X \to Y$ is called
+- *continuous at a point $x \in X$* if
+  
+  $$
+  \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen
+  $$
+
+- *continuous* if it is continuous at every point of $X$, that is
+  
+  $$
+  \forall x \in X \ \ \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen
+  $$
+
+- *uniformly continuous* if
+  
+  $$
+  \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x,x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen
+  $$
+
+- *Lipschitz continuous* if
+  
+  $$
+  \exists L \ge 0 \ \ \forall x,x' \in X : d_Y(f(x),f(x')) \le L \, d_X(x,x') 
+  $$
+{% enddefinition %}
+
+{% definition Open Ball, Closed Ball, Sphere %}
+Suppose $(X,d)$ is a metric space.
+The *open ball* with center $c \in X$ and radius $r>0$ is the set
+
+$$
+B(c,r) = \braces{x \in X : d(x,c) < r}.
+$$
+
+The *closed ball* with center $c \in X$ and radius $r>0$ is the set
+
+$$
+\overline{B}(c,r) = \braces{x \in X : d(x,c) \le r}.
+$$
+
+The *sphere* with center $c \in X$ and radius $r>0$ is the set
+
+$$
+S(c,r) = \braces{x \in X : d(x,c) = r}.
+$$
+{% enddefinition %}
+
+Observe that $S(c,r) = \overline{B}(c,r) \setminus B(c,r)$.
+
+{% definition Open Subset of a Metric Space %}
+A subset $O$ of a metric space is called *open* if for every point $x \in O$
+there exists an $\epsilon > 0$ such that $B(x,\epsilon) \subset O$.
+{% enddefinition %}
+
+{% proposition Metric Topology %}
+Let $(X,d)$ be a metric space.
+The collection of open subsets of $X$ forms a topology on $X$.
+This topology is called the *metric topology* on $X$ induced by $d$.
+{% endproposition %}
+
+{% proposition %}
+- Open balls in a metric space are open with respect to the metric topology.
+- Closed balls in a metric space are closed with respect to the metric topology.
+- The boundary (with respect to the metric topology) of an open or closed ball
+  is the sphere with the same center and radius. Not true!!!!
+- The collection of open balls in a metric space forms a basis of the metric topology.
+{% endproposition %}
+
+## Complete Metric Spaces
+
+- Definition
+- Banach Fixed-Point Theorem
+- Baire
+- Metric Completion
diff --git a/pages/general-topology/separation/index.md b/pages/general-topology/separation/index.md
new file mode 100644
index 0000000..b4916f5
--- /dev/null
+++ b/pages/general-topology/separation/index.md
@@ -0,0 +1,9 @@
+---
+title: Separation
+parent: General Topology
+nav_order: 4
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
diff --git a/pages/general-topology/separation/tietze-extension-theorem.md b/pages/general-topology/separation/tietze-extension-theorem.md
new file mode 100644
index 0000000..4b2eee3
--- /dev/null
+++ b/pages/general-topology/separation/tietze-extension-theorem.md
@@ -0,0 +1,14 @@
+---
+title: Tietze Extension Theorem
+parent: Separation
+grand_parent: General Topology
+nav_order: 2
+---
+
+# {{ page.title }}
+
+{% theorem %}
+{% endtheorem %}
+
+{% proof %}
+{% endproof %}
diff --git a/pages/general-topology/separation/urysohn-lemma.md b/pages/general-topology/separation/urysohn-lemma.md
new file mode 100644
index 0000000..b3f8208
--- /dev/null
+++ b/pages/general-topology/separation/urysohn-lemma.md
@@ -0,0 +1,14 @@
+---
+title: Urysohn Lemma
+parent: Separation
+grand_parent: General Topology
+nav_order: 1
+---
+
+# The {{ page.title }}
+
+{% theorem %}
+{% endtheorem %}
+
+{% proof %}
+{% endproof %}
diff --git a/pages/general-topology/topological-spaces.md b/pages/general-topology/topological-spaces.md
new file mode 100644
index 0000000..b0b1834
--- /dev/null
+++ b/pages/general-topology/topological-spaces.md
@@ -0,0 +1,146 @@
+---
+title: Topological Spaces
+parent: General Topology
+nav_order: 1
+---
+
+# {{ page.title }}
+
+## Elementary Concepts
+
+{% definition Topology, Topological Space %}
+A *topology* on a set $X$ is a collection $\mathcal{T}$ of subsets of $X$ such that \
+**(T1)** $\varnothing$ and $X$ belong to $\mathcal{T}$, \
+**(T2)** the union of any subcollection of $\mathcal{T}$ belongs to $\mathcal{T}$, \
+**(T3)** the intersection of any finite subcollection $\mathcal{T}$ belongs to $\mathcal{T}$. \
+A *topological space* is a pair $(X,\mathcal{T})$ consisting of
+a set $X$ and a topology $\mathcal{T}$ on $X$.
+{% enddefinition %}
+
+If one follows the convention that
+the union of the empty collection of subsets of $X$ is the empty subset of $X$,
+and its intersection is all of $X$,
+then **(T1)** is a consequence of **(T2)**, **(T3)**
+and can be omitted.
+
+If $(X,\mathcal{T})$ is a topological space,
+the elements of $X$ are called *points*
+and the elements of $\mathcal{T}$ are called the *open sets*.
+
+{% example %}
+On every set $X$ we have
+the *trivial* (or *indiscrete*) *topology* $\braces{\varnothing,X}$ and
+the *discrete topology* $\mathcal{P}(X)$.
+These collections are in fact topologies on $X$.
+{% endexample %}
+
+{% example %}
+If $X$ is any set,
+then the collection of all subsets of $X$
+whose complement is either finite or all of $X$
+is a topology on $X$;
+it is called the *finite complement topology*.
+The *countable complement topology* is defined analogously.
+{% endexample %}
+
+{% definition Comparison of Topologies %}
+Suppose $\mathcal{T}$ and $\mathcal{T}'$ are topologies on a set $X$.
+When $\mathcal{T} \subset \mathcal{T}'$,
+we say that $\mathcal{T}$ is *coarser* or *smaller* or *weaker* than $\mathcal{T}'$,
+and that $\mathcal{T}'$ is *finer* or *larger* or *stronger* than $\mathcal{T}$.
+If the inclusion is proper, then we say *strictly coarser* and so on.
+If either $\mathcal{T} \subset \mathcal{T}'$ or $\mathcal{T} \supset \mathcal{T}'$ holds,
+then the topologies are said to be *comparable*.
+{% enddefinition %}
+
+{% proposition Intersection of Topologies %}
+If $\braces{\mathcal{T}_{\alpha}}$ is a family of topologies on a set $X$,
+then $\bigcap_{\alpha} \mathcal{T}_{\alpha}$ is a topology on $X$.
+{% endproposition %}
+
+{% definition Generated Topology %}
+Suppose $\mathcal{A}$ is a collection of subsets of a set $X$.
+The *topology generated by $\mathcal{A}$* is
+the intersection of all topologies on $X$ containing $\mathcal{A}$.
+{% enddefinition %}
+
+By the previous proposition, the generated topology is indeed a topology.
+
+{% proposition %}
+The topology generated by a collection $\mathcal{A}$ of subsets of a set $X$
+is the smallest topology on $X$ containing $\mathcal{A}$.
+{% endproposition %}
+
+## Bases and Subbases
+
+{% definition Basis for a Topology %}
+A *basis for a topology* on a set $X$ is a collection $\mathcal{B}$ of subsets of $X$
+such that for every point $x \in X$ 
+- there exists $B \in \mathcal{B}$ such that $x \in B$,
+- if $x \in B_1 \cap B_2$ for $B_1, B_2 \in \mathcal{B}$,
+  then there exists a $B_3 \in \mathcal{B}$
+  such that $x \in B_3 \subset B_1 \cap B_2$.
+{% enddefinition %}
+
+{% theorem Topology Generated by a Basis %}
+If $X$ is set and $\mathcal{B}$ is a basis for a topology on $X$,
+then the topology generated by $\mathcal{B}$ equals
+- the collection of all subsets $S \subset X$ with the property
+  that for every $x \in S$ there exists a basis element $B \in \mathcal{B}$
+  such that $x \in B$ and $B \subset S$;
+- the collection of all arbitrary unions of elements of $\mathcal{B}$.
+{% endtheorem %}
+
+Let $\mathcal{T}$ be a topology on a set $X$.
+As one might expect,
+a collection $\mathcal{B}$ of subsets of $X$
+is said to be a *basis for the topology $\mathcal{T}$*,
+if $\mathcal{B}$ is basis for a topology on $X$ and
+the topology generated by $\mathcal{B}$ equals $\mathcal{T}$.
+
+{% example %}
+If $X$ is a set, then the collection of singletons $\braces{x}$, $x \in X$,
+is a basis for the discrete topology on $X$.
+{% endexample %}
+
+{% example %}
+If $(X,d)$ is a metric space,
+then the collection of open balls is a basis for the metric topology on $X$.
+{% endexample %}
+
+{% definition Subbasis for a Topology %}
+A *subbasis for a topology* on a set $X$ is a collection $\mathcal{S}$ of subsets of $X$
+such that for every point $x \in X$ there exists a $S \in \mathcal{S}$ such that $x \in S$.
+{% enddefinition %}
+
+{% theorem Topology Generated by a Subbasis %}
+If $X$ is set and $\mathcal{S}$ is a subbasis for a topology on $X$,
+then the topology generated by $\mathcal{S}$ equals
+- the collection of all arbitrary unions of finite intersections of elements of $\mathcal{S}$.
+{% endtheorem %}
+
+## Open and Closed Sets
+
+{% definition Open Set, Closed Set %}
+Suppose $(X,\mathcal{T})$ is a topological space.
+A subset $S$ of $X$
+is called *open* with respect to $\mathcal{T}$
+when it belongs to $\mathcal{T}$
+and it is called *closed* with respect to $\mathcal{T}$
+when its complement $X \setminus S$ belongs to $\mathcal{T}$.
+{% enddefinition %}
+
+A subset of a topological space is open
+if and only if its complement is closed.
+
+{% proposition %}
+Let $\mathcal{C}$ be the collection of closed subsets of a topological space. Then
+{: .mb-0 }
+- $X$ and $\varnothing$ belong to $\mathcal{C}$,
+-  the intersection of any subcollection of $\mathcal{C}$ belongs to $\mathcal{C}$,
+-  the union of any finite subcollection $\mathcal{C}$ belongs to $\mathcal{C}$.
+{% endproposition %}
+
+## The Subspace Topology
+
+
diff --git a/pages/general-topology/universal-constructions.md b/pages/general-topology/universal-constructions.md
new file mode 100644
index 0000000..827f730
--- /dev/null
+++ b/pages/general-topology/universal-constructions.md
@@ -0,0 +1,23 @@
+---
+title: Universal Constructions
+parent: General Topology
+nav_order: 1
+---
+
+# {{ page.title }}
+
+{% definition Initial Topology %}
+Suppose that $f_i : S \to X_i$, $i \in I$, is a family of maps,
+from a set $S$ into topological spaces $X_i$.
+The *initial topology* on $S$ induced by the family $(f_i)$
+is defined to be the weakest topology on $S$
+making all maps $f_i$ continuous.
+{% enddefinition %}
+
+{% theorem * Universal Property of the Initial Topology %}
+The initial topology on $S$ induced by the family $(f_i)$
+is the unique topology on $S$ with the property that
+for any topological space $T$,
+a mapping $g : T \to S$ is continuous if and only if
+all compositions $f_i \circ g : T \to X_i$ are continuous.
+{% endtheorem %}
diff --git a/pages/measure-and-integration/bochner-integral/index.md b/pages/measure-and-integration/bochner-integral/index.md
new file mode 100644
index 0000000..5517934
--- /dev/null
+++ b/pages/measure-and-integration/bochner-integral/index.md
@@ -0,0 +1,9 @@
+---
+title: Bochner Integral
+parent: Measure and Integration
+nav_order: 3
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
diff --git a/pages/measure-and-integration/index.md b/pages/measure-and-integration/index.md
new file mode 100644
index 0000000..841e941
--- /dev/null
+++ b/pages/measure-and-integration/index.md
@@ -0,0 +1,8 @@
+---
+title: Measure and Integration
+nav_order: 2
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
diff --git a/pages/measure-and-integration/lebesgue-integral/almost-everywhere.md b/pages/measure-and-integration/lebesgue-integral/almost-everywhere.md
new file mode 100644
index 0000000..a77cf9a
--- /dev/null
+++ b/pages/measure-and-integration/lebesgue-integral/almost-everywhere.md
@@ -0,0 +1,27 @@
+---
+title: Almost Everywhere
+parent: Lebesgue Integral
+grand_parent: Measure and Integration
+nav_order: 1
+---
+
+# {{ page.title }}
+
+{% definition Almost Everywhere %}
+We say that a property $P(x)$ depending on $x \in X$
+holds *almost everywhere* (abbreviated by *a.e.*) or for *almost all $x \in X$* if
+the set of points where it does not hold has measure zero.
+{% enddefinition %}
+
+In other words, $P(x)$ a.e. iff
+$\mu(\set{x \in X : \neg P(x)}) = 0$.
+
+{% theorem %}
+Let $f : X \to \overline{\RR}$ be a nonnegative measurable function. Then
+
+$$
+\int_X f \, d\mu = 0
+$$
+
+holds if and only if $f$ vanishes almost everywhere.
+{% endtheorem %}
diff --git a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md
new file mode 100644
index 0000000..67f0996
--- /dev/null
+++ b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md
@@ -0,0 +1,77 @@
+---
+title: Convergence Theorems
+parent: Lebesgue Integral
+grand_parent: Measure and Integration
+nav_order: 2
+---
+
+# {{ page.title }}
+
+For all statements on this page,
+assume that $(X,\mathcal{A},\mu)$ is a measure space.
+
+{% theorem * Monotone Convergence Theorem %}
+For each $n \in \NN$ let $f_n : X \to \overline{\RR}$ be a measurable function.
+If $0 \le f_n \le f_{n+1}$ almost everywhere, then
+
+$$
+\int_X \lim_{n \to \infty} f_n \, d\mu = \lim_{n \to \infty} \int_X f_n \, d\mu.
+$$
+{% endtheorem %}
+
+Note that the pointwise limit $\lim_{n \to \infty} f_n$ always exists and is measurable by this proposition.
+
+{% lemma * Fatou’s Lemma %}
+For each $n \in \NN$ let $f_n : X \to \overline{\RR}$ be a nonnegative measurable function. Then
+
+$$
+\int_X \liminf_{n \to \infty} f_n \, d\mu \le \liminf_{n \to \infty} \int_X f_n \, d\mu.
+$$
+{% endlemma %}
+
+In the following proof we omit $X$ and $d\mu$ for visual clarity.
+
+{% proof %}
+By definition, we have $\liminf_{n \to \infty} f_n = \lim_{n \to \infty} g_n$, where $g_n = \inf_{k \ge n} f_k$.
+Now $(g_n)$ is a monotonic sequence of nonnegative measurable functions.
+By the
+[Monotone Convergence Theorem](#monotone-convergence-theorem)
+
+$$
+\int \liminf_{n \to \infty} f_n = \lim_{n \to \infty} \int g_n.
+$$
+
+For all $k \ge n$ one has $g_n \le f_k$, hence
+$\int g_n \le \int f_k$ by the monotonicity of the integral.
+This implies
+
+$$
+\int g_n \le \inf_{k \ge n} \int f_k
+$$
+
+for all $n \in \NN$. In the limit $n \to \infty$ we obtain
+
+$$
+\lim_{n \to \infty} \int g_n
+\le \liminf_{n \to \infty} \int f_n
+$$
+
+thereby completing the proof.
+{% endproof %}
+
+{% theorem * Dominated Convergence Theorem %}
+Let $(X,\mathcal{A},\mu)$ be a measure space.
+For each $n \in \NN$ let $f_n : X \to \overline{\RR}$ (or $\CC$) be a measurable function.
+Suppose that the pointwise limit $f = \lim_{n \to \infty} f_n$ exists almost everywhere.
+Suppose further that there exists an integrable function $g : X \to \overline{\RR}$
+such that $\abs{f_n} \le g$ almost everywhere for all $n \in \NN$.
+Then the functions $f_n$ and $f$ are all integrable, and
+
+$$
+\lim_{n \to \infty} \int_X f_n \, d\mu = \int_X f \, d\mu.
+$$
+{% endtheorem %}
+
+{% proof %}
+TODO
+{% endproof %}
diff --git a/pages/measure-and-integration/lebesgue-integral/fubini-theorem.md b/pages/measure-and-integration/lebesgue-integral/fubini-theorem.md
new file mode 100644
index 0000000..6e5179c
--- /dev/null
+++ b/pages/measure-and-integration/lebesgue-integral/fubini-theorem.md
@@ -0,0 +1,14 @@
+---
+title: Fubini Theorem
+parent: Lebesgue Integral
+grand_parent: Measure and Integration
+nav_order: 2
+---
+
+# {{ page.title }}
+
+{% theorem %}
+{% endtheorem %}
+
+{% proof %}
+{% endproof %}
diff --git a/pages/measure-and-integration/lebesgue-integral/index.md b/pages/measure-and-integration/lebesgue-integral/index.md
new file mode 100644
index 0000000..a857d95
--- /dev/null
+++ b/pages/measure-and-integration/lebesgue-integral/index.md
@@ -0,0 +1,112 @@
+---
+title: Lebesgue Integral
+parent: Measure and Integration
+nav_order: 2
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
+
+For this entire section we fix a measure space $(X,\mathcal{A},\mu)$.
+
+## Integration of Nonnegative Step Functions
+
+{% definition %}
+Let $f : X \to \RR$ be a nonnegative step function
+with representation $f = \sum_{i=1}^n \alpha_i \chi_{A_i}$,
+where $\alpha_1, \ldots, \alpha_n \ge 0$ and
+$A_1, \ldots, A_n \in \mathcal{A}$.
+We define the *integral of $f$ on $X$ with respect to $\mu$* by
+
+$$
+\int_X f \, d\mu = \sum_{i=1}^n \alpha_i \, \mu(A_i) \in [0,\infty].
+$$
+{% enddefinition %}
+
+TODO: This does not depend on the representation of $f$.
+
+## Integration of Nonnegative Measurable Functions
+
+{% theorem Approximation by Step Functions %}
+Every nonnegative measurable function $f : X \to \overline{\RR}$
+is the pointwise limit of an increasing sequence $(s_n)$ of
+nonnegative step functions $s_n : X \to \RR$.
+{% endtheorem %}
+
+{% definition %}
+Let $f : X \to \overline{\RR}$ be a nonnegative measurable function
+and let $(s_n)$ be a sequence of nonnegative step functions
+with $s_n \uparrow f$.
+We define the *integral of $f$ on $X$ with respect to $\mu$* by
+
+$$
+\int_X f \, d\mu = \lim_{n \to \infty} \int_X s_n \, d\mu \in [0,\infty].
+$$
+{% enddefinition %}
+
+## Integrable Functions
+
+Recall that the positive and (flipped) negative parts
+of a function $f : X \to \overline{R}$ are defined by
+
+$$
+f^+ = \max(f,0) \qquad
+f^- = \max(-f,0),
+$$
+
+and that $f$ is measurable if and only if both $f^+$ and $f^-$ are measurable.
+We have $f = f^+\! - f^-$.
+
+{% definition Integrable Function, Lebesgue Integral %}
+A measurable function $f : X \to \overline{\RR}$ is said to be
+*integrable on $X$ with respect to $\mu$* if the integrals
+
+$$
+\int_X f^+ \, d\mu, \qquad \int_X f^- \, d\mu
+\tag{$*$}
+$$
+
+are both finite.
+In this case the *(Lebesgue) integral of $f$ on $X$ with respect to $\mu$* is defined as
+
+$$
+\int_X f \, d\mu = \int_X f^+ \, d\mu - \int_X f^- \, d\mu \in \RR.
+$$
+{% enddefinition %}
+
+Sometimes it is convenient to have a slightly more general notion of integrability:
+
+{% definition Quasi-Integrable Function %}
+A measurable function $f : X \to \overline{\RR}$ is said to be
+*quasi-integrable on $X$ with respect to $\mu$* if at least one of the integrals
+$(*)$ is finite.
+In this case the *integral of $f$ on $X$ with respect to $\mu$* is defined as
+
+$$
+\int_X f \, d\mu = \int_X f^+ \, d\mu - \int_X f^- \, d\mu \in \overline{\RR}.
+$$
+{% enddefinition %}
+
+{% definition %}
+A measurable function $f : X \to \CC$ is said to be
+*integrable on $X$ with respect to $\mu$* if
+$\Re f$ and $\Im f$ are integrable on $X$ with respect to $\mu$.
+In this case the *integral of $f$ on $X$ with respect to $\mu$* is defined as
+
+$$
+\int_X f \, d\mu = \int_X \Re f \, d\mu + i \int_X \Im f \, d\mu \in \CC.
+$$
+{% enddefinition %}
+
+## Integration on Measurable Subsets
+
+{% definition %}
+For any measurable subset $A \subset X$ we define
+the *integral on $A$* of a (quasi-)integrable function $f : X \to \overline{\RR}$ (or $\CC$) by
+
+$$
+\int_A f \, d\mu = 
+\int_X \chi_A f \, d\mu.
+$$
+{% enddefinition %}
diff --git a/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md
new file mode 100644
index 0000000..023c253
--- /dev/null
+++ b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md
@@ -0,0 +1,36 @@
+---
+title: The L<sup>p</sup> Spaces 
+parent: Lebesgue Integral
+grand_parent: Measure and Integration
+nav_order: 4
+---
+
+# {{ page.title }}
+
+{% definition %}
+Let $(X,\mathcal{A},\mu)$ be a measure space and let $p \in [1,\infty)$.
+We write $\mathscr{L}^p(X,\mathcal{A},\mu)$ for the set of all
+measurable functions $f : X \to \KK$ such that $\abs{f}^p$ is integrable.
+For such $f$ we write
+
+$$
+\norm{f}_p = {\bigg\lparen\int_X \abs{f}^p \, d\mu\bigg\rparen}^{\!1/p}.
+$$
+{% enddefinition %}
+
+{% proposition %}
+Endowed with pointwise addition and scalar multiplication
+$\mathscr{L}^p(X,\mathcal{A},\mu)$ becomes a vector space.
+{% endproposition %}
+
+{% proposition %}
+$\norm{\cdot}_p$ is a seminorm on $\mathscr{L}^p(X,\mathcal{A},\mu)$.
+{% endproposition %}
+
+{% theorem * Young Inequality %}
+Consider $p,q > 1$ such that $1/p + 1/q = 1$. Then
+
+$$
+a \cdot b \le \frac{a^p}{p} + \frac{b^q}{q} \qquad \forall a,b \ge 0.
+$$
+{% endtheorem %}
diff --git a/pages/measure-and-integration/lebesgue-integral/transformation-formula.md b/pages/measure-and-integration/lebesgue-integral/transformation-formula.md
new file mode 100644
index 0000000..6f02bc8
--- /dev/null
+++ b/pages/measure-and-integration/lebesgue-integral/transformation-formula.md
@@ -0,0 +1,14 @@
+---
+title: Transformation Formula
+parent: Lebesgue Integral
+grand_parent: Measure and Integration
+nav_order: 3
+---
+
+# {{ page.title }}
+
+{% theorem %}
+{% endtheorem %}
+
+{% proof %}
+{% endproof %}
diff --git a/pages/measure-and-integration/measure-theory/borels-sets.md b/pages/measure-and-integration/measure-theory/borels-sets.md
new file mode 100644
index 0000000..737a7c8
--- /dev/null
+++ b/pages/measure-and-integration/measure-theory/borels-sets.md
@@ -0,0 +1,33 @@
+---
+title: Borel Sets
+parent: Measure Theory
+grand_parent: Measure and Integration
+nav_order: 2
+---
+
+# {{ page.title }}
+
+{% definition Borel Sigma-Algebra, Borel Set %}
+The *Borel σ-algebra* $\mathcal{B}(X)$ on a topological space $X$ is
+the σ-algebra generated by its open sets.
+The elements of $\mathcal{B}(X)$ are called *Borel(-measurable) sets*.
+{% enddefinition %}
+
+That is, $\mathcal{B}(X) = \sigma(\mathcal{O})$,
+where $\mathcal{O}$ is the collection of open sets in $X$.
+It is also true that $\mathcal{B}(X) = \sigma(\mathcal{C})$,
+where $\mathcal{C}$ is the collection of closed sets in $X$.
+
+{% definition Borel Function %}
+If $(X,\mathcal{A})$ is a measure space
+and $Y$ is a topological space,
+then a function $f : X \to Y$ is called *measurable*,
+or a *Borel function*,
+if it is measurable with respect to $\mathcal{A}$ and
+the Borel σ-algebra on $Y$.
+{% enddefinition %}
+
+{% definition Borel Measure %}
+A *Borel measure* on a topological space $X$
+is any measure on the Borel σ-algebra of $X$.
+{% enddefinition %}
diff --git a/pages/measure-and-integration/measure-theory/index.md b/pages/measure-and-integration/measure-theory/index.md
new file mode 100644
index 0000000..575c945
--- /dev/null
+++ b/pages/measure-and-integration/measure-theory/index.md
@@ -0,0 +1,9 @@
+---
+title: Measure Theory
+parent: Measure and Integration
+nav_order: 1
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
diff --git a/pages/measure-and-integration/measure-theory/measurable-maps.md b/pages/measure-and-integration/measure-theory/measurable-maps.md
new file mode 100644
index 0000000..5b7a76e
--- /dev/null
+++ b/pages/measure-and-integration/measure-theory/measurable-maps.md
@@ -0,0 +1,27 @@
+---
+title: Measurable Maps
+parent: Measure Theory
+grand_parent: Measure and Integration
+nav_order: 3
+---
+
+# {{ page.title }}
+
+{% definition Measurable Map %}
+Suppose $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are measurable spaces.
+We say that a map $f: X \to Y$ is *measurable* (with respect to $\mathcal{A}$ and $\mathcal{B}$) if
+$f^{-1}(B) \in \mathcal{A}$ for all $B \in \mathcal{B}$.
+{% enddefinition %}
+
+{% proposition %}
+The composition of measurable maps is measurable.
+{% endproposition %}
+
+It is sufficient to check measurability for a generator:
+
+{% proposition %}
+Suppose that $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are measurable spaces,
+and that $\mathcal{E}$ is a generator of $\mathcal{B}$.
+Then a map $f : X \to Y$ is measurable iff
+$f^{-1}(E) \in \mathcal{A}$ for every $E \in \mathcal{E}$.
+{% endproposition %}
diff --git a/pages/measure-and-integration/measure-theory/measures.md b/pages/measure-and-integration/measure-theory/measures.md
new file mode 100644
index 0000000..637ab0c
--- /dev/null
+++ b/pages/measure-and-integration/measure-theory/measures.md
@@ -0,0 +1,29 @@
+---
+title: Measures
+parent: Measure Theory
+grand_parent: Measure and Integration
+nav_order: 4
+---
+
+# {{ page.title }}
+
+{% definition %}
+A *measure* on a σ-algebra $\mathcal{A}$ on a set $X$
+is mapping $\mu : \mathcal{A} \to [0,\infty]$ such that
+
+- $\mu(\varnothing) = 0$,
+- for every sequence $(A_n)_{n \in \NN}$ of
+  pairwise disjoint sets $A_n \in \mathcal{A}$
+  
+  $$
+  \mu \bigg\lparen \bigcup_{n=1}^{\infty} A_n \! \bigg\rparen
+  = \sum_{n=0}^{\infty} \mu(A_n).
+  $$
+{% enddefinition %}
+
+{% definition Measure Space %}
+A *measure space* is a triple $(X,\mathcal{A},\mu)$ of
+a set $X$,
+a σ-algebra $\mathcal{A}$ on $X$
+and a measure $\mu$ on $\mathcal{A}$.
+{% enddefinition %}
diff --git a/pages/measure-and-integration/measure-theory/sigma-algebras.md b/pages/measure-and-integration/measure-theory/sigma-algebras.md
new file mode 100644
index 0000000..5d22f6b
--- /dev/null
+++ b/pages/measure-and-integration/measure-theory/sigma-algebras.md
@@ -0,0 +1,50 @@
+---
+title: σ-Algebras
+parent: Measure Theory
+grand_parent: Measure and Integration
+nav_order: 1
+---
+
+# {{ page.title }}
+
+{% definition Sigma-Algebra, Measurable Space, Measurable Set %}
+A *σ-algebra* on a set $X$ is a collection $\mathcal{A}$ of subsets of $X$ such that
+
+- $X$ belongs to $\mathcal{A}$,
+- if $A \in \mathcal{A}$, then $X \setminus A \in \mathcal{A}$,
+- the union of any countable subcollection of $\mathcal{A}$ belongs to $\mathcal{A}$.
+
+A *measurable space* is a pair $(X,\mathcal{A})$ consisting of
+a set $X$ and a σ-algebra $\mathcal{A}$ on $X$. \
+The subsets of $X$ belonging to $\mathcal{A}$ are called *measurable sets*.
+{% enddefinition %}
+
+{% example %}
+On every set $X$ we have the σ-algebras $\braces{\varnothing,X}$ and $\mathcal{P}(X)$.
+{% endexample %}
+
+{% proposition %}
+If $\mathcal{A}$ is *σ-algebra* on a set $X$, then:
+
+- $\varnothing$ belongs to $\mathcal{A}$,
+- if $A,B \in \mathcal{A}$, then $B \setminus A \in \mathcal{A}$,
+- the intersection of any countable subcollection of $\mathcal{A}$ belongs to $\mathcal{A}$.
+{% endproposition %}
+
+## Generated {{ page.title }}
+
+{% proposition Intersection of σ-Algebras %}
+If $\braces{\mathcal{A}_i}$ is a family of σ-algebras on a set $X$,
+then $\bigcap_i \mathcal{A}_i$ is a σ-algebra on $X$.
+{% endproposition %}
+
+{% definition Generated σ-Algebras %}
+Suppose $\mathcal{E}$ is any collection of subsets of a set $X$.
+The *σ-algebra generated by $\mathcal{E}$*, denoted by $\sigma(\mathcal{E})$, is
+defined to be the intersection of all σ-algebras on $X$ containing $\mathcal{A}$.
+{% enddefinition %}
+
+By the previous proposition, $\sigma(\mathcal{E})$ is in fact a σ-algebra on $X$.
+
+## Products of {{ page.title }}
+
diff --git a/pages/measure-and-integration/measure-theory/signed-measures.md b/pages/measure-and-integration/measure-theory/signed-measures.md
new file mode 100644
index 0000000..77b2416
--- /dev/null
+++ b/pages/measure-and-integration/measure-theory/signed-measures.md
@@ -0,0 +1,33 @@
+---
+title: Signed Measures
+parent: Measure Theory
+grand_parent: Measure and Integration
+nav_order: 10
+---
+
+# {{ page.title }}
+
+{% definition Signed Measure %}
+A *signed measure* on a σ-algebra $\mathcal{A}$ on a set $X$
+is a mapping $\mu : \mathcal{A} \to [-\infty,\infty]$ such that
+{: .mb-0 }
+
+- $\mu(\varnothing) = 0$,
+- either there is no $A \in \mathcal{A}$ with $\mu(A) = -\infty$
+  or there is no $A \in \mathcal{A}$ with $\mu(A) = \infty$,
+- for every sequence $(A_n)_{n \in \NN}$ of
+  pairwise disjoint sets $A_n \in \mathcal{A}$
+  {: .my-0 }
+  
+  $$
+  \mu \bigg\lparen \bigcup_{n=1}^{\infty} A_n \! \bigg\rparen
+  = \sum_{n=0}^{\infty} \mu(A_n).
+  $$
+{% enddefinition %}
+
+{% definition Measure Space %}
+A *measure space* is a triple $(X,\mathcal{A},\mu)$ of
+a set $X$,
+a σ-algebra $\mathcal{A}$ on $X$
+and a measure $\mu$ on $\mathcal{A}$.
+{% enddefinition %}
diff --git a/pages/more-functional-analysis/fixed-point-theorems/index.md b/pages/more-functional-analysis/fixed-point-theorems/index.md
new file mode 100644
index 0000000..b90bf00
--- /dev/null
+++ b/pages/more-functional-analysis/fixed-point-theorems/index.md
@@ -0,0 +1,13 @@
+---
+title: Fixed-Point Theorems
+parent: More Functional Analysis
+nav_order: 1
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
+
+{% theorem * Banach Fixed-Point Theorem %}
+test
+{% endtheorem %}
diff --git a/pages/more-functional-analysis/index.md b/pages/more-functional-analysis/index.md
new file mode 100644
index 0000000..1af344d
--- /dev/null
+++ b/pages/more-functional-analysis/index.md
@@ -0,0 +1,8 @@
+---
+title: More Functional Analysis
+nav_order: 4
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
diff --git a/pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md b/pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md
new file mode 100644
index 0000000..be424c3
--- /dev/null
+++ b/pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md
@@ -0,0 +1,28 @@
+---
+title: Alaoglu–Bourbaki Theorem
+parent: Locally Convex Spaces
+grand_parent: More Functional Analysis
+nav_order: 1
+---
+
+# {{ page.title }}
+
+Let $X$ be locally convex space and
+let $U \subset X$ be a neighborhood of zero.
+Let $X'$ denote the continuous dual of $X$.
+Recall that there is a canonical pairing
+
+$$
+X \times X' \to \CC, \quad (x,f) \mapsto \angles{x,f} = f(x).
+$$
+
+The weak topology on $X'$ with respect to this pairing
+is called weak\* topology.
+It is the weakest topology on $X'$ such that
+all evaluation maps $\angles{x,\cdot} : X \to \CC$ are continuous.
+The polar of $U$ is the subset $U^{\circ} \subset X'$.
+The theorem asserts that $U^{\circ}$ is compact in the weak\* topology.
+
+{% theorem * Alaoglu–Bourbaki Theorem %}
+The polar of a neighborhood of zero in a locally convex space is weak\* compact.
+{% endtheorem %}
diff --git a/pages/more-functional-analysis/locally-convex-spaces/index.md b/pages/more-functional-analysis/locally-convex-spaces/index.md
new file mode 100644
index 0000000..990ec99
--- /dev/null
+++ b/pages/more-functional-analysis/locally-convex-spaces/index.md
@@ -0,0 +1,9 @@
+---
+title: Locally Convex Spaces
+parent: More Functional Analysis
+nav_order: 2
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
diff --git a/pages/more-functional-analysis/locally-convex-spaces/krein-milman-theorem.md b/pages/more-functional-analysis/locally-convex-spaces/krein-milman-theorem.md
new file mode 100644
index 0000000..58b9182
--- /dev/null
+++ b/pages/more-functional-analysis/locally-convex-spaces/krein-milman-theorem.md
@@ -0,0 +1,19 @@
+---
+title: Krein–Milman Theorem
+parent: Locally Convex Spaces
+grand_parent: More Functional Analysis
+nav_order: 2
+---
+
+# {{ page.title }}
+
+## Extreme Points
+
+{% definition Extreme Point %}
+Suppose $C$ is a convex subset of a vector space $X$.
+We say that an element $x \in C$ is an *extreme point* of $C$
+if
+{% enddefinition %}
+
+{% proof %}
+{% endproof %}
diff --git a/pages/more-functional-analysis/topological-vector-spaces/index.md b/pages/more-functional-analysis/topological-vector-spaces/index.md
new file mode 100644
index 0000000..745d53b
--- /dev/null
+++ b/pages/more-functional-analysis/topological-vector-spaces/index.md
@@ -0,0 +1,56 @@
+---
+title: Topological Vector Spaces
+parent: More Functional Analysis
+nav_order: 1
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
+
+Let $X$ be a set.
+A *property* of subsets of $X$ is a set $P \subset \mathcal{P}(X)$.
+We say that a subset $A \subset X$ has the property $P$, if $A \in P$.
+A property $P$ of subsets of $X$ is said to be *stable under arbitrary intersections*,
+if for every family $F$ of subsets of $X$ with property $P$,
+the intersection $\bigcap F$ has the property $P$.
+In other words, $P$ is stable under arbitrary intersections iff
+$\bigcap F \in P$ for every subset $F \subset P$.
+In this definition the family $F$ is allowed to be empty,
+hence $\bigcap \emptyset = X$ needs to have the property $P$.
+
+For example, in a topological space $X$ the property of being a closed subset of $X$
+is stable under arbitrary intersections.
+
+If $P$ is stable under arbitrary intersections,
+and $A$ is a subset of $X$, which may or may not have the property $P$,
+then we define the *$P$-hull* of $A$ to be
+the intersection of all supersets $B \supset A$
+having have the property $P$.
+By definition, the $P$-hull of $A$ has the property $P$.
+Moreover, it is the smallest superset of $A$ with property $P$
+in the following sense: If $B$ is any superset of $A$ with property $P$,
+then $B$ contains the $P$-hull of $A$.
+
+For example, the "closed"-hull of a subset $A$ of a topological space
+is the closure of $A$.
+
+There are the dual notions of being *stable under arbitrary unions*
+and *$P$-core* with obvious definitions.
+
+{% definition Convex, Balanced, Absolutely Convex %}
+Let $X$ be a vector space over the field $\KK$.
+A subset $A \subset X$ is said to be
+- *convex* if
+- *balanced* if
+- *absolutely convex* if
+{% enddefinition %}
+
+{% theorem %}
+These properties of subsets of $X$
+are stable under arbitrary intersections.
+Thus we obtain the notions of
+*convex hull* $\co A$,
+*balanced hull* $\bal A$, and
+*absolutely convex hull* $\aco A$.
+{% endtheorem %}
diff --git a/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md b/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md
new file mode 100644
index 0000000..277ecd3
--- /dev/null
+++ b/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md
@@ -0,0 +1,109 @@
+---
+title: Polar Topologies
+parent: Topological Vector Spaces
+grand_parent: More Functional Analysis
+nav_order: 1
+---
+
+# {{ page.title }}
+
+# Dual pairs of vector spaces
+
+Recall that a *bilinear form* on two vector spaces $V$ and $W$ over a field $\KK$
+is a mapping $b : V \times W \to \KK$ which is linear in each of its arguments,
+that is, which satisfies
+
+$$
+\begin{align*}
+b(v_1+v_2,w) &= b(v_1,w) + b(v_2,w) &
+b(v,w_1+w_2) &= b(v,w_1) + b(v,w_2) \\
+b(\lambda v, w) &= \lambda \, b(v,w) &
+b(v, \lambda w) &= \lambda \, b(v,w)
+\end{align*}
+$$
+
+for all vectors $v,v_1,v_2 \in V$, $w,w_1,w_2 \in W$ and all scalars $\lambda \in \KK$.
+
+We say that the bilinear form $b : V \times W \to \KK$ is *nondegenerate*, if it has the properties
+
+$$
+\begin{gather*}
+\forall v \in V : \qquad ( \forall w \in W : \angles{v,w} = 0 ) \implies v = 0 \\
+\forall w \in W : \qquad ( \forall v \in V : \angles{v,w} = 0 ) \implies w = 0 
+\end{gather*}
+$$
+
+If $V$ is a vector space over $\KK$,
+let us denote its *algebraic dual* by $V^*$.
+Given a bilinear form $V \times W \to \KK$, consider the mappings
+
+$$
+c : V \to W*, c(v)(w) = b(v,w)
+\tilde{c} : W \to V*, \tilde{c}(w)(v) = b(v,w)
+$$
+
+Then $b$ is nondegenerate if and only if
+both $c$ and $\tilde{c}$ are injective.
+
+
+{% definition Dual Pair %}
+A *dual pair* (or *dual system* or *duality*) $\angles{V,W}$ over a field $\KK$ is constituted by
+two vector spaces $V$ and $W$ over $\KK$
+and a nondegenerate bilinear form $\angles{\cdot,\cdot} : V \times W \to \KK$.
+{% enddefinition %}
+
+(We resist saying that a dual pair is a triple ...)
+
+{% definition Weak Topology %}
+Suppose $\angles{X,Y}$ is a dual pair of vector spaces over a field $\KK$.
+We define the *weak topology on $X$*, denoted by $\sigma(X,Y)$, as
+the [initial topology](/pages/general-topology/universal-constructions.html#initial-topology) induced by the maps
+$\angles{\cdot,y} : X \to \KK$, where $y \in Y$.
+Similarly, the *weak topology on $Y$*, denoted by $\sigma(Y,X)$, is
+the initial topology induced by the maps
+$\angles{x,\cdot} : Y \to \KK$, where $x \in X$.
+{% enddefinition %}
+
+{% theorem Weak Topologies are Locally Convex %}
+Suppose $\angles{X,Y}$ is a dual pair of vector spaces over a field $\KK$.
+TODO
+{% endtheorem %}
+
+## The Canonical Pairing
+
+TODO: Def & Theorem (weak rep)
+
+{% definition Polar Set %}
+Suppose $\angles{X,Y}$ is a dual pair of vector spaces.
+The *polar* of a subset $A \subset X$ is the set
+
+$$
+A^{\circ} = \braces{y \in Y : \abs{\angles{x,y}} \le 1 \ \forall x \in A}.
+$$
+
+The *polar* of a subset $B \subset Y$ is the set
+
+$$
+B^{\circ} = \braces{x \in X : \abs{\angles{x,y}} \le 1 \ \forall y \in B}.
+$$
+{% enddefinition %}
+
+Some authors define the polar with the condition $\Re \angles{x,y} \le 1$
+instead of $\abs{\angles{x,y}} \le 1$ and call *absolute polar* what we call polar.
+Some authors write $B_{\circ}$ for $B^{\circ}$.
+
+Note that the *bipolar* $A^{\circ\circ} = (A^{\circ})^{\circ}$ is a subset of $X$.
+
+{% theorem * Bipolar Theorem %}
+Suppose $\angles{X,Y}$ is a dual pair of vector spaces
+and $A \subset X$. Then
+
+$$
+A^{\circ\circ} = \overline{\aco(A)},
+$$
+
+where the closure is taken with respect to the weak topology on $X$, that is $\sigma(X,Y)$.
+{% endtheorem %}
+
+{% proof %}
+{% endproof %}
diff --git a/pages/operator-algebras/banach-algebras/index.md b/pages/operator-algebras/banach-algebras/index.md
index 2fb8f03..3335d78 100644
--- a/pages/operator-algebras/banach-algebras/index.md
+++ b/pages/operator-algebras/banach-algebras/index.md
@@ -47,25 +47,23 @@ We say that $\mathcal{A}$ is an *unital* Banach algebra, if $\mathcal{A}$ contai
 
 It is easy to see that a Banach algebra has at most one unit.
 
-{: .proposition-title #neumann-series }
-> Proposition (Neumann Series)
->
-> Let $\mathcal{A}$ be a unital Banach algebra
-> and let $x \in \mathcal{A}$ satisfy $\norm{x} < 1$.
-> Then $\mathbf{1}-x$ is invertible
-> and the inverse is given by the series
-> 
-> $$
-> (\mathbf{1}-x)^{-1} = \sum_{n=0}^{\infty} x^n,
-> $$
->
-> which converges absolutely in norm.
-> Moreover, we have the estimate
-> 
-> $$
-> \norm{(\mathbf{1}-x)^{-1}} \le \frac{1}{1 - \norm{x}}.
-> $$
-> {: .katex-display .mb-0 }
+{% proposition Neumann Series %}
+Let $\mathcal{A}$ be a unital Banach algebra
+and let $x \in \mathcal{A}$ satisfy $\norm{x} < 1$.
+Then $\mathbf{1}-x$ is invertible,
+and the inverse is given by the series
+
+$$
+(\mathbf{1}-x)^{-1} = \sum_{n=0}^{\infty} x^n,
+$$
+
+which converges absolutely in norm.
+Moreover, we have the estimate
+
+$$
+\norm{(\mathbf{1}-x)^{-1}} \le \frac{1}{1 - \norm{x}}.
+$$
+{% endproposition %}
 
 {% proof %}
 Since the Banach algebra norm is submultiplicative,
@@ -91,18 +89,17 @@ The estimate follows from $\norm{s} \le \sum \norm{x}^n = 1 / (1 - \norm{x})$.
 
 ## The Spectrum
 
-{: .definition-title }
-> Definition (Spectrum, Resolvent Set)
->
-> Suppose $x$ is an element of a unital Banach algebra $\mathcal{A}$.
-> {: .mb-0 }
-> 
-> {: .my-0 }
-> - The *spectrum* of $x$ is the set $\sigma(x) = \lbrace\lambda \in \CC : x - \lambda$ is not invertible in $\mathcal{A}\rbrace$. \
->   The elements of $\sigma(x)$ are called *spectral values* of $x$.
-> - The *resolvent set* of $x$ is the set $\rho (x) = \CC \setminus \sigma(x)$. \
->   For $\lambda \in \rho(x)$ the *resolvent* of $x$ is the algebra element $R_{\lambda} = (\lambda - x)^{-1}$. \
->   The mapping $R : \rho(x) \to \mathcal{A}$, $\lambda \mapsto R_{\lambda}$, is called *resolvent map*.
+{% definition Spectrum, Resolvent Set %}
+Suppose $x$ is an element of a unital Banach algebra $\mathcal{A}$.
+{: .mb-0 }
+
+{: .my-0 }
+- The *spectrum* of $x$ is the set $\sigma(x) = \lbrace\lambda \in \CC : x - \lambda$ is not invertible in $\mathcal{A}\rbrace$. \
+  The elements of $\sigma(x)$ are called *spectral values* of $x$.
+- The *resolvent set* of $x$ is the set $\rho (x) = \CC \setminus \sigma(x)$. \
+  For $\lambda \in \rho(x)$ the *resolvent* of $x$ is the algebra element $R_{\lambda} = (\lambda - x)^{-1}$. \
+  The mapping $R : \rho(x) \to \mathcal{A}$, $\lambda \mapsto R_{\lambda}$, is called *resolvent map*.
+{% enddefinition %}
 
 {% theorem %}
 Suppose $x$ is an element of a unital Banach algebra $\mathcal{A}$.
@@ -122,7 +119,7 @@ $$
 
 {% proof %}
 Let $\lambda$ be in the resolvent set of $x$.
-Then $\lambda - x$ is invertible and we have for all $\mu \in \CC$ 
+Then $\lambda - x$ is invertible, and we have for all $\mu \in \CC$ 
 
 $$
 \mu - x = \bigl(\mathbf{1} - (\lambda - \mu) (\lambda - x)^{-1}\bigr) (\lambda - x).
@@ -165,7 +162,7 @@ We assume that $\sigma(x)$ is empty
 and derive a contradiction.
 Observe that the resolvent map $R$ is defined on the whole complex plane.
 By [this corollary](#resolvent-map-is-analytic), $R$ is analytic, hence entire.
-Analytic functions are countinuous;
+Analytic functions are continuous;
 therefore $R$ is bounded on the compact disk $\abs{\lambda} \le 2 \norm{x}$.
 For $\abs{\lambda} > 2 \norm{x}$ we may expand $R_{\lambda}$ into a [Neumann series](#neumann-series),
 
@@ -188,13 +185,12 @@ $$
 This shows that $R$ is a bounded entire function. Now
 [Liouville's Theorem](/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.html#liouvilles-theorem)
 (for vector-valued functions) implies that $R$ is constant.
-This is contradictiory because XXX
+This is contradictory because XXX
 {% endproof %}
 
-{: .theorem-title }
-> Gelfand–Mazur Theorem
->
-> Every Banach algebra in which all nonzero elements are invertible is isometrically isomorphic to $\CC$.
+{% theorem * Gelfand–Mazur Theorem %}
+Every Banach algebra in which all nonzero elements are invertible is isometrically isomorphic to $\CC$.
+{% endtheorem %}
 
 {% proof %}
 For any Banach algebra $A$,
@@ -220,40 +216,39 @@ include:
 - $\mathcal{A}$ is a division algebra.
 - The underlying ring of $\mathcal{A}$ is a field.
 
-{: .theorem-title }
-> Spectral Radius Formula
->
-> For every Banach algebra element $x$ the spectral radius is given by
-> 
-> $$
-> r(x) = \lim_{n \to \infty} \norm{x^n}^{1/n}.
-> $$
-> {: .katex-display .mb-0 }
+{% theorem * Spectral Radius Formula %}
+For every Banach algebra element $x$ the spectral radius is given by
+
+$$
+r(x) = \lim_{n \to \infty} \norm{x^n}^{1/n}.
+$$
+{% endtheorem %}
 
 ## Gelfand’s Theory
 
-Proposition
+{% proposition %}
 Let $\mathcal{A}$ be a unital commutative Banach algebra.
 If $\phi$ is a nonzero multiplicative linear functional on $\mathcal{A}$,
 then its kernel $\ker \phi$ is a maximal ideal in $\mathcal{A}$.
 Every maximal ideal $\mathcal{I}$ in $\mathcal{A}$ is of the form
 $I = \ker \phi$ for some nonzero multiplicative linear functional $\phi$ on $\mathcal{A}$.
+{% endproposition %}
 
-In other words, the mapping $\phi \mapsto \ker \phi$ is gives a bijection
+In other words, the mapping $\phi \mapsto \ker \phi$ gives a bijection
 between the sets of nonzero multiplicative linear functionals and maximal ideals.
 
+{% definition %}
+The *Gelfand space* $\Gamma_{\mathcal{A}}$ of a unital commutative Banach algebra $\mathcal{A}$
+is the set of maximal ideals of $\mathcal{A}$; its topology is inherited from
+the weak* topology on the dual of $\mathcal{A}$ via the correspondence described above.
+{% enddefinition %}
 
-Definition
+{% definition %}
 The *maximal ideal space* $\mathcal{M}_{\mathcal{A}}$ of a unital commutative Banach algebra $\mathcal{A}$
 is the set of maximal ideals of $\mathcal{A}$; its topology is inherited from
-the weak* topology on the dual of $\mathcal{A}$ via the correspondece described above.
-
-Proposition
-The *maximal ideal space* of a unital commutative Banach algebra is a compact Hausdorff space.
-
-{% definition bla, blubb %}
-a
-b
+the weak* topology on the dual of $\mathcal{A}$ via the correspondence described above.
 {% enddefinition %}
 
-
+{% proposition %}
+The *Gelfand space* of a unital commutative Banach algebra is a compact Hausdorff space.
+{% endproposition %}
diff --git a/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md b/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md
index 05b1d4f..ea15f87 100644
--- a/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md
+++ b/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md
@@ -3,37 +3,33 @@ title: Positive Linear Functionals
 parent: C*-Algebras
 grand_parent: Operator Algebras
 nav_order: 1
-# cspell:words
 ---
 
 # {{ page.title }}
 
 all algebra are assumed to be unital
 
-{: .definition-title }
-> Hermitian Functional, Positive Functional, State
->
-> A linear functional on a $C^*$-algebra $\mathcal{A}$ is said to be
-> 
-> - *Hermitian* if $\phi(x*) = \overline{\phi(x)}$ for all $x \in \mathcal{A}$.
-> - *positive* if $\phi(x) \ge 0$ for all $x \ge 0$.
-> - a *state* if $\phi$ is positive and $\phi(\mathbf{1}) = 1$.
->
-
-{: .definition-title }
-> State
->
-> A norm-one positive linear functional on a $C^*$-algebra is called a *state*.
-
-{: .definition-title }
-> State Space
->
-> The *state space* of a $C^*$-algebra $\mathcal{A}$, denoted by $S(\mathcal{A})$, is the set of all states of $\mathcal{A}$.
+{% definition Hermitian Functional, Positive Functional, State %}
+A linear functional on a $C^*$-algebra $\mathcal{A}$ is said to be
+
+- *Hermitian* if $\phi(x*) = \overline{\phi(x)}$ for all $x \in \mathcal{A}$.
+- *positive* if $\phi(x) \ge 0$ for all $x \ge 0$.
+- a *state* if $\phi$ is positive and $\phi(\mathbf{1}) = 1$.
+{% enddefinition %}
+
+{% definition State %}
+A norm-one positive linear functional on a $C^*$-algebra is called a *state*.
+{% enddefinition %}
+
+{% definition State Space %}
+The *state space* of a $C^*$-algebra $\mathcal{A}$, denoted by $S(\mathcal{A})$, is the set of all states of $\mathcal{A}$.
+{% enddefinition %}
 
 Note that $S(\mathcal{A})$ is a subset of the unit ball in the dual space of $\mathcal{A}$.
 
-{: .proposition }
-> The state space of a $C^*$-algebra is convex and weak* compact.
+{% proposition %}
+The state space of a $C^*$-algebra is convex and weak* compact.
+{% endproposition %}
 
 {% proof %}
 {% endproof %}
diff --git a/pages/operator-algebras/c-star-algebras/states.md b/pages/operator-algebras/c-star-algebras/states.md
index 619bc9a..29cf5f5 100644
--- a/pages/operator-algebras/c-star-algebras/states.md
+++ b/pages/operator-algebras/c-star-algebras/states.md
@@ -3,27 +3,28 @@ title: States
 parent: C*-Algebras
 grand_parent: Operator Algebras
 nav_order: 1
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .definition-title }
-> Definition (State, State Space)
->
-> A norm-one [positive linear functional]( {% link pages/operator-algebras/c-star-algebras/positive-linear-functionals.md %} ) on a C\*-algebra is called a *state*.\
-> The *state space* $S(\mathcal{A})$ of a C\*-algebra $\mathcal{A}$ is the set of all its states.
+{% definition State, State Space %}
+A norm-one [positive linear functional]( {% link pages/operator-algebras/c-star-algebras/positive-linear-functionals.md %} ) on a C\*-algebra is called a *state*.\
+The *state space* $S(\mathcal{A})$ of a C\*-algebra $\mathcal{A}$ is the set of all its states.
+{% enddefinition %}
 
-Note that $S(\mathcal{A})$ is a subset of the unit ball in the dual space of $\mathcal{A}$.
+Note that $S(\mathcal{A})$ is a subset of the closed unit ball in the dual space of $\mathcal{A}$.
 
-{: .corollary }
-> A linear functional $\omega$ on a C\*-algebra is a state
-> if and only if $\omega(\mathbf{1}) = 1 = \norm{\omega}$.
+{% corollary %}
+A linear functional $\omega$ on a C\*-algebra is a state
+if and only if $\omega(\mathbf{1}) = 1 = \norm{\omega}$.
+{% endcorollary %}
 
-{: .proposition }
-> The state space of a C\*-algebra is convex and weak\* compact.
+{% proposition %}
+The state space of a C\*-algebra is convex and weak\* compact.
+{% endproposition %}
 
 {% proof %}
+Let $\mathcal{A}$ be a C\*-algebra and let $S(\mathcal{A})$ be its state space.
 First, we show convexity.
 Let $\omega_0, \omega_1$ be states on $\mathcal{A}$ and let $t \in (0,1)$.
 Consider the convex combination $\omega = (1-t)\omega_0 + t\omega_1$.
@@ -41,3 +42,8 @@ This means that $\omega_i(x) \to \omega(x)$ for every $x \in \mathcal{A}$.
 For all $i$ we have $\omega_i(x) \ge 0$ for $x \ge 0$ and $\omega_i(\mathbf{1}) = 1$; hence $\omega(x) \ge 0$ for $x \ge 0$ and $\omega(\mathbf{1}) = 1$. Thus $\omega$ is again a state.
 This shows that the state space is weak* closed, completing the proof.
 {% endproof %}
+
+TODO: state space is nonempty
+
+TODO: pure states
+
diff --git a/pages/operator-algebras/operator-topologies.md b/pages/operator-algebras/operator-topologies.md
new file mode 100644
index 0000000..2d7722e
--- /dev/null
+++ b/pages/operator-algebras/operator-topologies.md
@@ -0,0 +1,14 @@
+---
+title: Operator Topologies
+parent: Operator Algebras
+nav_order: 1
+---
+
+# {{ page.title }}
+
+{% definition Weak & Strong Operator Topology %}
+TODO
+{% enddefinition %}
+
+{% proof %}
+{% endproof %}
diff --git a/pages/quantum-field-theory/wightman-axioms/index.md b/pages/quantum-field-theory/wightman-axioms/index.md
index ecc204e..36d0e17 100644
--- a/pages/quantum-field-theory/wightman-axioms/index.md
+++ b/pages/quantum-field-theory/wightman-axioms/index.md
@@ -3,7 +3,6 @@ title: Wightman Axioms
 parent: Quantum Field Theory
 nav_order: 1
 has_children: true
-# cspell:words
 ---
 
 # {{ page.title }}
diff --git a/pages/quantum-field-theory/wightman-axioms/scalar-field.md b/pages/quantum-field-theory/wightman-axioms/scalar-field.md
index d37bcd0..4e7658f 100644
--- a/pages/quantum-field-theory/wightman-axioms/scalar-field.md
+++ b/pages/quantum-field-theory/wightman-axioms/scalar-field.md
@@ -3,7 +3,6 @@ title: Scalar Field
 parent: Wightman Axioms
 grand_parent: Quantum Field Theory
 nav_order: 1
-# cspell:words
 ---
 
 # Wightman Axioms
@@ -12,8 +11,6 @@ Also known as *Gårding–Wightman axioms*.
 
 ## Wightman Axioms for a Hermitian Scalar Field
 
-{: .axiom-title }
-> Axiom 1
->
-> j
-
+{% axiom %}
+TODO
+{% endaxiom %}
diff --git a/pages/spectral-theory/of-unbounded-operators/index.md b/pages/spectral-theory/of-unbounded-operators/index.md
index 0b2a6f9..af10626 100644
--- a/pages/spectral-theory/of-unbounded-operators/index.md
+++ b/pages/spectral-theory/of-unbounded-operators/index.md
@@ -3,7 +3,6 @@ title: of Unbounded Operators
 parent: Spectral Theory
 nav_order: 1
 has_children: true
-# cspell:words
 ---
 
 # {{ page.title }}
diff --git a/pages/spectral-theory/test/basic.md b/pages/spectral-theory/test/basic.md
index 8c42f6d..05405f4 100644
--- a/pages/spectral-theory/test/basic.md
+++ b/pages/spectral-theory/test/basic.md
@@ -3,9 +3,6 @@ title: Test
 parent: Test
 grand_parent: Spectral Theory
 nav_order: 2
-description: >
-  The
-# spellchecker:words Steinhaus preimages Baire pointwise
 ---
 
 # {{ page.title }}
diff --git a/pages/tomita-takesaki-theory/index.md b/pages/tomita-takesaki-theory/index.md
index dd2a312..28b73e6 100644
--- a/pages/tomita-takesaki-theory/index.md
+++ b/pages/tomita-takesaki-theory/index.md
@@ -3,7 +3,6 @@ title: Tomita Takesaki Theory
 nav_order: 10
 has_children: true
 published: false
-# cspell:words
 ---
 
 # {{ page.title }}
diff --git a/pages/tomita-takesaki-theory/standard-subspaces.md b/pages/tomita-takesaki-theory/standard-subspaces.md
index 970c51a..85135aa 100644
--- a/pages/tomita-takesaki-theory/standard-subspaces.md
+++ b/pages/tomita-takesaki-theory/standard-subspaces.md
@@ -2,18 +2,13 @@
 title: Standard Subspaces
 parent: Tomita Takesaki Theory
 nav_order: 1
-# cspell:words
 ---
 
 # {{ page.title }}
 
-{: .definition-title }
-> Definition (Cyclic, Separating, Standard Subspace)
->
-> A closed real linear subspace $H$ of a complex Hilbert space $\hilb{H}$ is called
-> * *cyclic*, if $H+iH$ is dense in $\hilb{H}$,
-> * *separating*, if $H \cap iH = \braces{0}$, and
-> * *standard*, if $H$ is cyclic and separating.
-
-**Proof:**
-{{ site.qed }}
+{% definition Cyclic, Separating, Standard Subspace %}
+A closed real linear subspace $H$ of a complex Hilbert space $\hilb{H}$ is called
+* *cyclic*, if $H+iH$ is dense in $\hilb{H}$,
+* *separating*, if $H \cap iH = \braces{0}$, and
+* *standard*, if $H$ is cyclic and separating.
+{% enddefinition %}
diff --git a/pages/unbounded-operators/adjoint-operators.md b/pages/unbounded-operators/adjoint-operators.md
index a93e6d4..1b9fb0c 100644
--- a/pages/unbounded-operators/adjoint-operators.md
+++ b/pages/unbounded-operators/adjoint-operators.md
@@ -7,8 +7,6 @@ description: >
   The Hellinger–Toeplitz Theorem states that an everywhere-defined symmetric
   operator on a Hilbert space is bounded. We give a proof using the Uniform
   Boundedness Theorem. We give another proof using the Closed Graph Theorem.
-# spellchecker:dictionaries latex
-# spellchecker:words Hellinger Toeplitz innerp hilb Schwarz functionals enspace Riesz
 ---
 
 # {{ page.title }}
diff --git a/pages/unbounded-operators/graph-and-closedness.md b/pages/unbounded-operators/graph-and-closedness.md
index a9bf738..04a6789 100644
--- a/pages/unbounded-operators/graph-and-closedness.md
+++ b/pages/unbounded-operators/graph-and-closedness.md
@@ -6,17 +6,12 @@ description: >
   The Hellinger–Toeplitz Theorem states that an everywhere-defined symmetric
   operator on a Hilbert space is bounded. We give a proof using the Uniform
   Boundedness Theorem. We give another proof using the Closed Graph Theorem.
-# spellchecker:dictionaries latex
-# spellchecker:words Hellinger Toeplitz innerp hilb Schwarz functionals enspace Riesz
 ---
 
 # {{ page.title }}
 
-
-{: .definition-title }
-
-> Definition (Graph of an Operator)
->
-> The *graph* of an operator $T$ in a Hilbert space $\hilb{H}$
-> is the set of all pairs $(x,y) \in \hilb{H}\times\hilb{H}$
-> where $x$ lies in the domain of $T$ and $y=Tx$.
+{% definition Graph of an Operator %}
+The *graph* of an operator $T$ in a Hilbert space $\hilb{H}$
+is the set of all pairs $(x,y) \in \hilb{H}\times\hilb{H}$
+where $x$ lies in the domain of $T$ and $y=Tx$.
+{% enddefinition %}
diff --git a/pages/unbounded-operators/hellinger-toeplitz-theorem.md b/pages/unbounded-operators/hellinger-toeplitz-theorem.md
index 07d6b81..fea54be 100644
--- a/pages/unbounded-operators/hellinger-toeplitz-theorem.md
+++ b/pages/unbounded-operators/hellinger-toeplitz-theorem.md
@@ -6,7 +6,6 @@ description: >
   The Hellinger–Toeplitz Theorem states that an everywhere-defined symmetric
   operator on a Hilbert space is bounded. We give a proof using the Uniform
   Boundedness Theorem. We give another proof using the Closed Graph Theorem.
-# cspell:words Hellinger Toeplitz Schwarz Riesz functionals
 ---
 
 # {{ page.title }}
@@ -25,10 +24,9 @@ $$
 \innerp{Tx}{y} = \innerp{x}{Ty} \quad \forall x,y \in D(T).
 $$
 
-{: .theorem-title }
-> Hellinger–Toeplitz theorem
->
-> An everywhere-defined symmetric operator on a Hilbert space is bounded.
+{% theorem * Hellinger–Toeplitz Theorem %}
+An everywhere-defined symmetric operator on a Hilbert space is bounded.
+{% endtheorem %}
 
 Consequently, a symmetric Hilbert space operator
 that is (truly) unbounded
@@ -75,7 +73,6 @@ Divide by $\norm{Tx_n}$ (if nonzero)
 to obtain $\norm{Tx_n} \le \norm{f_n}$ for all but finitely many $n$.
 Thus $(\norm{Tx_n})$ is a bounded sequence,
 contradicting $\norm{Tx_n} \to \infty$.
-{{ site.qed }}
 
 ---
 
@@ -113,4 +110,3 @@ $$
 A sequence of complex numbers has at most one limit,
 hence $\innerp{Tx}{y} = \innerp{z}{y}$ for all $y$.
 By the Riesz representation theorem, $Tx=z$.
-{{ site.qed }}
diff --git a/pages/unbounded-operators/quadratic-forms.md b/pages/unbounded-operators/quadratic-forms.md
index 5831b88..cb1c44a 100644
--- a/pages/unbounded-operators/quadratic-forms.md
+++ b/pages/unbounded-operators/quadratic-forms.md
@@ -7,17 +7,12 @@ description: >
   The Hellinger–Toeplitz Theorem states that an everywhere-defined symmetric
   operator on a Hilbert space is bounded. We give a proof using the Uniform
   Boundedness Theorem. We give another proof using the Closed Graph Theorem.
-# spellchecker:dictionaries latex
-# spellchecker:words Hellinger Toeplitz innerp hilb Schwarz functionals enspace Riesz
 ---
 
 # {{ page.title }}
 
-
-{: .definition-title }
-
-> Definition (Graph of an Operator)
->
-> The *graph* of an operator $T$ in a Hilbert space $\hilb{H}$
-> is the set of all pairs $(x,y) \in \hilb{H}\times\hilb{H}$
-> where $x$ lies in the domain of $T$ and $y=Tx$.
+{% definition Graph of an Operator %}
+The *graph* of an operator $T$ in a Hilbert space $\hilb{H}$
+is the set of all pairs $(x,y) \in \hilb{H}\times\hilb{H}$
+where $x$ lies in the domain of $T$ and $y=Tx$.
+{% enddefinition %}