Update

29 files changed, 133 insertions, 88 deletions

diff --git a/.cspell.yaml b/.cspell.yaml
index eff8b31..b4f5223 100644
--- a/.cspell.yaml
+++ b/.cspell.yaml
@@ -1,18 +1,19 @@
-version: "0.2"
-language: "en_US"
+version: 0.2
+language: en_US
 minWordLength: 3
 dictionaryDefinitions:
   - name: names
     addWords: true
-    path: "./.cspell/my-cspell-dicts/names.txt"
+    path: ./.cspell/my-cspell-dicts/names.txt
   - name: math
     addWords: true
-    path: "./.cspell/my-cspell-dicts/math.txt"
+    path: ./.cspell/my-cspell-dicts/math.txt
   - name: liquid
     addWords: true
-    path: "./.cspell/liquid.txt"
+    path: ./.cspell/liquid.txt
 ignorePaths:
-  - "./.cspell/"
+  - ./.cspell/
+  - ./assets/
 dictionaries:
   - latex
   - names
@@ -38,5 +39,4 @@ words:
   - mathscr
   - enspace
   - callouts
-  - Closedness
   - vcs
diff --git a/.cspell/liquid.txt b/.cspell/liquid.txt
index 155a881..4f435d6 100644
--- a/.cspell/liquid.txt
+++ b/.cspell/liquid.txt
@@ -1,4 +1,5 @@
 endaxiom
+endcomment
 endcorollary
 enddefinition
 endexample
diff --git a/Gemfile b/Gemfile
index b6cef77..2f559a1 100644
--- a/Gemfile
+++ b/Gemfile
@@ -5,6 +5,7 @@ source 'https://rubygems.org'
 gem 'jekyll'
 gem 'jekyll-include-cache'
 gem 'jekyll-scholar'
+gem 'jekyll-sitemap'
 gem 'just-the-docs'
 gem 'kramdown-math-katex'
 gem 'mini_racer'
diff --git a/_bibliography/functional-analysis-basics.bib b/_bibliography/functional-analysis-basics.bib
index 45bcd37..3bfae4b 100644
--- a/_bibliography/functional-analysis-basics.bib
+++ b/_bibliography/functional-analysis-basics.bib
@@ -1,6 +1,6 @@
 @book{kreyszig,
-   title =     {Introductory Functional Analysis With Applications},
-   author =    {Erwin Kreyszig},
-   publisher = {John Wiley & Sons},
-   year =      {1978},
+  title      = {Introductory Functional Analysis With Applications},
+  author     = {Erwin Kreyszig},
+  publisher  = {John Wiley & Sons},
+  year       = {1978},
 }
diff --git a/_bibliography/general-topology.bib b/_bibliography/general-topology.bib
index 7118fd3..df0ba21 100644
--- a/_bibliography/general-topology.bib
+++ b/_bibliography/general-topology.bib
@@ -1,8 +1,8 @@
 @book{munkres,
-   title =     {Topology},
-   author =    {James Munkres},
-   publisher = {Pearson Education Limited},
-   isbn =      {978-1-292-02362-5},
-   year =      {2014},
-   edition =   {2},
+  title      = {Topology},
+  author     = {James Munkres},
+  publisher  = {Pearson Education Limited},
+  isbn       = {978-1-292-02362-5},
+  year       = {2014},
+  edition    = {2},
 }
diff --git a/_config.yaml b/_config.yaml
index 172c410..52c48eb 100644
--- a/_config.yaml
+++ b/_config.yaml
@@ -1,35 +1,30 @@
 title: jxir's math pages
-description: Desc
+description: jxir's math pages
 
 source: .
 destination: /srv/http/
 url: https://jxir.de
-# baseurl: /masters-thesis
 permalink: /:basename:output_ext
 
+# incremental: true
+
 theme: just-the-docs
 plugins:
   - jekyll-include-cache
   - jekyll-scholar
+  - jekyll-sitemap
 
 defaults:
   - scope:
-      path: "pages"
+      path: pages
     values:
       layout: default
 
-search:
-  tokenizer_separator: /[\s\-\u2013/]+/
-
-# color_scheme: dark
-# incremental: true
-
 kramdown:
   input: GFMKatex
-  # parse_block_html: true
   math_engine: katex
   math_engine_opts:
-    output: "html"
+    output: html
     macros:
       "\\NN": "\\mathbb{N}"
       "\\ZZ": "\\mathbb{Z}"
@@ -80,5 +75,8 @@ callouts:
     title: Axiom
     color: yellow
 
+search:
+  tokenizer_separator: /[\s\-\u2013/]+/
+
 scholar:
   style: chicago-author-date
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 a2fa53d..e915097 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
@@ -3,6 +3,7 @@ title: The Calculus of Residues
 parent: One Complex Variable
 grand_parent: Complex Analysis
 nav_order: 4
+published: false
 ---
 
 # {{ page.title }}
diff --git a/pages/complex-analysis/several-complex-variables/index.md b/pages/complex-analysis/several-complex-variables/index.md
index 803eea4..c630e80 100644
--- a/pages/complex-analysis/several-complex-variables/index.md
+++ b/pages/complex-analysis/several-complex-variables/index.md
@@ -3,6 +3,7 @@ title: Several Complex Variables
 parent: Complex Analysis
 nav_order: 2
 has_children: true
+published: false
 ---
 
 # {{ page.title }}
diff --git a/pages/distribution-theory/index.md b/pages/distribution-theory/index.md
index 08a3ab2..5d1b7bf 100644
--- a/pages/distribution-theory/index.md
+++ b/pages/distribution-theory/index.md
@@ -3,7 +3,7 @@ title: Distribution Theory
 nav_order: 5
 has_children: true
 has_toc: false
-published: true
+published: false
 ---
 
 # {{ page.title }}
diff --git a/pages/functional-analysis-basics/banach-alaoglu-theorem.md b/pages/functional-analysis-basics/banach-alaoglu-theorem.md
index 0913776..1b6ff81 100644
--- a/pages/functional-analysis-basics/banach-alaoglu-theorem.md
+++ b/pages/functional-analysis-basics/banach-alaoglu-theorem.md
@@ -1,7 +1,7 @@
 ---
 title: Banach–Alaoglu Theorem
 parent: Functional Analysis Basics
-nav_order: 3
+nav_order: 5
 ---
 
 # {{ page.title }}
diff --git a/pages/functional-analysis-basics/compact-operators.md b/pages/functional-analysis-basics/compact-operators.md
index 92e94ba..dc20bad 100644
--- a/pages/functional-analysis-basics/compact-operators.md
+++ b/pages/functional-analysis-basics/compact-operators.md
@@ -1,7 +1,7 @@
 ---
 title: Compact Operators
 parent: Functional Analysis Basics
-nav_order: 4
+nav_order: 6
 published: false
 ---
 
diff --git a/pages/functional-analysis-basics/hilbert-spaces.md b/pages/functional-analysis-basics/hilbert-spaces.md
index b3ef52b..a77e5c7 100644
--- a/pages/functional-analysis-basics/hilbert-spaces.md
+++ b/pages/functional-analysis-basics/hilbert-spaces.md
@@ -1,7 +1,8 @@
 ---
 title: Hilbert Spaces
 parent: Functional Analysis Basics
-nav_order: 1
+nav_order: 7
+published: false
 ---
 
 # {{ page.title }}
diff --git a/pages/functional-analysis-basics/normed-spaces/index.md b/pages/functional-analysis-basics/normed-spaces/index.md
new file mode 100644
index 0000000..c92d8c1
--- /dev/null
+++ b/pages/functional-analysis-basics/normed-spaces/index.md
@@ -0,0 +1,9 @@
+---
+title: Normed Spaces
+parent: Functional Analysis Basics
+nav_order: 1
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
diff --git a/pages/functional-analysis-basics/reflexive-spaces.md b/pages/functional-analysis-basics/reflexive-spaces.md
index 58ca1d3..69fefc4 100644
--- a/pages/functional-analysis-basics/reflexive-spaces.md
+++ b/pages/functional-analysis-basics/reflexive-spaces.md
@@ -1,7 +1,9 @@
 ---
 title: Reflexive Spaces
 parent: Functional Analysis Basics
-nav_order: 2
+nav_order: 4
+description: >
+  A normed space is said to be reflexive if the canonical embedding into its bidual is surjective.
 ---
 
 # {{ page.title }}
@@ -28,10 +30,30 @@ The canonical embedding $C : X \to X''$ of a normed space into its bidual
 is well-defined and an embedding of normed spaces.
 {% endlemma %}
 
+In particular, $C$ is isometric, hence injective.
+
 {% proof %}
-{% endproof %}
+We have to show that, for any given $x \in X$,
+$g_x$ is a bounded linear functional on $X'$.
+Linearity follows from the fact that
+the vector space structure on $X'$ is given by pointwise operations.
+To see that $g_x$ is bounded, observe that
 
-In particular, $C$ is isometric, hence injective.
+$$
+\abs{g_x(f)} = \abs{f(x)} \le \norm{f} \norm{x}
+$$
+
+holds for all $f \in X'$.
+Moreover, this implies that $\norm{g_x} \le \norm{x}$.
+Thanks to
+[Hahn–Banach](/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.html#hahn-banach-theorem-existence-of-functionals),
+we know that there exists a bounded linear functional
+$f \in X'$ with $\norm{f} = 1$ such that $f(x) = \norm{x}$;
+hence, $\norm{g_x} = \norm{x}$.
+This means that the mapping $x \mapsto g_x$ is isometric.
+Clearly, this mapping is also linear, and thus an embedding
+of normed spaces.
+{% endproof %}
 
 {% definition Reflexivity %}
 A normed space is said to be *reflexive*
@@ -40,7 +62,7 @@ is surjective.
 {% enddefinition %}
 
 If a normed space $X$ is reflexive,
-then $X$ is isomorphic with $X''$, its bidual.
+then $X$ is isometrically isomorphic with $X''$, its bidual.
 James gives a counterexample for the converse statement.
 
 {% theorem %}
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 1140e45..c88608b 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
@@ -3,6 +3,10 @@ title: Uniform Boundedness Theorem
 parent: The Fundamental Four
 grand_parent: Functional Analysis Basics
 nav_order: 2
+description: >
+  The Uniform Boundedness Theorem states that a pointwise bounded collection of
+  bounded linear operators from a Banach space into a normed space must be
+  uniformly bounded. We give a proof based on Baire’s Category Theorem.
 ---
 
 # {{ page.title }}
@@ -84,7 +88,9 @@ $$
 
 If $X$ is not complete, this may be false.
 
+{% comment %}
 TODO:
 - strong operator convergence
 - Kreyszig 4.9-5
 - Haase 15.6
+{% endcomment %}
diff --git a/pages/functional-analysis-basics/weak-convergence.md b/pages/functional-analysis-basics/weak-convergence.md
new file mode 100644
index 0000000..6e7f417
--- /dev/null
+++ b/pages/functional-analysis-basics/weak-convergence.md
@@ -0,0 +1,13 @@
+---
+title: Weak Convergence
+parent: Functional Analysis Basics
+nav_order: 3
+---
+
+# {{ page.title }}
+
+{% theorem %}
+{% endtheorem %}
+
+{% proof %}
+{% endproof %}
diff --git a/pages/general-topology/connectedness.md b/pages/general-topology/connectedness.md
index b422775..c57dfdc 100644
--- a/pages/general-topology/connectedness.md
+++ b/pages/general-topology/connectedness.md
@@ -2,6 +2,7 @@
 title: Connectedness
 parent: General Topology
 nav_order: 5
+published: false
 ---
 
 # {{ page.title }}
diff --git a/pages/general-topology/separation/index.md b/pages/general-topology/separation/index.md
index b4916f5..d5fdd5b 100644
--- a/pages/general-topology/separation/index.md
+++ b/pages/general-topology/separation/index.md
@@ -4,6 +4,7 @@ parent: General Topology
 nav_order: 4
 has_children: true
 has_toc: false
+published: false
 ---
 
 # {{ page.title }}
diff --git a/pages/measure-and-integration/bochner-integral/index.md b/pages/measure-and-integration/bochner-integral/index.md
index 5517934..ff9e442 100644
--- a/pages/measure-and-integration/bochner-integral/index.md
+++ b/pages/measure-and-integration/bochner-integral/index.md
@@ -4,6 +4,7 @@ parent: Measure and Integration
 nav_order: 3
 has_children: true
 has_toc: false
+published: false
 ---
 
 # {{ page.title }}
diff --git a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md
index f9ebc4a..1a34820 100644
--- a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md
+++ b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md
@@ -3,6 +3,10 @@ title: Convergence Theorems
 parent: Lebesgue Integral
 grand_parent: Measure and Integration
 nav_order: 2
+description: >
+We state and prove the most important convergence theorems of Lebesgue
+integration theory such as the Monotone Convergence Theorem, Fatou’s Lemma, and the
+Dominated Convergence Theorem.
 ---
 
 # {{ page.title }}
diff --git a/pages/measure-and-integration/lebesgue-integral/fubini-theorem.md b/pages/measure-and-integration/lebesgue-integral/fubini-theorem.md
index 6e5179c..c87607a 100644
--- a/pages/measure-and-integration/lebesgue-integral/fubini-theorem.md
+++ b/pages/measure-and-integration/lebesgue-integral/fubini-theorem.md
@@ -3,6 +3,7 @@ title: Fubini Theorem
 parent: Lebesgue Integral
 grand_parent: Measure and Integration
 nav_order: 2
+published: false
 ---
 
 # {{ page.title }}
diff --git a/pages/measure-and-integration/lebesgue-integral/transformation-formula.md b/pages/measure-and-integration/lebesgue-integral/transformation-formula.md
index 6f02bc8..00441c3 100644
--- a/pages/measure-and-integration/lebesgue-integral/transformation-formula.md
+++ b/pages/measure-and-integration/lebesgue-integral/transformation-formula.md
@@ -3,6 +3,7 @@ title: Transformation Formula
 parent: Lebesgue Integral
 grand_parent: Measure and Integration
 nav_order: 3
+published: false
 ---
 
 # {{ page.title }}
diff --git a/pages/more-functional-analysis/fixed-point-theorems/index.md b/pages/more-functional-analysis/fixed-point-theorems/index.md
index b90bf00..1065c70 100644
--- a/pages/more-functional-analysis/fixed-point-theorems/index.md
+++ b/pages/more-functional-analysis/fixed-point-theorems/index.md
@@ -4,6 +4,7 @@ parent: More Functional Analysis
 nav_order: 1
 has_children: true
 has_toc: false
+published: false
 ---
 
 # {{ page.title }}
diff --git a/pages/operator-algebras/banach-algebras/index.md b/pages/operator-algebras/banach-algebras/index.md
index 9d70df8..3a427a7 100644
--- a/pages/operator-algebras/banach-algebras/index.md
+++ b/pages/operator-algebras/banach-algebras/index.md
@@ -157,9 +157,10 @@ is (strongly) analytic.
 
 ---
 
-{: .proposition #spectrum-is-not-empty }
-> Suppose $x$ is an element of a unital Banach algebra.
-> Then its spectrum $\sigma(x)$ is not empty.
+{% proposition %}
+Suppose $x$ is an element of a unital Banach algebra.
+Then its spectrum $\sigma(x)$ is not empty.
+{% endproposition %}
 
 {% proof %}
 We assume that $\sigma(x)$ is empty
diff --git a/pages/spectral-theory/test/basic.md b/pages/spectral-theory/test/basic.md
index b1015d1..9fa409b 100644
--- a/pages/spectral-theory/test/basic.md
+++ b/pages/spectral-theory/test/basic.md
@@ -7,34 +7,33 @@ nav_order: 2
 
 # {{ page.title }}
 
-{: .definition-title }
-> Definition (resolvent operator, regular value, resolvent set, spectrum, spectral value)
->
-> Let $T : \dom{T} \to X$ be an operator in a complex normed space $X$.
-> We write
->
-> $$
-> T_{\lambda} = T - \lambda = T - \lambda I,
-> $$
->
-> where $\lambda$ is a complex number and
-> $I$ is the identical operator on the domain of $T$.
-> If the operator $T_{\lambda}$ is injective,
-> that is, it has an inverse $T_{\lambda}^{-1}$
-> (with domain $\ran{T_{\lambda}}$),
-> then we call
->
-> $$
-> R_{\lambda}(T) = T_{\lambda}^{-1} = (T - \lambda)^{-1} = (T - \lambda I)^{-1}
-> $$
->
-> the *resolvent operator* of $T$ for $\lambda$.
-> A *regular value* of $T$ is a complex number $\lambda$ for which the resolvent $R_{\lambda}(T)$ exists,
-> has dense domain and is bounded.
-> The set of all regular values of $T$ is called the *resolvent set* of $T$ and denoted $\rho(T)$.
-> The complement of the resolvent set in the complex plane
-> is called the *spectrum* of $T$ and denoted $\sigma(T)$.
-> The elements of the spectrum of $T$ are called the *spectral values* of $T$.
+{% definition %}
+Let $T : \dom{T} \to X$ be an operator in a complex normed space $X$.
+We write
+
+$$
+T_{\lambda} = T - \lambda = T - \lambda I,
+$$
+
+where $\lambda$ is a complex number and
+$I$ is the identical operator on the domain of $T$.
+If the operator $T_{\lambda}$ is injective,
+that is, it has an inverse $T_{\lambda}^{-1}$
+(with domain $\ran{T_{\lambda}}$),
+then we call
+
+$$
+R_{\lambda}(T) = T_{\lambda}^{-1} = (T - \lambda)^{-1} = (T - \lambda I)^{-1}
+$$
+
+the *resolvent operator* of $T$ for $\lambda$.
+A *regular value* of $T$ is a complex number $\lambda$ for which the resolvent $R_{\lambda}(T)$ exists,
+has dense domain and is bounded.
+The set of all regular values of $T$ is called the *resolvent set* of $T$ and denoted $\rho(T)$.
+The complement of the resolvent set in the complex plane
+is called the *spectrum* of $T$ and denoted $\sigma(T)$.
+The elements of the spectrum of $T$ are called the *spectral values* of $T$.
+{% enddefinition %}
 
 {% definition Point Spectrum, Residual Spectrum, Continuous Spectrum %}
 Let $T : \dom{T} \to X$ be an operator in a complex normed space $X$.
diff --git a/pages/unbounded-operators/adjoint-operators.md b/pages/unbounded-operators/adjoint-operators.md
index 96933db..b1c9207 100644
--- a/pages/unbounded-operators/adjoint-operators.md
+++ b/pages/unbounded-operators/adjoint-operators.md
@@ -1,12 +1,8 @@
 ---
 title: Adjoint Operators
 parent: Unbounded Operators
-nav_order: 1
+nav_order: 3
 published: false
-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.
 ---
 
 # {{ page.title }}
diff --git a/pages/unbounded-operators/graph-and-closedness.md b/pages/unbounded-operators/graph-and-closedness.md
index 04a6789..73b00a0 100644
--- a/pages/unbounded-operators/graph-and-closedness.md
+++ b/pages/unbounded-operators/graph-and-closedness.md
@@ -1,11 +1,7 @@
 ---
 title: Graph and Closedness
 parent: Unbounded Operators
-nav_order: 1
-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.
+nav_order: 2
 ---
 
 # {{ page.title }}
diff --git a/pages/unbounded-operators/hellinger-toeplitz-theorem.md b/pages/unbounded-operators/hellinger-toeplitz-theorem.md
index 09046f4..67cdeea 100644
--- a/pages/unbounded-operators/hellinger-toeplitz-theorem.md
+++ b/pages/unbounded-operators/hellinger-toeplitz-theorem.md
@@ -1,7 +1,7 @@
 ---
 title: Hellinger–Toeplitz Theorem
 parent: Unbounded Operators
-nav_order: 10
+nav_order: 1
 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
diff --git a/pages/unbounded-operators/quadratic-forms.md b/pages/unbounded-operators/quadratic-forms.md
index cb1c44a..e2183ff 100644
--- a/pages/unbounded-operators/quadratic-forms.md
+++ b/pages/unbounded-operators/quadratic-forms.md
@@ -3,16 +3,6 @@ title: Quadratic Forms
 parent: Unbounded Operators
 nav_order: 5
 published: false
-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.
 ---
 
 # {{ page.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 $y=Tx$.
-{% enddefinition %}