From 7c66b227a494748e2a546fb85317accd00aebe53 Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Thu, 15 Feb 2024 05:11:07 +0100
Subject: Update

---
 .../cauchys-integral-formula.md                    |  8 +++--
 .../one-complex-variable/cauchys-theorem.md        |  2 +-
 .../one-complex-variable/power-series.md           |  5 +--
 pages/distribution-theory/index.md                 |  2 +-
 .../banach-alaoglu-theorem.md                      |  4 ++-
 .../functional-analysis-basics/reflexive-spaces.md |  4 +--
 .../the-fundamental-four/closed-graph-theorem.md   |  7 ++--
 .../the-fundamental-four/hahn-banach-theorem.md    | 14 ++------
 .../the-fundamental-four/open-mapping-theorem.md   |  4 +--
 .../uniform-boundedness-theorem.md                 |  6 ++--
 pages/general-topology/compactness/index.md        |  4 +--
 .../general-topology/continuity-and-convergence.md |  2 +-
 pages/general-topology/metric-spaces/index.md      | 24 +++++++------
 pages/general-topology/topological-spaces.md       | 12 +++----
 .../lebesgue-integral/convergence-theorems.md      |  6 ++--
 .../lebesgue-integral/index.md                     |  2 +-
 .../lebesgue-integral/the-lp-spaces.md             |  2 +-
 .../measure-theory/measures.md                     |  2 +-
 .../measure-theory/sigma-algebras.md               |  1 -
 .../measure-theory/signed-measures.md              |  2 +-
 .../topological-vector-spaces/index.md             |  2 +-
 .../topological-vector-spaces/polar-topologies.md  |  9 +++--
 pages/operator-algebras/banach-algebras/index.md   | 42 ++++++++++++----------
 .../c-star-algebras/positive-linear-functionals.md |  4 ++-
 pages/operator-algebras/c-star-algebras/states.md  | 17 +++++----
 pages/spectral-theory/test/basic.md                | 32 +++++++----------
 pages/unbounded-operators/adjoint-operators.md     |  1 -
 .../hellinger-toeplitz-theorem.md                  |  2 +-
 28 files changed, 113 insertions(+), 109 deletions(-)

(limited to 'pages')

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 3cf81f7..6ac0803 100644
--- a/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md
+++ b/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md
@@ -66,19 +66,21 @@ $$
 Note that the supremum is finite (and is attained),
 because $f$ is continuous and the circle is compact.
 Clearly, the integral evaluates to $2 \pi r / r^{n+1}$
-and the right hand side of the inequality reduces to the desired expression.
+and the right-hand side of the inequality reduces to the desired expression.
 {% endproof %}
 
 ---
 
-Recall that an *entire* function is a holomorphic function that is defined everywhere in the complex plane.
+Recall that an *entire* function is a holomorphic function
+that is defined everywhere in the complex plane.
 
 {% 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 $M > 0$.
+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](#cauchy-s-estimate)
 for all disks centered at any point $a \in \CC$ and with any radius $r>0$.
diff --git a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
index 2445b8b..6d78e89 100644
--- a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
+++ b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
@@ -14,7 +14,7 @@ If $\gamma_0$, $\gamma_1$ are homotopic closed curves in $G$, then
 
 $$
 \int_{\gamma_0} \! f(z) \, dz =
-\int_{\gamma_1} \! f(z) \, dz 
+\int_{\gamma_1} \! f(z) \, dz
 $$
 
 If $\gamma$ is a null-homotopic closed curve in $G$, then
diff --git a/pages/complex-analysis/one-complex-variable/power-series.md b/pages/complex-analysis/one-complex-variable/power-series.md
index 31793ab..4876d30 100644
--- a/pages/complex-analysis/one-complex-variable/power-series.md
+++ b/pages/complex-analysis/one-complex-variable/power-series.md
@@ -21,7 +21,8 @@ $a$ 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 $z = a + w$.
+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 %}
 
@@ -35,7 +36,7 @@ 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 $\abs{z-a} < R$ 
+  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 $R=\infty$).
 
diff --git a/pages/distribution-theory/index.md b/pages/distribution-theory/index.md
index 3055c8f..08a3ab2 100644
--- a/pages/distribution-theory/index.md
+++ b/pages/distribution-theory/index.md
@@ -19,7 +19,7 @@ an unbounded linear operator $\Phi(f)$ in $\hilb{H}$ such that
 
 {: .my-0 }
 - there is a dense linear subspace $\mathcal{D}$ of $\hilb{H}$ that
-is contained in the domain of all the $\Phi(f)$ 
+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/functional-analysis-basics/banach-alaoglu-theorem.md b/pages/functional-analysis-basics/banach-alaoglu-theorem.md
index 91906cd..0913776 100644
--- a/pages/functional-analysis-basics/banach-alaoglu-theorem.md
+++ b/pages/functional-analysis-basics/banach-alaoglu-theorem.md
@@ -19,4 +19,6 @@ The {{ page.title }} is a special case of the following result:
 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.
+See
+[Alaoglu–Bourbaki Theorem](/pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.html)
+for more information.
diff --git a/pages/functional-analysis-basics/reflexive-spaces.md b/pages/functional-analysis-basics/reflexive-spaces.md
index 781fb1f..58ca1d3 100644
--- a/pages/functional-analysis-basics/reflexive-spaces.md
+++ b/pages/functional-analysis-basics/reflexive-spaces.md
@@ -17,7 +17,7 @@ $$
 where the functional $g_x$ on $X'$ is defined by
 
 $$
-g_x(f) = f(x) \quad \text{for $f \in X'$,} 
+g_x(f) = f(x) \quad \text{for $f \in X'$,}
 $$
 
 is called the *canonical embedding* of $X$ into its bidual $X''$.
@@ -79,7 +79,7 @@ C : X \longrightarrow X'', \quad C(x)(f) = f(x), \quad x \in X, f \in X',
 $$
 
 is an isomorphism.
-Therefore, the the dual map
+Therefore, the dual map
 
 $$
 C' : X''' \longrightarrow X', \quad C'(h)(x) = h(C(x)), \quad x \in X, h \in X''',
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 f6a9783..e0ec62b 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
@@ -33,10 +33,11 @@ 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 $\norm{(x,Tx)} = \norm{x} + \norm{Tx}$.
+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
+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.
+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 18cf64a..a2602ac 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
@@ -6,7 +6,6 @@ nav_order: 1
 ---
 
 # {{ page.title }}
-{: .no_toc }
 
 In fact, there are multiple theorems and corollaries
 which bear the name Hahn–Banach.
@@ -14,15 +13,6 @@ All have in common that
 they guarantee the existence of linear functionals
 with various additional properties.
 
-<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
@@ -146,7 +136,7 @@ we define a functional $f_0$ by $f_0(\alpha x) = \alpha \norm{x}$ for $\alpha \i
 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 $\norm{f} = \norm{f_0} = 1$.
+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 $X'$.
@@ -155,7 +145,7 @@ Recall that for a normed space $X$ we denote its (topological) dual space by $X'
 For every element $x$ of a real or complex normed space $X$ one has
 
 $$
-\norm{x} = \sup_{f \in X' \setminus \braces{0}} \frac{\abs{f(x)}}{\norm{f}} 
+\norm{x} = \sup_{f \in X' \setminus \braces{0}} \frac{\abs{f(x)}}{\norm{f}}
 $$
 
 and the supremum is attained.
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 b191bb2..e7f2b70 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
@@ -30,10 +30,10 @@ This remains true, if we take closures:
 $\bigcup \overline{mTB_X} = Y$.
 Hence, we have written the space $Y$,
 which is assumed to have a complete norm,
-as the union of countably many closed sets. It follows form the
+as the union of countably many closed sets. It follows from the
 [Baire Category Theorem]({% link pages/general-topology/baire-spaces.md %})
 that $\overline{mTB_X}$ has nonempty interior for some $m$.
-Thus there are $q \in Y$ and $\alpha > 0$
+Thus, there are $q \in Y$ and $\alpha > 0$
 such that $q + \alpha B_Y \subset \overline{mTB_X}$.
 Choose a $p \in X$ with $Tp=q$.
 It is a well known fact, that in a normed space
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 47ddd3f..1140e45 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
@@ -14,11 +14,13 @@ 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 $x \in X$,
+- *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 $\norm{Tx} \le \norm{T} \norm{x}$.
+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 %}
diff --git a/pages/general-topology/compactness/index.md b/pages/general-topology/compactness/index.md
index 37e9b4d..6c2e274 100644
--- a/pages/general-topology/compactness/index.md
+++ b/pages/general-topology/compactness/index.md
@@ -26,7 +26,8 @@ if and only if it has the following property:
   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.
+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
@@ -43,4 +44,3 @@ if and only if $\mathcal{A}^c$ consists of closed subsets of $X$.
 {% definition Finite Intersection Property%}
 TODO
 {% enddefinition %}
-
diff --git a/pages/general-topology/continuity-and-convergence.md b/pages/general-topology/continuity-and-convergence.md
index 7ae4534..57e5ca9 100644
--- a/pages/general-topology/continuity-and-convergence.md
+++ b/pages/general-topology/continuity-and-convergence.md
@@ -1,5 +1,5 @@
 ---
-title: Continuity & Convergence
+title: Continuity &amp; Convergence
 parent: General Topology
 nav_order: 2
 ---
diff --git a/pages/general-topology/metric-spaces/index.md b/pages/general-topology/metric-spaces/index.md
index c0dc45a..52b2b4c 100644
--- a/pages/general-topology/metric-spaces/index.md
+++ b/pages/general-topology/metric-spaces/index.md
@@ -46,13 +46,13 @@ 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*
-  
+
   $$
   \abs{d(v,w) - d(x,y)} \le d(v,x) + d(w,y)
   $$
@@ -141,27 +141,31 @@ every sequence in $X$ has at most one limit.
 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
-  
+
   $$
-  \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen
+  \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X :
+  \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen
   $$
 
 - *continuous* if it is continuous at every point of $X$, that is
-  
+
   $$
-  \forall x \in X \ \ \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen
+  \forall x \in X \ \ \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X :
+  \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen
   $$
 
 - *uniformly continuous* if
-  
+
   $$
-  \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x,x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen
+  \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x,x' \in X :
+  \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen
   $$
 
 - *Lipschitz continuous* if
-  
+
   $$
-  \exists L \ge 0 \ \ \forall x,x' \in X : d_Y(f(x),f(x')) \le L \, d_X(x,x') 
+  \exists L \ge 0 \ \ \forall x,x' \in X :
+  d_Y(f(x),f(x')) \le L \, d_X(x,x')
   $$
 {% enddefinition %}
 
diff --git a/pages/general-topology/topological-spaces.md b/pages/general-topology/topological-spaces.md
index b0b1834..cb0c30b 100644
--- a/pages/general-topology/topological-spaces.md
+++ b/pages/general-topology/topological-spaces.md
@@ -75,7 +75,8 @@ is the smallest topology on $X$ containing $\mathcal{A}$.
 
 {% 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 $x \in X$ 
+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 $B_3 \in \mathcal{B}$
@@ -85,6 +86,7 @@ such that for every point $x \in X$
 {% 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 $B \subset S$;
@@ -125,7 +127,7 @@ then the topology generated by $\mathcal{S}$ equals
 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 $\mathcal{T}$
+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 %}
@@ -137,10 +139,8 @@ if and only if its complement is closed.
 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 $\mathcal{C}$.
+- 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/measure-and-integration/lebesgue-integral/convergence-theorems.md b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md
index 67f0996..f9ebc4a 100644
--- a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md
+++ b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md
@@ -32,10 +32,10 @@ $$
 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 $g_n = \inf_{k \ge n} f_k$.
+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)
+By the [Monotone Convergence Theorem](#monotone-convergence-theorem)
 
 $$
 \int \liminf_{n \to \infty} f_n = \lim_{n \to \infty} \int g_n.
diff --git a/pages/measure-and-integration/lebesgue-integral/index.md b/pages/measure-and-integration/lebesgue-integral/index.md
index a857d95..3418e10 100644
--- a/pages/measure-and-integration/lebesgue-integral/index.md
+++ b/pages/measure-and-integration/lebesgue-integral/index.md
@@ -106,7 +106,7 @@ 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
 
 $$
-\int_A f \, d\mu = 
+\int_A f \, d\mu =
 \int_X \chi_A f \, d\mu.
 $$
 {% 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
index 023c253..8482e87 100644
--- a/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md
+++ b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md
@@ -1,5 +1,5 @@
 ---
-title: The L<sup>p</sup> Spaces 
+title: The L<sup>p</sup> Spaces
 parent: Lebesgue Integral
 grand_parent: Measure and Integration
 nav_order: 4
diff --git a/pages/measure-and-integration/measure-theory/measures.md b/pages/measure-and-integration/measure-theory/measures.md
index 637ab0c..c843881 100644
--- a/pages/measure-and-integration/measure-theory/measures.md
+++ b/pages/measure-and-integration/measure-theory/measures.md
@@ -14,7 +14,7 @@ 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).
diff --git a/pages/measure-and-integration/measure-theory/sigma-algebras.md b/pages/measure-and-integration/measure-theory/sigma-algebras.md
index 5d22f6b..8f58f09 100644
--- a/pages/measure-and-integration/measure-theory/sigma-algebras.md
+++ b/pages/measure-and-integration/measure-theory/sigma-algebras.md
@@ -47,4 +47,3 @@ defined to be the intersection of all σ-algebras on $X$ containing $\mathcal{A}
 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
index 77b2416..657a28f 100644
--- a/pages/measure-and-integration/measure-theory/signed-measures.md
+++ b/pages/measure-and-integration/measure-theory/signed-measures.md
@@ -18,7 +18,7 @@ is a mapping $\mu : \mathcal{A} \to [-\infty,\infty]$ such that
 - for every sequence $(A_n)_{n \in \NN}$ of
   pairwise disjoint sets $A_n \in \mathcal{A}$
   {: .my-0 }
-  
+
   $$
   \mu \bigg\lparen \bigcup_{n=1}^{\infty} A_n \! \bigg\rparen
   = \sum_{n=0}^{\infty} \mu(A_n).
diff --git a/pages/more-functional-analysis/topological-vector-spaces/index.md b/pages/more-functional-analysis/topological-vector-spaces/index.md
index 745d53b..a3e6220 100644
--- a/pages/more-functional-analysis/topological-vector-spaces/index.md
+++ b/pages/more-functional-analysis/topological-vector-spaces/index.md
@@ -49,7 +49,7 @@ A subset $A \subset X$ is said to be
 {% theorem %}
 These properties of subsets of $X$
 are stable under arbitrary intersections.
-Thus we obtain the notions of
+Thus, we obtain the notions of
 *convex hull* $\co A$,
 *balanced hull* $\bal A$, and
 *absolutely convex hull* $\aco A$.
diff --git a/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md b/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md
index 277ecd3..ea0b22b 100644
--- a/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md
+++ b/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md
@@ -29,7 +29,7 @@ We say that the bilinear form $b : V \times W \to \KK$ is *nondegenerate*, if it
 $$
 \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 
+\forall w \in W : \qquad ( \forall v \in V : \angles{v,w} = 0 ) \implies w = 0
 \end{gather*}
 $$
 
@@ -45,7 +45,6 @@ $$
 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 $\KK$
@@ -56,9 +55,9 @@ and a nondegenerate bilinear form $\angles{\cdot,\cdot} : V \times W \to \KK$.
 
 {% 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 $y \in Y$.
+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 $x \in X$.
diff --git a/pages/operator-algebras/banach-algebras/index.md b/pages/operator-algebras/banach-algebras/index.md
index 3335d78..9d70df8 100644
--- a/pages/operator-algebras/banach-algebras/index.md
+++ b/pages/operator-algebras/banach-algebras/index.md
@@ -80,7 +80,7 @@ $$
 (\mathbf{1}-x) s_n = s_n (\mathbf{1}-x) = \mathbf{1} - x^{n+1}.
 $$
 
-In the limit $n \to \infty$ we obtain $(\mathbf{1}-x) s = s (\mathbf{1}-x) = \mathbf{1}$, 
+In the limit $n \to \infty$ we obtain $(\mathbf{1}-x) s = s (\mathbf{1}-x) = \mathbf{1}$,
 because multiplication in a Banach algebra is continuous, and because $y^n \to 0$ when $\norm{y} < 1$.
 This proves that $s$ is the inverse of $\mathbf{1}-x$.
 
@@ -94,10 +94,12 @@ 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 *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 $R_{\lambda} = (\lambda - x)^{-1}$. \
+  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 %}
 
@@ -119,7 +121,7 @@ $$
 
 {% 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).
@@ -138,20 +140,22 @@ the claimed formula for its inverse follows by an application of
 the rule $(ab)^{-1} = b^{-1} a^{-1}$ for invertible $a,b \in \mathcal{A}$.
 {% endproof %}
 
-{: .corollary #resolvent-set-is-open #spectrum-is-closed }
-> The resolvent set $\rho(x)$ is open and the spectrum $\sigma(x)$ is closed.
+{% corollary %}
+The resolvent set $\rho(x)$ is open and the spectrum $\sigma(x)$ is closed.
+{% endcorollary %}
 
-{: .corollary #resolvent-map-is-analytic }
-> Suppose $x$ is an element of a unital Banach algebra $\mathcal{A}$.
-> The resolvent map
->
-> $$
-> R : \rho(x) \longrightarrow \mathcal{A}, \quad \lambda \longmapsto R_{\lambda} = (\lambda - x)^{-1},
-> $$
->
-> is (strongly) analytic.
+{% corollary %}
+Suppose $x$ is an element of a unital Banach algebra $\mathcal{A}$.
+The resolvent map
+
+$$
+R : \rho(x) \longrightarrow \mathcal{A}, \quad \lambda \longmapsto R_{\lambda} = (\lambda - x)^{-1},
+$$
+
+is (strongly) analytic.
+{% endcorollary %}
 
- ---
+---
 
 {: .proposition #spectrum-is-not-empty }
 > Suppose $x$ is an element of a unital Banach algebra.
@@ -169,7 +173,7 @@ For $\abs{\lambda} > 2 \norm{x}$ we may expand $R_{\lambda}$ into a [Neumann ser
 $$
 R_{\lambda}
 = (\lambda - x)^{-1}
-= \lambda^{-1} (\mathbf{1} - \lambda^{-1} x)^{-1} 
+= \lambda^{-1} (\mathbf{1} - \lambda^{-1} x)^{-1}
 = \lambda^{-1} \sum_{n=0}^{\infty} (\lambda^{-1} x)^n,
 $$
 
@@ -177,7 +181,7 @@ and make the estimate
 
 $$
 \norm{R_{\lambda}}
-\le \abs{\lambda}^{-1} (1 - \norm{\lambda^{-1} x})^{-1} 
+\le \abs{\lambda}^{-1} (1 - \norm{\lambda^{-1} x})^{-1}
 = (\abs{\lambda} - \norm{x})^{-1}
 < \norm{x}^{-1}.
 $$
@@ -197,7 +201,7 @@ For any Banach algebra $A$,
 the mapping $\varphi : \CC \to A$, $\lambda \mapsto \lambda \mathbf{1}$,
 is linear, multiplicative and isometric, hence injective.
 Let $x$ be any element of $A$.
-Since its 
+Since its
 [spectrum is not empty](/pages/operator-algebras/banach-algebras/index.html#spectrum-is-not-empty),
 there must exist a complex number $\lambda$
 such that $x - \lambda \mathbf{1}$ is not invertible.
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 ea15f87..bdf7b3d 100644
--- a/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md
+++ b/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md
@@ -22,7 +22,9 @@ 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 $\mathcal{A}$.
+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 $\mathcal{A}$.
diff --git a/pages/operator-algebras/c-star-algebras/states.md b/pages/operator-algebras/c-star-algebras/states.md
index 29cf5f5..a483915 100644
--- a/pages/operator-algebras/c-star-algebras/states.md
+++ b/pages/operator-algebras/c-star-algebras/states.md
@@ -8,7 +8,9 @@ nav_order: 1
 # {{ page.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*.\
+A norm-one
+[positive linear functional](/pages/operator-algebras/c-star-algebras/positive-linear-functionals.html)
+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 %}
 
@@ -30,20 +32,23 @@ Let $\omega_0, \omega_1$ be states on $\mathcal{A}$ and let $t \in (0,1)$.
 Consider the convex combination $\omega = (1-t)\omega_0 + t\omega_1$.
 Clearly, $\omega$ is linear and $\omega(\mathbf{1}) = 1$.
 By the triangle inequality, $\norm{\omega} \le 1$.
-It follows from the lemma above that $\omega$ lies in $S(\mathcal{A})$. This proves that $S(\mathcal{A})$ is convex.
+It follows from the lemma above that $\omega$ lies in $S(\mathcal{A})$.
+This proves that $S(\mathcal{A})$ is convex.
 
 Next we show weak\* compactness. Since $S(\mathcal{A})$ is contained
 in the closed unit ball in the dual of $\mathcal{A}$,
 which is weak\* compact by the
-[Banach–Alaoglu Theorem]({% link pages/functional-analysis-basics/banach-alaoglu-theorem.md %}),
+[Banach–Alaoglu Theorem](/pages/functional-analysis-basics/banach-alaoglu-theorem.html),
 it will suffice to show that $S(\mathcal{A})$ is weak\* closed.
-Let $(\omega_i)$ be a net of states that weak\* converges to some bounded linear functional $\omega$ on $\mathcal{A}$.
+Let $(\omega_i)$ be a net of states
+that weak\* converges to some bounded linear functional $\omega$ on $\mathcal{A}$.
 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.
+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/spectral-theory/test/basic.md b/pages/spectral-theory/test/basic.md
index 05405f4..b1015d1 100644
--- a/pages/spectral-theory/test/basic.md
+++ b/pages/spectral-theory/test/basic.md
@@ -32,25 +32,19 @@ nav_order: 2
 > 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 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-title }
-> Definition (point spectrum, residual spectrum, continuous spectrum)
->
-> Let $T : \dom{T} \to X$ be an operator in a complex normed space $X$.
-> The *point spectrum* $\pspec{T}$ is the set of all $\lambda \in \CC$
-> for which the resolvent $R_\lambda(T)$ does not exist.
-> The *residual spectrum* $\rspec{T}$ is the set of all $\lambda \in \CC$
-> for which the resolvent $R_\lambda(T)$ exists, but is not densely defined.
-> The *continuous spectrum* $\cspec{T}$ is the set of all $\lambda \in \CC$
-> for which the resolvent $R_\lambda(T)$ exists and is densely defined, but unbounded.
-
-| If $R_\lambda(T)$ exists, | is densely defined | and is bounded ... | ... then $\lambda$ belongs to the |
-|:-------------------------:|:------------------:|:------------------:|-----------------------------------|
-|             ✗             |          -         |          -         | point spectrum $\pspec{T}$        |
-|             ✓             |          ✗         |          ?         | residual spectrum $\rspec{T}$     |
-|             ✓             |          ✓         |          ✗         | continuous spectrum $\cspec{T}$   |
-|             ✓             |          ✓         |          ✓         | resolvent set $\rho(T)$           |
+{% definition Point Spectrum, Residual Spectrum, Continuous Spectrum %}
+Let $T : \dom{T} \to X$ be an operator in a complex normed space $X$.
+The *point spectrum* $\pspec{T}$ is the set of all $\lambda \in \CC$
+for which the resolvent $R_\lambda(T)$ does not exist.
+The *residual spectrum* $\rspec{T}$ is the set of all $\lambda \in \CC$
+for which the resolvent $R_\lambda(T)$ exists, but is not densely defined.
+The *continuous spectrum* $\cspec{T}$ is the set of all $\lambda \in \CC$
+for which the resolvent $R_\lambda(T)$ exists and is densely defined, but unbounded.
+{% enddefinition %}
 
-By definition, the sets $\pspec{T}$, $\rspec{T}$, $\cspec{T}$ and $\rho(T)$ form a partition of the complex plane.
+By definition, the sets $\pspec{T}$, $\rspec{T}$, $\cspec{T}$ and $\rho(T)$
+form a partition of the complex plane.
diff --git a/pages/unbounded-operators/adjoint-operators.md b/pages/unbounded-operators/adjoint-operators.md
index 1b9fb0c..96933db 100644
--- a/pages/unbounded-operators/adjoint-operators.md
+++ b/pages/unbounded-operators/adjoint-operators.md
@@ -10,4 +10,3 @@ description: >
 ---
 
 # {{ page.title }}
-
diff --git a/pages/unbounded-operators/hellinger-toeplitz-theorem.md b/pages/unbounded-operators/hellinger-toeplitz-theorem.md
index fea54be..09046f4 100644
--- a/pages/unbounded-operators/hellinger-toeplitz-theorem.md
+++ b/pages/unbounded-operators/hellinger-toeplitz-theorem.md
@@ -18,7 +18,7 @@ Conventions:
 - Operators are linear and possibly unbounded.
 
 Recall that an operator $T : D(T) \to \hilb{H}$ in a Hilbert space $\hilb{H}$
-is called *symmetric*, if is has the property
+is called *symmetric*, if it has the property
 
 $$
 \innerp{Tx}{y} = \innerp{x}{Ty} \quad \forall x,y \in D(T).
-- 
cgit v1.2.3-54-g00ecf