Update

11 files changed, 223 insertions, 16 deletions

diff --git a/.cspell.yaml b/.cspell.yaml
index b4f5223..7375c55 100644
--- a/.cspell.yaml
+++ b/.cspell.yaml
@@ -11,9 +11,6 @@ dictionaryDefinitions:
   - name: liquid
     addWords: true
     path: ./.cspell/liquid.txt
-ignorePaths:
-  - ./.cspell/
-  - ./assets/
 dictionaries:
   - latex
   - names
@@ -40,3 +37,6 @@ words:
   - enspace
   - callouts
   - vcs
+ignorePaths:
+  - ./.cspell/
+  - ./assets/
diff --git a/.cspell/my-cspell-dicts b/.cspell/my-cspell-dicts
-Subproject 1e177028a4a8a45a9927479811411ee2beb5c59
+Subproject 87b975f2f6ae630991ac532e47e96c1bce2a4a5
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 6ac0803..f7414d5 100644
--- a/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md
+++ b/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md
@@ -20,7 +20,7 @@ $$
 {% proof %}
 {% endproof %}
 
-{% theorem * Cauchy's Integral Formula (Generalization) %}
+{% 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 $G$,
@@ -44,7 +44,7 @@ and is often used to compute the integral.
 
 ## Many Consequences
 
-{% theorem * Cauchy's Estimate %}
+{% theorem * Cauchy’s Estimate %}
 Let $f$ be holomorphic on an open set containing the disc with center $a$ and radius $r>0$.
 Then
 
@@ -74,7 +74,7 @@ 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 * Liouville's Theorem %}
+{% theorem * Liouville’s Theorem %}
 Every bounded entire function is constant.
 {% endtheorem %}
 
@@ -82,7 +82,7 @@ Every bounded entire function is constant.
 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)
+[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 6d78e89..eb040ca 100644
--- a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
+++ b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
@@ -7,7 +7,7 @@ nav_order: 2
 
 # {{ page.title }}
 
-{% theorem Cauchy's Theorem (Homotopy Version) %}
+{% 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
@@ -29,7 +29,7 @@ $$
 
 {{ page.title }} has a converse:
 
-{% theorem * Morera's Theorem %}
+{% 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 $G$.
diff --git a/pages/functional-analysis-basics/inner-product-spaces.md b/pages/functional-analysis-basics/inner-product-spaces.md
new file mode 100644
index 0000000..5f34a8a
--- /dev/null
+++ b/pages/functional-analysis-basics/inner-product-spaces.md
@@ -0,0 +1,177 @@
+---
+title: Inner Product Spaces
+parent: Functional Analysis Basics
+nav_order: 1
+---
+
+# {{ page.title }}
+
+{% definition Inner Product Space %}
+An *inner product* on a real or complex vector space $X$
+is a mapping
+
+$$
+\innerp{\cdot}{\cdot} : X \times X \to \KK
+$$
+
+that is
+
+- linear in its second argument
+- conjugate symmetric
+- nondegenerate
+
+An *inner product space* is a pair $(X,\innerp{\cdot}{\cdot})$
+consisting of a real or complex vector space $X$
+and an inner product $\innerp{\cdot}{\cdot}$ on $X$.
+{% enddefinition %}
+
+{% proposition Norm Induced by an Inner Product %}
+If $\innerp{\cdot}{\cdot}$ is an inner product
+on a real or complex vector space $X$, then
+
+$$
+\norm{x} = \sqrt{\innerp{x}{x}} \qquad \forall x \in X
+$$
+
+defines a norm on $X$.
+{% endproposition %}
+
+In this sense, every inner product space is also a normed space.
+As a consequence it is also a metric space and a topolgical space.
+
+The next theorem shows how the inner product can be recovered from the norm.
+
+{% theorem * Polarization Identity %}
+For all vectors $x$ and $y$ of a real inner product space
+
+$$
+4 \innerp{x}{y} = \norm{x+y}^2 - \norm{x-y}^2.
+$$
+
+For all vectors $x$ and $y$ of a complex inner product space
+
+$$
+4 \innerp{x}{y} = \norm{x+y}^2 - \norm{x-y}^2 + i \norm{x-iy}^2 - i \norm{x+iy}^2.
+$$
+{% endtheorem %}
+
+Note that the complex polarization identity takes the slightly different form
+
+$$
+4 \innerp{x}{y} = \norm{x+y}^2 - \norm{x-y}^2 + i \norm{x\mathrel{\color{red}+}iy}^2 - i \norm{x\mathrel{\color{red}-}iy}^2,
+$$
+
+if we follow the convention that the inner product is conjugate linear in its second argument.
+
+{% proof %}
+In the real case, the inner product is symmetric, and we have
+
+$$
+\norm{x \pm y}^2 = \norm{x}^2 \pm 2 \innerp{x}{y} + \norm{y}^2
+$$
+
+for all vectors $x$ and $y$.
+Taking the difference yields the desired result.
+
+In the complex case, the inner product is conjugate symmetric, and we have
+
+$$
+\norm{x \pm y}^2 = \norm{x}^2 \pm 2 \Re \innerp{x}{y} + \norm{y}^2
+$$
+
+for all vectors $x$ and $y$. This implies
+
+$$
+\begin{aligned}
+\norm{x + \phantom{i}y}^2 - \norm{x - \phantom{i}y}^2 &= 4 \Re \innerp{x}{y}, \\
+\norm{x - iy}^2 - \norm{x + iy}^2 &= 4 \Im \innerp{x}{y}.
+\end{aligned}
+$$
+
+The second equation follows from the first by
+substituting $y$ with $-iy$ and
+using that $\Re \innerp{x}{-iy} = \Re (-i\innerp{x}{y}) = \Im \innerp{x}{y}$.
+To obtain the polarization Identity, multiply the second equation with $i$ and then add it to the first.
+{% endproof %}
+
+{% theorem * General Polarization Identity %}
+Let $X$ be a complex inner product space.
+Let $\zeta$ be a $n$-th root of unity with $\zeta \ne 1$ and $\zeta^2 \ne 1$.
+Then
+
+$$
+\innerp{x}{y} = \frac{1}{n} \sum_{k=0}^{n-1} \zeta^k \norm{x + \zeta^k y}^2 \qquad \forall x,y \in X.
+$$
+{% endtheorem %}
+
+As a special case, for $\zeta = i$ and $n=4$, we obtain
+
+$$
+\innerp{x}{y} = \frac{1}{4} \sum_{k=0}^{3} i^k \norm{x + i^k y}^2.
+$$
+
+{% proof %}
+TODO
+{% endproof %}
+
+For an arbitrary normed space,
+the polarization identity does not, in general,
+define an inner product.
+The following theorem, gives a condition for when it does.
+
+{% theorem * Parallelogram Law %}
+Let $X$ be a real or complex normed space.
+A norm $\norm{\cdot}$ on $X$ is induced by
+an inner product $\innerp{\cdot}{\cdot}$ on $X$,
+if and only if $\norm{\cdot}$ satisfies the *parallelogram law*
+
+$$
+\norm{x+y}^2 + \norm{x-y}^2 = 2 \norm{x}^2 + 2 \norm{y}^2 \qquad \forall x,y \in X.
+$$
+
+In this case, the inner product is uniquely determined by $\norm{\cdot}$ and given by the polarization identity.
+{% endtheorem %}
+
+{% theorem * Cauchy–Schwarz Inequality %}
+For all vectors $x$ and $y$ of an inner product space (with inner product $\innerp{\cdot}{\cdot}$ and induced norm $\norm{\cdot}$)
+
+$$
+\abs{\innerp{x}{y}} \le \norm{x} \norm{y},
+$$
+
+and equality holds precisely when $x$ and $y$ are linearly dependent.
+{% endtheorem %}
+
+Expressed only in terms of the inner product, the Cauchy–Schwarz Inequality reads
+
+$$
+\abs{\innerp{x}{y}}^2 \le \innerp{x}{x} \innerp{y}{y}.
+$$
+
+{% proof %}
+TODO
+{% endproof %}
+
+{% corollary Continuity of the Inner Product %}
+The inner product is jointly norm continous.
+{% endcorollary %}
+
+## Orthogonality
+
+{% definition Orthogonal Vectors %}
+Two vectors $x$ and $y$ of an inner product space $(X,\innerp{\cdot}{\cdot})$
+are said to be *orthogonal* or *perpendicular* if $\innerp{x}{y} = 0$,
+and this fact is indicated by writing $x \perp y$.
+{% enddefinition %}
+
+{% theorem * Pythagoras’ Theorem %}
+For all vectors $x$ and $y$ of an inner product space we have
+
+$$
+x \perp y \iff \norm{x+y}^2 = \norm{x}^2 + \norm{y}^2.
+$$
+{% endtheorem %}
+
+{% proof %}
+Immediate.
+{% endproof %}
diff --git a/pages/general-topology/metric-spaces/index.md b/pages/general-topology/metric-spaces/index.md
index 52b2b4c..6aef400 100644
--- a/pages/general-topology/metric-spaces/index.md
+++ b/pages/general-topology/metric-spaces/index.md
@@ -60,8 +60,16 @@ Let $(X,d)$ be a (semi-)metric space.
 
 The proofs are straightforward.
 
+{% definition Isometry %}
+Suppose $(X,d_X)$ and $(Y,d_Y)$ are metric spaces.
+We say that a mapping $f : X \to Y$ is *isometric* or an *isometry* if it obeys
+$d_Y \big\lparen f(x),f(x') \big\rparen = d_X(x,x')$ for all $x,x' \in X$.
+{% enddefinition %}
+
+As a consequence of **(M1)**,
+every isometry is injective.
+
 TODO
-- isometry
 - metric induced by a norm
 - metric product
 
diff --git a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md
index 1a34820..6808280 100644
--- a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md
+++ b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md
@@ -4,9 +4,9 @@ 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.
+  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/the-lp-spaces.md b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md
index 8482e87..0424117 100644
--- a/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md
+++ b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md
@@ -23,6 +23,28 @@ Endowed with pointwise addition and scalar multiplication
 $\mathscr{L}^p(X,\mathcal{A},\mu)$ becomes a vector space.
 {% endproposition %}
 
+{% proof %}
+We show that $\mathscr{L}^p := \mathscr{L}^p(X,\mathcal{A},\mu)$ is a linear subspace of
+the vector space of all $\KK$-valued functions on $X$.
+The set $\mathscr{L}^p$ is nonempty since
+it contains the zero function.
+Now, suppose $f$ and $g$ are in $\mathscr{L}^p$.
+Then the sum $f+g$ is measurable, because $f$ and $g$ are measurable.
+Moreover, the function $\abs{f+g}^p$ is integrable, because we have the estimate
+
+$$
+\abs{f+g}^p
+\le (\abs{f} + \abs{g})^p
+\le \big\lparen 2 \max(\abs{f},\abs{g}) \big\rparen^p
+\le 2^p (\abs{f}^p + \abs{g}^p),
+$$
+
+where $\abs{f}^p$ and $\abs{g}^p$ are integrable.
+This proves that $f+g$ lies in $\mathscr{L}^p$.
+Finally, it is easy to see that $\alpha f$ lies in $\mathscr{L}^p$
+for any scalar $\alpha \in \KK$.
+{% endproof %}
+
 {% proposition %}
 $\norm{\cdot}_p$ is a seminorm on $\mathscr{L}^p(X,\mathcal{A},\mu)$.
 {% endproposition %}
diff --git a/pages/measure-and-integration/measure-theory/borels-sets.md b/pages/measure-and-integration/measure-theory/borels-sets.md
index 737a7c8..0cdb142 100644
--- a/pages/measure-and-integration/measure-theory/borels-sets.md
+++ b/pages/measure-and-integration/measure-theory/borels-sets.md
@@ -2,7 +2,7 @@
 title: Borel Sets
 parent: Measure Theory
 grand_parent: Measure and Integration
-nav_order: 2
+nav_order: 3
 ---
 
 # {{ page.title }}
diff --git a/pages/measure-and-integration/measure-theory/measurable-maps.md b/pages/measure-and-integration/measure-theory/measurable-maps.md
index 5b7a76e..dc9f4d7 100644
--- a/pages/measure-and-integration/measure-theory/measurable-maps.md
+++ b/pages/measure-and-integration/measure-theory/measurable-maps.md
@@ -2,7 +2,7 @@
 title: Measurable Maps
 parent: Measure Theory
 grand_parent: Measure and Integration
-nav_order: 3
+nav_order: 2
 ---
 
 # {{ page.title }}
diff --git a/pages/operator-algebras/banach-algebras/index.md b/pages/operator-algebras/banach-algebras/index.md
index 3a427a7..5f33816 100644
--- a/pages/operator-algebras/banach-algebras/index.md
+++ b/pages/operator-algebras/banach-algebras/index.md
@@ -188,7 +188,7 @@ $$
 $$
 
 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)
+[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 contradictory because XXX
 {% endproof %}