From 73445885d54edffcc9ae74525887b529a3f96165 Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Sat, 23 Mar 2024 18:08:44 +0100
Subject: Update

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+---
+title: Inner Product Spaces
+parent: Functional Analysis Basics
+nav_order: 6
+has_children: true
+has_toc: false
+---
+
+# {{ page.title }}
+
+{% definition Inner Product Space %}
+An *inner product* (or *scalar 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
+  
+  $$
+  \innerp{x}{y+z} = \innerp{x}{y} + \innerp{x}{z} \qquad
+  \innerp{x}{\alpha y} = \alpha \innerp{x}{y}
+  $$
+
+- conjugate symmetric
+  
+  $$
+  \overline{\innerp{x}{y}} = \innerp{x}{y}
+  $$
+
+- nondegenerate
+
+An *inner product space* (or *pre-Hilbert 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 topological 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 * Stewart’s Theorem %}
+Let $x$, $y$, $z$ be vectors of an inner product space.
+If $x$, $y$ and $z$ are colinear and $y$ lies inbetween $x$ and $y$,
+then we have
+
+$$
+\norm{p-x}^2 \norm{y-z} + \norm{p-z}^2 \norm{x-y} =
+\big\lparen \norm{p-y}^2 + \norm{x-y} \norm{y-z} \big\rparen \norm{x-z}
+$$
+{% 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 continuous.
+{% endcorollary %}
-- 
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