From fadd6961c92393d86de69ec468f0a15a2f320252 Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Wed, 25 Sep 2024 00:26:13 +0200
Subject: weiter

---
 convolution.tex | 28 ++++++++++++++++------------
 1 file changed, 16 insertions(+), 12 deletions(-)

(limited to 'convolution.tex')

diff --git a/convolution.tex b/convolution.tex
index eee6e16..66120f2 100644
--- a/convolution.tex
+++ b/convolution.tex
@@ -42,14 +42,18 @@ as this will facilitate our proof of the convolution formula.
 We consider a $\sigma$-finite measure space $(X,\SigmaAlgebra{A},\mu)$,
 a separable Fréchet space $Y$ (over $\CC$) and the task is
 to define the integral of functions $f \vcentcolon X \to Y$.
-Recall that a measure space is said to be \emph{$\sigma$-finite}
+Recall that a measure space is said to be
+\emph{$\sigma$-finite}
+%\index{sigma-finite@$sigma$-finite} TODO: fix @
+%\nomenclature[A]{$\mathcal{A}'$}{commutant of $\mathcal{A}$}
 if it can be exhausted by a countable number of measurable subsets of finite measure.
-By \emph{Fréchet space} we mean a complete Hausdorff locally convex (topological vector) space
+By \emph{Fréchet space}\index{Fréchet space}
+we mean a complete Hausdorff locally convex (topological vector) space
 which possesses countable neighborhood bases.
 We will make use of a countable family $P@@$ of seminorms that generates the topology of $@@Y$.
 A topological space is called \emph{separable} if it contains a countable dense subset.
 
-A function $f \vcentcolon X \to Y$ will be called \emph{simple}
+A function $f \vcentcolon X \to Y$ will be called \emph{simple}\index{simple function}
 if it is of the form $\sum_{i=1}^n \chi_{A_i} y_i$
 where $n \in \NN$, $A_i \in \SigmaAlgebra{A}$ with $\mu(A_i) < \infty$, and $y_i \in Y$.
 Naturally, the \emph{integral} of $f$ is defined to be the vector $\int f = \sum_{i=1}^n \mu(A_i) y_i \in Y$.
@@ -60,14 +64,14 @@ if it is the $\mu$-almost everywhere pointwise limit of simple functions.
   Suppose $(X,\SigmaAlgebra{A},\mu)$ is a $\sigma$-finite measure space,
   and $Y@@$ is a separable Fréchet space
   whose topology is generated by a family $P@@$ of seminorms.
-A strongly measurable function $f \vcentcolon X \to Y$ is called \emph{(generalized Bochner) integrable}
+  A strongly measurable function $f \vcentcolon X \to Y$ is called \emph{(generalized Bochner) integrable}
 if there exists a sequence $(f_n)$ of simple functions such that
 \begin{equation}
   \label{equation:bochner-integrable}
   \lim_{n \to \infty} \int_X p \circ (f_n - f) \, d\mu = 0
   \qquad \forall p \in P.
 \end{equation}
-In this case, the \emph{(generalized Bochner) integral} of $f$ is defined by
+  In this case, the \emph{(generalized Bochner) integral}\index{Bochner integral!generalized} of $f$ is defined by
 \begin{equation}
   \label{equation:bochner-integral}
   \int_X f \ d\mu \defequal
@@ -226,7 +230,7 @@ Denote by $\TestFunctions{\RR^n}$  the vector space of all functions $f \vcentco
 such that the derivatives $\partial^{\alpha} f$ exist and are continuous for all multi-indices $\alpha \in \NN^n$.
 Recall that the space $\SchwartzFunctions{\RR^n}$ of \emph{Schwartz functions} is defined to be the vector space
 \begin{equation*}
-  \SchwartzFunctions{\RR^n,X} \defequal \braces{f \in \TestFunctions{\RR^n} \vcentcolon \norm{f}_{\alpha,\beta} < \infty \ \forall \alpha,\beta \in \NN^n}
+  \SchwartzFunctions{\RR^n} \defequal \braces{f \in \TestFunctions{\RR^n} \vcentcolon \norm{f}_{\alpha,\beta} < \infty \ \forall \alpha,\beta \in \NN^n}
 \end{equation*}
 equipped with the locally convex topology induced by the family of seminorms
 \begin{equation*}
@@ -247,7 +251,7 @@ is defined in the same way as in \cref{definition:convolution-distribution-test-
   \end{equation*}
 
 \begin{proposition}{Vector-Valued Convolution Formula}{vector-valued-convolution-formula}
-  Let $v \in \TemperedDistributions{\RR^n\!,X}$ be a tempered distribution with values in a separable Fréchet space $X$, and
+  Let $v \in \TemperedDistributions{\RR^n\!,Y}$ be a tempered distribution with values in a separable Fréchet space $Y$, and
   let $f \in \SchwartzFunctions{\RR^n}$ be a Schwartz test function. Then one has
   \begin{equation*}
     (v * f)(g) = \int v(\tau_x \tilde{f}@@) g(x) \, dx \qquad g \in \SchwartzFunctions{\RR^n}.
@@ -256,7 +260,7 @@ is defined in the same way as in \cref{definition:convolution-distribution-test-
 
 \begin{proof}
   We fix a Schwartz function $g$, and consider the finite measure $\mu = \abs{g} \lambda$ on $\RR^n$,
-  where $\lambda(x) = dx$ is the Lebesgue measure.
+  where $\lambda = dx$ is the Lebesgue measure.
   We show that the mapping $x \mapsto \tau_x \tilde{f}$ is a generalized Bochner $\mu$-integrable function $\RR^n \to \SchwartzFunctions{\RR^n}$
   using \cref{theorem:generalized-bochner}.
   For all $\alpha,\beta \in \NN^n$ we see by substituting $x+y$ for $y$ that
@@ -275,9 +279,9 @@ is defined in the same way as in \cref{definition:convolution-distribution-test-
   because $g$ is Schwartz class.
   Hence, $x \mapsto \tau_x \tilde{f}$ defines an integrable function.
   
-  The mapping $v \vcentcolon \SchwartzFunctions{\RR^n} \to X$ is linear and continuous by definition.
+  The mapping $v \vcentcolon \SchwartzFunctions{\RR^n} \to Y$ is linear and continuous by definition.
   By \cref{theorem:integral-commutes-with-operator},
-  the composite mapping $x \mapsto v(\tau_x \tilde{f})$ is a $\mu$-integrable function $\RR^n \to X$, and
+  the composite mapping $x \mapsto v(\tau_x \tilde{f})$ is a $\mu$-integrable function $\RR^n \to Y$, and
   \begin{equation}
     \label{equation:general-bochner-appears}
     \int v(\tau_x \tilde{f}) \, d\mu(x) = v \parens[\bigg]{\int \tau_x \tilde{f} \, d\mu(x)}
@@ -293,11 +297,11 @@ is defined in the same way as in \cref{definition:convolution-distribution-test-
   and the proof is complete.
 \end{proof}
 
-Let us point out that even in the special case that $X$ is a Banach space
+Let us point out that even in the special case that $Y$ is a Banach space
 the integral on the right hand side of~\eqref{equation:general-bochner-appears}
 only has meaning as a generalized Bochner integral,
 since the integrand takes values in $\SchwartzFunctions{\RR^n}$,
 which is not a Banach space.
 We could not have performed this step with the ordinary Bochner integral.
 
-%\nomenclature[B]{$\BoundedLinearOperators{X,Y}$}{bounded linear operators from $X$ to $Y$\nomnorefpage}
+% vim: syntax=mytex
-- 
cgit v1.2.3-70-g09d2

