From daa8ca73cd506b68668cad9e78516fbb7c779954 Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Sun, 24 Nov 2024 11:03:54 -0500
Subject: [PATCH] {examples,mjo-linear_algebra}.tex: add \Isom for isometries
---
examples.tex | 7 +++++--
mjo-linear_algebra.tex | 29 +++++++++++++++++++++++++++++
2 files changed, 34 insertions(+), 2 deletions(-)
diff --git a/examples.tex b/examples.tex
index 9096b70..accec5d 100644
--- a/examples.tex
+++ b/examples.tex
@@ -254,8 +254,11 @@
\end{align*}
\normalsize
- If $V$ is an algebra, then $\Der{V}$ is the space of all
- (linear) derivations on $V$.
+ If $V$ is an algebra, then $\Der{V}$ is the space of all (linear)
+ derivations on $V$. We also have the group of isometries on $V$,
+ if $V$ has a metric: $\Isom{V}$. More generally, if $V$ and $W$
+ are both metric spaces, then we can represent the isometries from
+ one to the other by $\Isom[W]{V}$.
\end{section}
\begin{section}{Listing}
diff --git a/mjo-linear_algebra.tex b/mjo-linear_algebra.tex
index 062121f..32d4d6c 100644
--- a/mjo-linear_algebra.tex
+++ b/mjo-linear_algebra.tex
@@ -196,4 +196,33 @@
}
\fi
+
+% The set of all isometries from its first argument to its second
+% (optional) argument, both assumed to be at least normed spaces, and
+% more likely Hilbert spaces. The norms are implicit, i.e. not
+% included in the arguments. If the optional argument is omitted, you
+% get the isometries from the first argument to itself AKA its
+% isometry group.
+\newcommand*{\Isom}[2][]{
+ \operatorname{Isom}\of{ {#2}
+ \if\relax\detokenize{#1}\relax
+ {}%
+ \else
+ {,{#1}}%
+ \fi
+ }
+}
+\ifdefined\newglossaryentry
+ \newglossaryentry{Isom}{
+ name={\ensuremath{\Isom{V}}},
+ description={the group of isometries on $V$},
+ sort=Isom
+ }
+ \newglossaryentry{Isom2}{
+ name={\ensuremath{\Isom[W]{V}}},
+ description={the set of isometries from $V$ to $W$},
+ sort=Isom
+ }
+\fi
+
\fi
--
2.45.3