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

---
 analytic2.tex | 9 +++++----
 1 file changed, 5 insertions(+), 4 deletions(-)

(limited to 'analytic2.tex')

diff --git a/analytic2.tex b/analytic2.tex
index 13b6700..0648130 100644
--- a/analytic2.tex
+++ b/analytic2.tex
@@ -22,7 +22,7 @@ that is, in the open disc with radius $t$ centered in the origin of the complex
 This is a well-known consequence of the convergence behavior of power series.
 
 \begin{definition}{Analyticity of Vector-Valued Functions}{}
-  Let $G \subset \CC$ be open and let $\hilb{H}$ be a Hilbert space.
+  Let $G \subset \CC$ be open and let $\hilb{X}$ be a Banach space.
   A function $f : G \to \hilb{H}$ is called
   \begin{itemize}
     \item \emph{strongly analytic} at $a \in G$, if the limit
@@ -38,6 +38,10 @@ This is a well-known consequence of the convergence behavior of power series.
   \end{itemize}
 \end{definition}
 
+\begin{lemma}{Uniform Boundedness Theorem}{uniform-boundedness-theorem}
+  If a collection of bounded linear operators from a Banach space into a normed space is pointwise bounded, then it is uniformly bounded.
+\end{lemma}
+
 \begin{lemma}{Equivalence of Weak and Strong Analyticity}{}
   Let $G \subset \CC$ be open.
   Then a Banach space-valued function is strongly analytic on $G$ if and only if it is weakly analytic on $G$.
@@ -130,6 +134,3 @@ This is a well-known consequence of the convergence behavior of power series.
   \end{equation*}
   has a positive radius of convergence $t>0$.
 \end{myproof}
-
-\chapterbib
-\cleardoublepage
-- 
cgit v1.2.3-70-g09d2

