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author | Justin Gassner <justin.gassner@mailbox.org> | 2024-09-25 00:26:13 +0200 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-09-25 00:26:13 +0200 |
commit | fadd6961c92393d86de69ec468f0a15a2f320252 (patch) | |
tree | 44e742f0fdb606f28cb009ec4f471c7a28c9c196 /analytic2.tex | |
parent | e1a26e4528eb7b9c2f462562c8265cf963f34dfb (diff) | |
download | master-fadd6961c92393d86de69ec468f0a15a2f320252.tar.zst |
Diffstat (limited to 'analytic2.tex')
-rw-r--r-- | analytic2.tex | 9 |
1 files changed, 5 insertions, 4 deletions
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 |