This commit is contained in:
Bokuan Li
2026-03-06 14:06:15 -05:00
parent 173727665b
commit 5034bc4220
109 changed files with 1184 additions and 410 deletions

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@@ -17,11 +17,12 @@
\begin{proof}
(1) $\Rightarrow$ (2): TODO
(2) $\Rightarrow$ (3): By \ref{proposition:measurable-simple-separable-norm}.
(2) $\Rightarrow$ (3): By \autoref{proposition:measurable-simple-separable-norm}.
(3) $\Rightarrow$ (1): For each $\phi \in E^*$, $\phi \circ f = \limv{n}\phi \circ f_n$ is measurable by \ref{proposition:limit-measurable}. Since
(3) $\Rightarrow$ (1): For each $\phi \in E^*$, $\phi \circ f = \limv{n}\phi \circ f_n$ is measurable by \autoref{proposition:limit-measurable}. Since
\[
f(X) \subset \ol{\bigcup_{n \in \natp}f_n(X)}
\]
and each $f_n$ is finitely-valued, $f(X)$ is separable.
\end{proof}