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RETRACTION: forgot to mention separable.

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Bokuan Li
2026-10-29 20:39:41 -04:00
parent bbfd061257
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src/fa/convex/legendre.tex
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@@ -269,7 +269,7 @@
\begin{corollary} \begin{corollary}
\label{corollary:separable-legendre} \label{corollary:separable-legendre}
Let $\dpn{E, F}{\lambda}$ be a duality over $\real$ and $f: E \to (-\infty, \infty]$ with $f \ne \infty$ be convex and lower semicontinuous, then there exists $\seq{(\phi_n, \alpha_n)} \subset F \times \real$ such that for each $x \in E$, Let $\dpn{E, F}{\lambda}$ be a duality over $\real$ with $E$ being separable, and $f: E \to (-\infty, \infty]$ with $f \ne \infty$ be convex and lower semicontinuous, then there exists $\seq{(\phi_n, \alpha_n)} \subset F \times \real$ such that for each $x \in E$,
\[ \[
f(x) = \sup_{n \in \natp} \dpn{x, \phi_n}{\lambda} - \alpha_n f(x) = \sup_{n \in \natp} \dpn{x, \phi_n}{\lambda} - \alpha_n
\] \]