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Updated the legendre corollary.
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Bokuan Li
2026-06-25 13:32:44 -04:00
parent 2c4a8c9d22
commit b7f93c1110
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src/fa/convex/legendre.tex
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@@ -257,7 +257,7 @@
\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 $\real$ 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 $x \in E$,
\[ \[
f(x) = \sup_{n \in \natp} \dpn{x, \phi_n}{\lambda} - \alpha_n f(x) = \sup_{n \in \natp} \dpn{x, \phi_n}{\lambda} - \alpha_n
\] \]
@@ -265,7 +265,7 @@
\begin{proof} \begin{proof}
For each $(\phi, \alpha) \in F \times \real$, denote $(\phi, \alpha) \le f$ if $\dpn{\cdot, \phi}{\lambda} - \alpha \le f$. By the \hyperref[Fenchel-Moreau Theorem]{theorem:fenchel-moreau}, For each $(\phi, \alpha) \in F \times \real$, denote $(\phi, \alpha) \le f$ if $\dpn{\cdot, \phi}{\lambda} - \alpha \le f$. By the \hyperref[Fenchel-Moreau Theorem]{theorem:fenchel-moreau},
\[ \[
f^{**}(x) = \sup\bracs{\dpn{x, \phi}{\lambda} - \alpha|(\phi, \alpha) \in F \times \real, (\phi, \alpha) \le f} f(x) = f^{**}(x) = \sup\bracs{\dpn{x, \phi}{\lambda} - \alpha|(\phi, \alpha) \in F \times \real, (\phi, \alpha) \le f}
\] \]
for all $x \in E$. By \autoref{proposition:separable-dual}, for all $x \in E$. By \autoref{proposition:separable-dual},