From b7f93c1110e2fe0aceb6084ae075e87c5aaee692 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Thu, 25 Jun 2026 13:32:44 -0400 Subject: [PATCH] Updated the legendre corollary. --- src/fa/convex/legendre.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/src/fa/convex/legendre.tex b/src/fa/convex/legendre.tex index 4709822..15098b3 100644 --- a/src/fa/convex/legendre.tex +++ b/src/fa/convex/legendre.tex @@ -257,7 +257,7 @@ \begin{corollary} \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 $x \in E$, \[ f(x) = \sup_{n \in \natp} \dpn{x, \phi_n}{\lambda} - \alpha_n \] @@ -265,7 +265,7 @@ \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}, \[ - f^{**}(x) = \sup\bracs{\dpn{x, \phi}{\lambda} - \alpha|(\phi, \alpha) \in F \times \real, (\phi, \alpha) \le f} + f(x) = f^{**}(x) = \sup\bracs{\dpn{x, \phi}{\lambda} - \alpha|(\phi, \alpha) \in F \times \real, (\phi, \alpha) \le f} \] for all $x \in E$. By \autoref{proposition:separable-dual},