From 25d4c7eadd4acadfb7fa61130512a0db1134e374 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Thu, 29 Oct 2026 20:39:41 -0400 Subject: [PATCH] RETRACTION: forgot to mention separable. --- src/fa/convex/legendre.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/fa/convex/legendre.tex b/src/fa/convex/legendre.tex index f10889d..5f1a7de 100644 --- a/src/fa/convex/legendre.tex +++ b/src/fa/convex/legendre.tex @@ -269,7 +269,7 @@ \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 $\real$ 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 $x \in E$, \[ f(x) = \sup_{n \in \natp} \dpn{x, \phi_n}{\lambda} - \alpha_n \]