RETRACTION: forgot to mention separable.

2026-10-29 20:39:41 -04:00
parent bbfd061257
commit 25d4c7eadd
1 changed files with 1 additions and 1 deletions
@@ -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 $x \in E$, 
    \[
    f(x) = \sup_{n \in \natp} \dpn{x, \phi_n}{\lambda} - \alpha_n
    \]