On the global shape of continuous convex functions on Banach spaces

TL;DR

The paper investigates the global structure of continuous convex functions on Banach spaces, proving a unique decomposition for separable spaces and highlighting limitations in nonseparable cases.

Contribution

It establishes a unique decomposition of convex functions on separable Banach spaces and demonstrates the failure of this structure in nonseparable spaces.

Findings

Existence of a unique closed linear subspace for convex functions on separable spaces.

Decomposition of convex functions into a sum involving a convex function with infinite limit behavior.

Failure of this decomposition in nonseparable Banach spaces, even in Hilbert spaces.

Abstract

We make some remarks on the global shape of continuous convex functions defined on a Banach space $Z$. Among other results we prove that if $Z$ is separable then for every continuous convex function $f:Z\to\mathbb{R}$ there exist a unique closed linear subspace $Y_f$ of $Z$ such that, for the quotient space $X_f :=Z/Y_{f}$ and the natural projection $\pi:Z\to X_f$, the function $f$ can be written in the form $$ f(z)=\varphi(\pi(z)) +\ell(z) \textrm{ for all } z\in Z, $$ where $\ell_{f}\in X^{*}$ and $\varphi:X_f\to\mathbb{R}$ is a convex function such that $\lim_{t\to\infty}\varphi(x+tv)=\infty$ for every $x, v\in X_f$ with $v\neq 0$. This kind of result is generally false if $Z$ is nonseparable (even in the Hilbertian case $Z=\ell_{2}(\Gamma)$ with $\Gamma$ an uncountable set).