Banach-space convex subgradients and dual pairings

Let $X$ be a real Banach space and let $X^{\ast }$ be its continuous real dual. The dual pairing is evaluation, written $⟨\xi ,x{⟩}_{X^{\ast },X}:=\xi (x)$ for $\xi \in X^{\ast }$ and $x\in X$. Let $R:X\to (-\mathrm{\infty },+\mathrm{\infty }]$ be a proper convex functional: $R$ never takes the value $-\mathrm{\infty }$, its effective domain $\mathrm{dom}R:=\{x\in X:R(x)<+\mathrm{\infty }\}$ is nonempty, and $R((1-t)x+tz)\le (1-t)R(x)+tR(z)$ for $x,z\in X$ and $t\in [0,1]$, with $0\cdot (+\mathrm{\infty }):=0$ for this convex-combination formula. For $x\in \mathrm{dom}R$, its subdifferential is

$\mathrm{\partial }R(x):=\{\xi \in X^{\ast }:R(z)\ge R(x)+⟨\xi ,z-x{⟩}_{X^{\ast },X}\text{ for every }z\in X\}\mathrm{.}$

An element of $\mathrm{\partial }R(x)$ is a subgradient of $R$ at $x$. Set $\mathrm{\partial }R(x):=\mathrm{\varnothing }$ when $x\mathrm{\notin }\mathrm{dom}R$. The subdifferential may be empty at a point of $\mathrm{dom}R$; no nonemptiness assertion is part of this definition.

Real Banach spaces

The continuous dual and evaluation pairing

Convex extended-real functionals

Properness and effective domain

Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, §1

Jonathan M. Borwein and Jon D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples, Chapter 4