Let $X$ be a Banach space, and let $X^*$ be the dual space of $X$ . For a function $f \colon X \rightarrow \mathbb{R}$ , and $x\in X$ , let us define $$ \partial f(x) = \{r^* \in X^* \; : f(x) - f(y) \leq r^\ast(x - y) \; \ \mbox{for all} \ y \in X\}. $$ If $\partial f(x)$ is non-empty, then $f$ is subdifferentiable at $x \in X$ , and if $\partial f(x)$ is non-empty for all $x$ , then $f$ is subdifferentiable [1,2].

Bibliography

1

C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing Company, 2002.

2

R.T. Rockafellar, Convex Analysis, Princeton University Press, 1996.