# subdifferentiable mapping

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].

## References

• 1 C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing Company, 2002.
• 2 R.T. Rockafellar, Convex Analysis, Princeton University Press, 1996.
Title subdifferentiable mapping SubdifferentiableMapping 2013-03-22 14:31:19 2013-03-22 14:31:19 matte (1858) matte (1858) 13 matte (1858) Definition msc 39B62 msc 52-00