subdifferentiable mapping

Let $X$ be a Banach space, and let $X^{*}$ be the dual space of $X$. For a function $f:X\to ℝ$, and $x\in X$, let us define

$$\partial f(x)=\{r^{*}\in X^{*}:f(x)-f(y)\le r^{\ast }(x-y)\text{for all}y\in X\}.$$

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].

Title	subdifferentiable mapping
Canonical name	SubdifferentiableMapping
Date of creation	2013-03-22 14:31:19
Last modified on	2013-03-22 14:31:19
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	13
Author	matte (1858)
Entry type	Definition
Classification	msc 39B62
Classification	msc 52-00