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Some remarks on the space of differences of sublinear functions. (English) Zbl 0826.49011
For a real Banach space $$X$$ (with topological dual $$X'$$) let $${\mathcal D}(X)$$ denote the space of differences of real-valued sublinear functions. The authors show that if $$X$$ is separable, then there exists a countable family of seminorms such that $${\mathcal D}(X)$$ becomes a Fréchet space. In particular, if $$X= \mathbb{R}^n$$, the construction yields a norm such that $${\mathcal D}(\mathbb{R}^n)$$ becomes a Banach space.
The following problem is also considered. Let $$f: U\to \mathbb{R}$$, $$U\subseteq X$$ open, be directionally differentiable at $$x_0\in U$$, with $${df\over dg}\bigl|_{x_0}$$ denoting the directional derivative of $$f$$ at $$x_0$$ in the direction $$g$$. A sublinear functional $$p: X\to \mathbb{R}$$ is called an upper convex approximation (u.c.a.) of $$f$$ at $$x_0$$ if $p(g)\geq {df\over dg}\Bigl|_{x_0}\qquad\text{for all}\quad g\in X,$ and a family $$\Phi_{f, x_0}$$ of u.c.a. of $$f$$ at $$x_0$$ is called exhaustive if $\inf_{p\in \Phi_{f, x_0}} p(g)= {df\over dg}\Bigl|_{x_0}\qquad\text{for all}\quad g\in X.$ The authors show that the family of all $$p: g\mapsto \kappa|g|- \langle l, g\rangle$$, $$g\in X$$, $$\kappa\in \mathbb{R}_+$$, $$l\in X'$$, contains an exhausitve family of u.c.a. of $$f$$ at $$x_0$$ provided that $$f$$ is quasidifferentiable at $$x_0$$ and $$X'$$ is smooth.

##### MSC:
 49J52 Nonsmooth analysis 26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives 90C30 Nonlinear programming
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