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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.

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