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Erschienen in:
2013 | OriginalPaper | Buchkapitel
8. Additional exercises for Part I
verfasst von : Francis Clarke
Erschienen in: Functional Analysis, Calculus of Variations and Optimal Control
Verlag: Springer London
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Abstract
The chapter consists of several dozen exercises on functional analysis, the first three of which are the following:
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Give an example of a locally Lipschitz function \(f:X\to\,{\mathbb{R}}\) defined on a Hilbert space X which is not bounded below on the unit ball. Could such a function be convex?
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Let A be a bounded subset of a normed space X. Prove that \({\text{co}}\,\big(\partial A\big)\,\supset\, { \text{cl}\,}\, A\,. \)
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Let X be a normed space, and let A be an open subset having the property that each boundary point x of A admits a supporting hyperplane; that is, there exist 0 ≠ ζ x ∈ X ∗ and \(c_{x}\in\,{\mathbb{R}}\) such thatProve that A is convex. Prove that the result remains valid if the hypothesis “A is open” is replaced by “A is closed and has nonempty interior.”$$\langle \, \zeta_{\,x}\,, x\,\rangle \: = \:c_x\:,\;\; \langle \, \zeta_{\,x}\,, u\,\rangle \: \leqslant \:\, c_x~\; \forall \,u\in\, A\,. $$