Niveau: Supérieur, Master Chapter 8 Additional Exercises for Part I Analyse Master 1 : Cours de Francis Clarke (2011) 8.1 Exercise. Give an example of a lower semicontinuous function defined on a Hilbert space which is not bounded below on the unit ball. 8.2 Exercise. Let A be a bounded subset of a normed space X . Prove that co ?A ? cl A. 8.3 Exercise. Let X be an infinite dimensional Banach space. Prove that any vector space basis for X is not countable. By considering ?c , observe that this fact fails for infinite dimensional normed spaces that are not complete. 8.4 Exercise. Let ?n be a sequence of real numbers, and let 1 p ?. Suppose that, for every x = (x1,x2, . . .) in l p, we have ?n1 |?n| |xn| < ?. Prove that the sequence ? belongs to l q, where q is the exponent conjugate to p. 8.5 Exercise. We give a direct definition of the normal cone when A is a subset of Rn, one that does not explicitly invoke polarity to the tangent cone. Let ? ? A. Show that ? ? NA(?) if and only if, for every ? > 0, there is a neighborhood V of ? such that ? ,u?? ? u?? ?u ? A ? V.

• convex
• space basis
• let
• tangent cone
• tbx ? ?
• uniform approxi- mation
• banach space
• hessian matrix ?2

Clarke, Convex, Tangent cone, Banach space

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