# Introduction to Optimization Theory in a Hilbert Space by A.V. Balakrishnan

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By A.V. Balakrishnan

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Also, and Yo = lim y Similarly, C'l als for C'l u v = TT STII>(T- s)Bu (s)ds 0 n 0 is also seen to be closed. and C'l v respecti vely. ) denote the support function- u v Then we can state a set of neces sary and suffi- cient conditions for the game to be completed in time T: Theorem Let T be a game completion time, starting with x(O) at time zero. A necessary and sufficient condition for this is that for every unit vector e in E k : f (e) - f (-e) < d-[e,TTII>(T)x(O)] v u Proof (1. 42) Before we give a formal proof, let us explore the condition for completion.

P. 1> - , 0, xeA Since, in particular then n ,'Lgk[CP(x'Yk)-c] < gkPk 1 it follow s that gk 2 ° Hence it follows that, (dividing through by Sup cp(x,y) - c < o,X e A or, x Inf Sup yeB xeA cp(x, y) < Sup cp(x, y') < c - xeA which is a contradiction of the definition of c. We have only to note now that the sets A are compact in the weak topology in H. property", it follows that n A y y y and A, being convex closed and bounded, Since the A is not empty. } be any countable subcollection. \ y1 For each n there exists x such that n n x e n A n i=l Yi Then, since A is bounded, we can find a subsequence x But each A y being convex and closed, we know that x {xj } e Ayk for every k.

Problem Show that if T is compact. so are T*. T*T. TT* and AT. TB where A and B are linear bounded. Problem Let L be a compact linear transformation mapping" H into H. closed bounded convex set in H. Let C be a Show that the set LC is closed. Problem Let L be a compact operator mapping H into H. convex set in H. ) Show that there is a (unique) minimizing element of minimal norm, which is given in fact by Limit €->o (L*L + el) -1 L*y Problem If T is compact, then the range of T is separable (has a countable dense subset) • Spectral Properties of Compact Operators A fundamental property of compact operators relates to their spectra.