By Stefano Pigola, Marco Rigoli, Alberto G. Setti

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A>GXM|*||=i,jG/? OGXM|ci>||z=i,je/? 252) Proof. 252). 16. 253) 44 1. 4. where HQ ^^ denotes the set of all hyperplanes that strictly separate G and XQ. 4b). 5 (which is very convenient for applications in various concrete spaces, since for these spaces the general form of continuous linear functions O e Z* is well known and simple). This basic idea, which we shall call the reduction principle, will be applied later also to nonconvex approximation and will be extended to convex and nonconvex optimization. *

82) 20 1. Preliminaries and the bipolar of C is C°° := (C°)°. The classical "bipolar theorem" states that, for any subset C of a locally convex space X, we have C°° = co(C U {0}); hence, a set C containing 0 is closed and convex if and only if C°° = C. + = [0, +oo], it is necessary to give a precise meaning to expressions like oo — oo and 0 X 00. , Moreau [164]) that the usual addition -\- on R = (—oo, +oo) admits two natural extensions to R, + and +, called upper and lower addition, respectively, defined by a+b = afb = a-\-b if either/?

110), we have 3/(0) = {CD G X*| CD < / } = supp/. 118) where 5x* = {O G Z * | | | 0 | | < 1}, the unit ball of X*. 8. 99) is attained for some OQ G X* if and only if 4>o e df(zo). g.. Holmes [106, p. 5. If X is a locally convex space and f,h:X^^R are convex functions such that one of them is continuous at some point 6>/dom / Pi dom h, then Hf + h)(xo) = df(xo) + dh(xo) {xo e X). 9. It is easy to see that here the + signs are just the usual sums. We recall that if X is a linear space and / : X -> R is a.