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By William M. Boothby (Editor)

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We have r* = fco ifeR''). 100) Hence, for a function f \ X -> R, we have f = /** (or, equivalently, f = fco) if and only if f is the supremum of a set of continuous affine functions. For any function / : X ^- R, where X is a linear space, the ("effective") domain of / is the set d o m / := {x e X\ f(x) < +oo}. 101) We recall that if X is a locally convex space, then the support set supp / of any function / : X -> /? is the subset of X* defined by s u p p / := {O G X * | 0 < / } . 102) Also, the (X*, R)-support set Supp / of any function / : X ^- /?

3 Duality for best approximation by elements of convex sets 43 with s u p O ( G ) < <^(xo) or supOCG) < O(jco) omitted, fails (in this example, supOo(G) =d >0= o(xo)). 249) OeX*\{0} yeX supO(G)<(jco) 0(y)=supO(G) where HG,XO denotes the set of all hyperplanes that quasi-support the set G and that strictly separate G and JCQ. 248) has a similar interpretation, with "strictly separate" replaced by "separate" and sup 0 ( G ) < 4>(jco) replaced by sup 0 ( G ) < O(xo). 242). 251). 8. Let X be a normed linear space, G a convex subset ofX, C G .

65)), where ^ G X*\{0}, s u p O ( C ) G R, quasi-supports the set C and does not contain C (respectively, int C). 8 below). 4. 67) Then, for any XQ ^ V^j, we have d — O(xo) dist(xo, V^ c y^j, j , we have dist(xo, // dist(xo, V^,d)' On the other hand, for any y G V<^^d the segment [jc, y] contains the a point y' G //o,^, so dist(xo, H^^d) £ Ik — y'W S \\x — JII, whence dist(jco, //,d) = dist(xo, H<^,d) = [TTT; (1-69) ll^ll But since XQ ^ V(D,J.

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