Download e-book for kindle: Abstract Convex Analysis (Wiley-Interscience and Canadian by Ivan Singer

By Ivan Singer

ISBN-10: 0471160156

ISBN-13: 9780471160151

This e-book examines summary convex research and offers the result of contemporary examine, in particular on parametrizations of Minkowski style dualities and of conjugations of variety Lau. It explains the most recommendations via circumstances and exact proofs.

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Additional info for Abstract Convex Analysis (Wiley-Interscience and Canadian Mathematics Series of Monographs and Texts)

Example text

Let us recall that if X = R", or, more generally, if X is a locally convex space with conjugate space X*, and f : X —> R the Fenchel conjugate of f is the function f* : X* —> T? defined by , f* (w) = sup [w(x) — f (x)) X (w E X * ). 46) by an arbitrary "coupling function" go : X x W —> R. 47) EX where ± is the "lower addition" on R [205], that is, the extension of the usual addition + by the convention: -koo ± (—oc) = —oc ± (-Foc) = —oc. 46), such as those "of type Lau" (in which go takes only the values 0 and —oc).

3 Abstract Convexity of Elements of Complete Lattices 13 be a complete lattice, and let M be a subset of E. An element e G E is said to be convex with respect to M (or, briefly, M-convex) if there exists a subset M' of M such that e = sup M'. 38) yields that —oc is convex with respect to M. Also, it is obvious (by taking M' to be a singleton {m}), that every ni G »1 is M-convex. 9) of convexity of subsets G of X with respect to a given family M of subsets of X. , for lattice (E, = (Rx , f2(x) for all x G X), we , f2 X —> T?

Let us recall that if X = R", or, more generally, if X is a locally convex space with conjugate space X*, and f : X —> R the Fenchel conjugate of f is the function f* : X* —> T? defined by , f* (w) = sup [w(x) — f (x)) X (w E X * ). 46) by an arbitrary "coupling function" go : X x W —> R. 47) EX where ± is the "lower addition" on R [205], that is, the extension of the usual addition + by the convention: -koo ± (—oc) = —oc ± (-Foc) = —oc. 46), such as those "of type Lau" (in which go takes only the values 0 and —oc).

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Abstract Convex Analysis (Wiley-Interscience and Canadian Mathematics Series of Monographs and Texts) by Ivan Singer


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