Optimization of functions on certain subsets of Banach spaces

Abstract

To motivate our results we recall the following theorem of Bishop and Phelps [_3] : Theorem. Let D be a closed, bounded and convex subset of a Banach space X. Then the set of support functionals of D (that is, the set of elements of the dual X* of X that attain their suprema on D) is a norm dense subset of X*.Phrasing this another way, if x* is any element of X* and 6 is any positive number then there exists a y* in X* with ily*H <c5 such that x*+y* attains its supremum on D. Of course, here, and in everything that follows, we may make the obvious dual statements about functions that attain their infimum.The problem we are interested in is the following: (Q) let D be a subset of a Banach space X, Yanother Banach space, andf : D--, Ya function from D to Y; find such functions f and sets D such that for every 6 >0, there exists a continuous, linear function T:X~ Y of rank one with [I TI[ <6 and such that f+ Tattains its supremum on D. We give a complete answer to this problem. Basically, the most important thing is the set D. If it satisfies a certain property (what we call an RNP set) thenfmay be any reasonable mapping and Yarbitrary. Conversely, if D is not well-behaved (in our sense) thenfmay be very nice (for example, continuous and convex), Y the real numbers, and f will not have close, norm attaining approximations even in a sense more general than that of (Q).As examples of the sort of results we obtain let us recall the following theorem of Lindenstrauss [8] : Theorem. Let X and Y be Banach spaces, W a weakly compact subset of X, and let 5~ (X, I1) denote the space of continuous, .linear functions from X to Y with the usual operator norm. Then the set of elements of ,~(X, Y) that attain their suprema in norm on W is a norm dense subset of f(X, Y).Recently, in a remarkable result Bourgain [5] has shown that for RNP sets (definition below) the characterization is exact: Theorem 1. Let D be a closed, bounded, convex and symmetric subset of the Banach space X. Then D is an RNP set if and only iffor every Banach space ~ the subset of

Суб- И Суперградиенты Огибающих, Полунепрерывных Замыканий И Пределов Последовательностей Функций

“…Здесь мы используем эту теорему, чтобы доказать существование таких точек минимума для некоторых произвольно малых линейных возмущений исходной оптимизационной задачи. Именно, мы доказы-ваем вариант гладкого вариационного принципа Стегалла [46] в предположении, что двойственное пространство X * является гладким. Детальное введение в ва-риационные принципы можно найти в книге Р. Р. Фелпса [38] и работах [6], [28], [27].…”

“…Здесь мы демонстрируем применение теоремы 4.1 для доказательства клас-сического результата [46] о существовании единственной проекции на замкну-тое подмножество гильбертова пространства для почти всех точек x. Теорема 4.4. Пусть S ⊂ H -замкнутое подмножество гильбертова про-странства.…”

Norm attaining and numerical radius attaining operator

“…Tite second autitor wants to express itis gratitude to tite organizing committee of tite Functional Analysis Meeting (El Escorial, June [13][14][15][16][17][18]1988) for inviting itim to tite meeting and giving hm a nice opportunity to expose tite results in titis paper.…”

Solutions de viscosité des equations de Hamilton-Jacobi en dimension infinie. Cas stationnaire

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