A note on norm attaining functionals

Abstract

Abstract. We are concerned in this paper with the density of functionals which do not attain their norms in Banach spaces. Some earlier results given for separable spaces are extended to the nonseparable case. We obtain that a Banach space X is reflexive if and only if it satisfies any of the following properties: (i) X admits a norm · with the Mazur Intersection Property and the set NA · of all norm attaining functionals of X * contains an open set, (ii) the set NA 1 · of all norm one elements of NA · contains a (relative) weak* open set of the unit sphere, (iii) X * has C * P CP and NA 1 · contains a (relative) weak open set of the unit sphere, (iv) X is W CG, X * has CP CP and NA 1 · contains a (relative) weak open set of the unit sphere. Finally, if X is separable, then X is reflexive if and only if NA 1 · contains a (relative) weak open set of the unit sphere.

James' weak compactness theorem: an exposition

“…A special case of this conjecture was proven in [21]. For some further results in this direction see [6].…”

One-sided James' compactness theorem

“…If we combine condition (d) and inequality (13) we deduce that (x * m ) m∈N is bounded. Fix x * ∞ a weak * -cluster point of the sequence (x * m ) m∈N .…”

“…Baire's category theorem implies that there are M ∈ N, a ∈ L D and δ > 0 such that a + δB E ⊂ H M , which yields that (13) b * , a + δ b * ≤ M for every b * ∈ co (B).…”

“…The well-known James' theorem [12] claims that a bounded closed convex set C in a Banach space E is weakly compact if and only if every x * ∈ E * attains its maximum on C. There are many reasons why this theorem has attracted the attention of so many researchers: the great number of applications, the search for simpler proofs (see [9], [14], [15], [16]) and the concern of strengthening it (see [3], [19]); in this sense, there are results (see [5], [13]) showing that in order to prove that B E is weakly compact (i.e. E reflexive) we do not have to check that every functional of E * attains its maximum on B E , but only those functionals belonging to a certain set which is "big enough" in a topological sense (for instance, a relative weak * -open subset of the unit sphere of E * ).…”

Связность И Солнечность В Задачах Наилучшего И Почти Наилучшего Приближения

“…Вообще, если X нерефлексивно, то множество функционалов из X * , не достигающих своей нормы, достаточно велико. Отметим следующий результат (M. Хименес-Севилла, Х. Морено [128]). Пусть Xсепарабельное банахово пространство.…”