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In this paper we apply the Bishop-Phelps Theorem to show that if X is a Banach space and G µ X is a maximal subspace so that G ? = fx ⁄ 2 X ⁄ jx ⁄ (y) = 0; 8y 2 Gg is an Lisummand in X ⁄ , then L 1 (›;G) is contained in a maximal proximinal subspace of L 1 (›;X).

In this work we introduce the concept of convex numerical radius for a continuous and linear operator in a Banach space, which generalizes that of the classical numerical radius. Besides studying some of its properties, we give a version of James's sup theorem in terms of convex numerical radius attaining operators.

A selection theorem concerning support points of convex sets in a Banach space is proved. As a corollary we obtain the following result. Denote by \({\mathcal{BCC}(X)}\) the metric space of all nonempty bounded closed convex sets in a Banach space X. Then there exists a continuous mapping \({S : \mathcal{BCC}(X) \rightarrow X}\) such that S(K) is a support point of K for each \({K \in \mathcal{BCC}(X)}\). Moreover, it is possible to prescribe the values of S on a closed discrete subset of \({\mathcal{BCC}(X)}\).

We prove that if the Bishop–Phelps Theorem is correct for a uniform dual algebra R of operators in a Hilbert space, then the algebra R is selfadjoint.

The Bishop-Phelps Theorem asserts that the set of functionals which attain the maximum value on a closed bounded convex subsetS of a real Banach spaceX is norm dense inX*. We show that this statement cannot be extended to general complex Banach spaces by constructing a closed bounded convex set with no support points.

We answer a question posed by R. Aron, C. Finet and E. Werner, on the bilinear version of the Bishop-Phelps theorem, by exhibiting an example of a Banach spaceX such that the set of norm-attaining bilinear forms onX×X is not dense in the space of all continuous bilinear forms.

Banach contributed to the theory of orthogonal series and made innovations in the theory of measure and integration, but his most important contribution was in functional analysis. Of his published works, his Theorie des operations lineaires (1932; “Theory of Linear Operations”) is the most important. Banach and his coworkers summarized the previously developed concepts and theorems of functional analysis and integrated them into a comprehensive system. Banach himself introduced the concept of [complete] normed linear spaces, which are now known as Banach spaces. He also proved several fundamental theorems in the field, and his applications of theory inspired much of the work in functional analysis for the next few decades . (Encyclopedia Britannica) Continuity and subdifferentials Throughout the rest of the book X denotes a normed or a Banach space. We assume familiarity with basic results and language from functional analysis, and we will often draw upon the classical spaces l p , c 0 and others for examples. Notwithstanding, we will sketch some of the classic convexity theorems in our presentation. We write X * for the real dual space of continuous linear functionals, and denote the unit ball and sphere by B X ≔ { x ∈ X : ∥ x ∥ ≤ 1} and S X ≔ { x ∈ X : ∥ x ∥ = 1}, respectively. We say that a subset C of X is convex if, for any x, y ∈ C and any λ ∈ [0, 1], λ x + (1 - λ) y ∈ C .

Let K be a bounded closed convex subset of a real Banach space of dimension at least two. Then the set of the support points of K is pathwise connected and the set Σ1(K) of the norm-one support functionals of K is uncountable in each nonempty open set that intersects the dual unit sphere. In particular, the set Σ 1(K) is always uncountable, which answers a question posed by L. Zajicek.

Using an operator-theoretical approach we show that the famous Bishop-Phelps theorem cannot be extended to general complex Banach spaces. Quaestiones Mathematicae 23(2000), 187–191

What follows is a history of the Bishop-Phelps theorem on the density of support functionals, together with a sketch of its proof This is followed by descriptions of a number of extensions and applications, plus two basic open questions.

In this report1, we present an order-theoretic version of the Cantor theorem. This result, which is based on the interplay of the notions of partial order and of completeness, permits to give a unified and simplified account to a long list of results related to the Bishop– Phelps theorem. We survey briefly only its simplest applications and refer the reader to [10] for a complete presentation of the results.

In this paper we show that the Bishop–Phelps–Bollobas theorem fails for bilinear forms on l 1 × l 1 , while it holds for linear operators from l 1 to l ∞ .

We give a new and a simple proof of proximinality for M-ideals. Unlike the known proofs, our proof derives proximinality of M-ideals directly from the definition of an M-ideal, using the Bishop-Phelps theorem.

We give some general constructions of supportless convex subsets of normed spaces and pose a number of open questions.

The following problem is considered. LetX andY be Banach spaces. Are those operators fromX toY which attain their norm on the unit cell ofX, norm dense in the space of all operators fromX toY? It is proved that this is always the case ifX is reflexive. In general the answer is negative and it depends on some convexity and smoothness properties of the unit cells inX andY. As an application a refinement of the Krein-Milman theorem and Mazur’s theorem concerning the density of smooth points, in the case of weakly compact sets in a separable space, is obtained.

Let Y be a Banach space and (Ω,Σ,μ) be a σ-finite measure space, where Σ is an infinite σ-algebra of measurable subsets of Ω. We show that if the couple (L1(μ),Y) has the Bishop–Phelps–Bollobás property for operators, then Y has the AHSP. Further, for a Banach space Y with the Radon–Nikodým property, we prove that the couple (L1(μ),Y) has the Bishop–Phelps–Bollobás property for operators if and only if Y has the AHSP.

Motivated by the recent work on conditional risk measures, this paper studies the Ekelandʼs variational principle for a proper, lower semicontinuous and lower bounded L¯0-valued function, where L¯0 is the set of equivalence classes of extended real-valued random variables on a probability space. First, we prove a general form of Ekelandʼs variational principle for such a function defined on a complete random metric space. Then, we give a more precise form of Ekelandʼs variational principle for such a local function on a complete random normed module. Finally, as applications, we establish the Bishop–Phelps theorem in a complete random normed module under the framework of random conjugate spaces.

In this paper, we adopt a new approach so that we can prove that the Bishop–Phelps theorem in complete random normed modules still holds under the (ε,λ)-topology, which solves an open problem posed in [T.X. Guo, Y.J. Yang, Ekelandʼs variational principle for an L¯0-valued function on a complete random metric space, J. Math. Anal. Appl. 389 (2012) 1–14].

We prove the Bishop–Phelps–Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop–Phelps–Bollobás theorem holds for operators from ℓ1 into Y. Several examples of classes of such spaces are provided. For instance, the Bishop–Phelps–Bollobás theorem holds when the range space is finite-dimensional, an L1(μ)-space for a σ-finite measure μ, a C(K)-space for a compact Hausdorff space K, or a uniformly convex Banach space.

In this paper we show that the Bishop–Phelps–Bollobás theorem fails for bilinear forms on l1×l1, while it holds for linear operators from l1 to l∞.

A general formulation of the completeness argument used in the Bishop-Phelps Theorem and many other places has been given by Ekeland. It is shown that Ekeland's formulation characterizes complete metric spaces. A central idea in the proof of the Bishop-Phelps Theorem is the use of norm completeness and a partial ordering to produce a point where a linear functional attains its supremum on a closed bounded convex set. In fact, this completeness technique is useful in many situations as has been described in the surveys of Phelps [7] and Brezis-Browder [1]. Recently Ekeland [3] has given a very general formulation of this technique and has applied it to a wide variety of problems [4]. In the present note we show that Ekeland's formulation is actually equivalent to completeness for metric spaces. Ekeland's Theorem may be stated as follows: THEOREM 1. Let (M, d) be a complete metric space, and F: M -R U { + oo} a lower semicontinuous function, F 5 + x, bounded from below. Let > 0 be given and a point u & M such that