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In this paper, we introduce a concept of norm-attainment in the projective symmetric tensor product of a Banach space X, which turns out to be naturally related to the classical norm-attainment of N -homogeneous polynomials on X. Due to this relation, we can prove that there exist symmetric tensors that do not attain their norms, which allows us to study the problem of when the set of norm-attaining elements in is dense. We show that the set of all normattaining symmetric tensors is dense in for a large set of Banach spaces such as Lp-spaces, isometric L1-predual spaces or Banach spaces with monotone Schauder basis, among others. Next, we prove that if X* satisfies the Radon-Nikodým and approximation properties, then the set of all norm-attaining symmetric tensors in is dense. From these techniques, we can present new examples of Banach spaces X and Y such that the set of all norm-attaining tensors in the projective tensor product is dense, answering positively an open question from the paper [10]. [ABSTRACT FROM AUTHOR]

In this paper, we study geometric properties of the set of group invariant continuous linear operators between Banach spaces. In particular, we present group invariant versions of the Hahn–Banach separation theorems and elementary properties of the invariant operators. This allows us to contextualize our main applications in the theory of norm-attaining operators; we establish group invariant versions of the properties α of Schachermayer and β of Lindenstrauss, and present relevant results from this theory in this (much wider) setting. In particular, we generalize Bourgain's result, which says that if X has the Radon–Nikodým property, then X has the G-Bishop–Phelps property for G-invariant operators whenever G ⊆ L (X) is a compact group of isometries on X. [ABSTRACT FROM AUTHOR]

In this paper we study the structure of the set Hom(X, ℝ) of all lattice homomorphisms from a Banach lattice X into ℝ. Using the relation among lattice homomorphisms and disjoint families, we prove that the topological dual of the free Banach lattice FBL(A) generated by a set A contains a disjoint family of cardinality 2|A|, answering a question of B. de Pagter and A.W. Wickstead. We also deal with norm-attaining lattice homomorphisms. For classical Banach lattices, as c0, Lp- and C(K)-spaces, every lattice homomorphism on it attains its norm, which shows, in particular, that there is no James theorem for this class of functions. We prove that, indeed, every lattice homomorphism on X and C(K, X) attains its norm whenever X has order continuous norm. On the other hand, we provide what seems to be the first example in the literature of a lattice homomorphism which does not attain its norm. In general, we study the existence and characterization of lattice homomorphisms not attaining their norm in free Banach lattices. As a consequence, it is shown that no Bishop-Phelps type theorem holds true in the Banach lattice setting, i.e. not every lattice homomorphism can be approximated by norm-attaining lattice homomorphisms. [ABSTRACT FROM AUTHOR]

Given two Banach spaces X and Y, we introduce and study a concept of norm-attainment in the space of nuclear operators N (X , Y) and in the projective tensor product space X ⊗ ^ π Y . We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which attain their norms in N (X , Y) and in X ⊗ ^ π Y is dense or not. We prove that, for both concepts, the density of norm-attaining elements holds for a large class of Banach spaces X and Y which, in particular, covers all classical Banach spaces. Nevertheless, we present Banach spaces X and Y failing the approximation property in such a way that the class of elements in X ⊗ ^ π Y which attain their projective norms is not dense. We also discuss some relations and applications of our work to the classical theory of norm-attaining operators throughout the paper. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in giving two characterizations for the so-called property L o , o , a local vector valued Bollobás type theorem. We say that (X, Y) has this property whenever given ε > 0 and an operador T : X → Y , there is η = η (ε , T) such that if x satisfies ‖ T (x) ‖ > 1 - η , then there exists x 0 ∈ S X such that x 0 ≈ x and T itself attains its norm at x 0 . This can be seen as a strong (although local) Bollobás theorem for operators. We prove that the pair (X, Y) has the L o , o for compact operators if and only if so does (X , K) for linear functionals. This generalizes at once some results due to D. Sain and J. Talponen. Moreover, we present a complete characterization for when (X ⊗ ^ π Y , K) satisfies the L o , o for linear functionals under strict convexity or Kadec–Klee property assumptions in one of the spaces. As a consequence, we generalize some results in the literature related to the strongly subdifferentiability of the projective tensor product and show that (L p (μ) × L q (ν) ; K) cannot satisfy the L o , o for bilinear forms. [ABSTRACT FROM AUTHOR]

Introduction In the last few years there is a growing interest in the theory of quasi-metric spaces and other related structures such as quasi-normed cones and asymmetric normed linear spaces, because such a theory provides an important tool in the study of several problems in theoretical computer science, approximation theory, applied physics, convex analysis and optimization. Many works on general topology and functional analysis have recently been obtained in order to extend the well-known results of the classical theory of normed linear spaces to the framework of asymmetric normed linear spaces and quasi-normed cones. ‎An abstract cone is analogous to a real vector space‎, ‎except that we take as the set‎ ‎of scalars‎. ‎ In 2004, O‎. ‎Valero introduced the normed cones and proved some closed graph and open mapping results for normed cones. Also Valero defined and studied some properties of quotient normed cones. P. Selinger studied the norm properties of a cone with its order properties and proved Hahn-Banach theorems in these cones under the appropriate conditions. Valero and his colleagues discussed the metrizability of the unit ball of the dual of a normed cone and the isometries of normed cones. Other properties are investigated in a series of papers by Romaguera, Sanchez Perez and Valero. The Bishop-Phelps theorem is a fundamental theorem in functional analysis which has many applications in the geometry of Banach spaces and optimization theory. The classical Bishop-Phelps theorem states that “the set of support functionals for a closed bounded convex subset of a real Banach space X, is norm dense in and the set of support points of is dense in the boundary of ". Indeed, E. Bishop and R. R. Phelps answer a question posed by ‎Victor Klee in 1958. We give an analogue to the normed cones‎, in fact we show that in a continuous normed cone the set of support points of a closed convex set is a dense subset of the boundary under the appropriate hypothesis. Conclusion In this paper the notion of support points of convex sets in normed cones is introduced and it is shown that in a continuous normed cone, under the appropriate conditions, the set of support points of a bounded Scott-closed convex set is nonempty. We also present a Bishop-Phelps type Theorem for normed cones../files/site1/files/52/5.pdf

In this paper, the notion of c-support points of a set in a semitopological cone is introduced. It is shown that any nonempty convex Scott closed bounded set has a c-support point in a cancellative bd-cone under certain condition. We also introduce the notion of wd-cone and then we prove a Bishop-Phelps type theorem for wd-cones, especially for normed cones, under appropriate conditions. Finally, using of the Bishop-Phelps technique, we obtain a result about the fixed points of a mapping on s-cones [ABSTRACT FROM AUTHOR]

Abstract We study the Bishop–Phelps–Bollobás property for operators from ℓ ∞ 4 to a Banach space. For this reason we introduce an appropriate geometric property, namely the AHSp- ℓ ∞ 4. We prove that spaces Y satisfying AHSp- ℓ ∞ 4 are precisely those spaces Y such that (ℓ ∞ 4 , Y) has the Bishop–Phelps–Bollobás property. We also provide classes of Banach spaces satisfying this condition. For instance, finite-dimensional spaces, uniformly convex spaces, C 0 (L) and L 1 (μ) satisfy AHSp- ℓ ∞ 4. [ABSTRACT FROM AUTHOR]

We study the Bishop–Phelps–Bollobás property and the Bishop–Phelps–Bollobás property for numerical radius. Our main aim is to extend some known results about norm or numerical radius attaining operators to multilinear and polynomial cases. We characterize the pair ( ℓ 1 ( X ) , Y ) to have the BPBp for bilinear forms and prove that on L 1 ( μ ) the numerical radius and the norm of a multilinear mapping are the same. We also show that L 1 ( μ ) fails the BPBp-nu for multilinear mappings although L 1 ( μ ) satisfies it in the operator case for every measure μ . [ABSTRACT FROM AUTHOR]

In this paper we introduce two Bishop-Phelps-Bollobás type properties for bounded linear operators between two Banach spaces X and Y: property 1 and property 2. These properties are motivated by a Kim-Lee result which states, under our notation, that a Banach space X is uniformly convex if and only if the pair [ABSTRACT FROM AUTHOR]

Using a diffeomorphism between the unit sphere and a closed hyperplane of an infinite dimensional Banach space, we introduce the differentiation of a function defined on the unit sphere, and show that a continuous linear functional attains its norm if and only if it has a critical point on the unit sphere. Furthermore, we provide a strong version of the Bishop–Phelps–Bollobás theorem for a Lipschitz smooth Banach space. [ABSTRACT FROM AUTHOR]

In this article, we study a version of the Bishop–Phelps–Bollobás property. We investigate a pair of Banach spaces ( X , Y ) such that every operator from X into Y is approximated by operators which attain their norm at the same point where the original operator almost attains its norm. In this case, we say that such a pair has the Bishop–Phelps–Bollobás point property (BPBpp). We characterize uniform smoothness in terms of BPBpp and we give some examples of pairs ( X , Y ) which have and fail this property. Some stability results are obtained about ℓ 1 and ℓ ∞ sums of Banach spaces and we also study this property for bilinear mappings. [ABSTRACT FROM AUTHOR]

We introduce a generalized approximate hyperplane series property for a pair ( X , Y ) of Banach spaces to characterize when ( ℓ 1 ( X ) , Y ) has the Bishop–Phelps–Bollobás property. In particular, we show that ( X , Y ) has this property if X , Y are finite-dimensional, if X is a C ( K ) space and Y is a Hilbert space, or if X is Asplund and Y = C 0 ( L ) , where K is a compact Hausdorff space and L is a locally compact Hausdorff space. [ABSTRACT FROM AUTHOR]

Abstract: We show that the space of bounded linear operators between spaces of continuous functions on compact Hausdorff topological spaces has the Bishop–Phelps–Bollobás property. A similar result is also proved for the class of compact operators from the space of continuous functions vanishing at infinity on a locally compact and Hausdorff topological space into a uniformly convex space, and for the class of compact operators from a Banach space into a predual of an -space. [Copyright &y& Elsevier]

Abstract: We introduce the notion of -closedness for any bounded convex set and investigate the relationship between -closedness and local closedness. Moreover we study some properties of -convex sets. By using -closedness, local closedness and -convexity, we give two main theorems on the domination property and the existence of efficient points in locally convex spaces. From these two main theorems we deduce a number of corollaries, which improve the related known results. As an application of the main results, we obtain several support point theorems in locally convex spaces, which generalize the famous Bishop–Phelps theorem. [Copyright &y& Elsevier]

Acosta et al. in 2008 gave a characterization of a Banach space Y (called an approximate hyperplane series property, or AHSP for short) guaranteeing exactly that a quantitative version of the Bishop-Phelps theorem holds for bounded operators from ...1 to the space Y . In this note, we give two new examples of spaces having the AHSP: the almost CL-spaces and the class of Banach spaces Y whose dual Y* is uniformly strongly subdifferentiable on some boundary of Y . We then calculate the precise parameters associated to almost CL-spaces. [ABSTRACT FROM AUTHOR]

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 < ɛ < 1/2. Then for every bounded linear operator T: H → H and x ∈ H with | T| = 1 = | x| such that | Tx| > 1 − g3, there exist x ∈ H and a bounded linear operator S: H → H with | S| = 1 = | x| such that [ABSTRACT FROM AUTHOR]

Abstract: 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 -topology, which solves an open problem posed in [T.X. Guo, Y.J. Yang, Ekelandʼs variational principle for an -valued function on a complete random metric space, J. Math. Anal. Appl. 389 (2012) 1–14]. [Copyright &y& Elsevier]

Abstract: Motivated by the recent work on conditional risk measures, this paper studies the Ekelandʼs variational principle for a proper, lower semicontinuous and lower bounded -valued function, where 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. [Copyright &y& Elsevier]

Abstract: Let Y be a Banach space and be a σ-finite measure space, where Σ is an infinite σ-algebra of measurable subsets of Ω. We show that if the couple 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 has the Bishop–Phelps–Bollobás property for operators if and only if Y has the AHSP. [Copyright &y& Elsevier]

In this paper we apply the Bishop-Phelps Theorem to show that if X is a Banach space and G ⊆ X is a maximal subspace so that G⊥ = {x* ϵ X*∣x*(y) = 0; ∀y ϵ G} is an L-summand in X*, then L1(Ω,G) is contained in a maximal proximinal subspace of L1(Ω;,X). [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: We show a modified version of Georgiev''s parametric smooth variational principle, and we use it to derive new support properties of convex functions and sets. For example, our results imply that, for any proper l.s.c. convex nonaffine function h on a Banach space Y, is pathwise connected and has cardinality at least continuum. If, in addition, Y is Fréchet-smooth renormable, then is pathwise connected and locally pathwise connected. Analogous properties for support points and normalized support functionals of closed convex sets are proved; they extend and strengthen recent results proved by C. De Bernardi and the author for bounded closed convex sets. [Copyright &y& Elsevier]

Abstract: 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 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 -space for a σ-finite measure μ, a -space for a compact Hausdorff space K, or a uniformly convex Banach space. [Copyright &y& Elsevier]

We characterize real Banach spaces Y such that the pair (ℓ ∞ n , Y) has the Bishop-Phelps-Bollobás property for operators. To this purpose it is essential using an appropriate basis of the domain space R n. As a consequence of the mentioned characterization, we provide examples of spaces Y satisfying such property. For instance, finite-dimensional spaces, uniformly convex spaces, uniform algebras and L 1 (μ) (μ a positive measure) satisfy the previous property. [ABSTRACT FROM AUTHOR]