Bishop Phelps Theorem Research Articles

Group invariant variational principles

In this paper we introduce a group invariant version of the well-known Ekeland variational principle. To achieve this, we define the concept of convexity with respect to a group and establish a version of the theorem within this framework. Additionally, we present several consequences of the group invariant Ekeland variational principle, including Palais-Smale minimizing sequences, the Brønsted-Rockafellar theorem, and a characterization of the linear and continuous group invariant functionals space. Moreover, we provide an alternative proof of the Bishop-Phelps theorem and proofs for the group-invariant Hahn-Banach separating theorems. Finally, we discuss some implications and applications of these results.

A multiset version of James's theorem

Let A and B be closed, convex and bounded subsets of a weakly sequentially complete Banach space E which both are not weakly compact. Then there is a linear form x0⁎∈E⁎ which does not attain its supremum on A and on B. In particular, given any bounded subset A⊂E, if every x⁎∈E⁎ either attains its supremum or infimum on A, then A is weakly relatively compact.The same happens for a finite family of closed, convex bounded but not weakly compact subsets.The result only remains true in arbitrary Banach space assuming, for any two σ(E⁎⁎,E⁎)-cluster points of A and B in E⁎⁎∖E, the fact that they generate vector subspace without nonzero vectors of E, i.e.: whenx0⁎⁎∈A‾σ(E⁎⁎,E⁎)∖A and y0⁎⁎∈B‾σ(E⁎⁎,E⁎)∖B verifyco({x0⁎⁎,−x0⁎⁎,y0⁎⁎,−y0⁎⁎})∩E={0}.A known example shows that a multiset analogue for the Bishop-Phelps theorem is not true.

A group invariant Bishop-Phelps theorem

We show that for any Banach space and any compact topological group G ⊂ L ( X ) G\subset L(X) such that the norm of X X is G G -invariant, the set of norm attaining G G -invariant functionals on X X is dense in the set of all G G -invariant functionals on X X , where a mapping f f is called G G -invariant if for every x ∈ X x\in X and every g ∈ G g\in G , f ( g ( x ) ) = f ( x ) f\big (g(x)\big )=f(x) . In contrast, we show also that the analog of Bollobás result does not hold in general. A version of Bollobás and James’ theorems is also presented.

Modulus support functionals, Rajchman measures and peak functions

In 2000 Victor Lomonosov suggested a counterexample to the complex version of the Bishop–Phelps theorem on modulus support functionals. We discuss the \(c_0\)-analog of that example and demonstrate that the set of sup-attaining functionals is non-trivial, thus answering an open question, asked in Kadets et al. (The mathematical legacy of Victor Lomonosov. Operator theory. Advances in analysis and geometry 2. De Gruyter, Berlin, 157–187, 2020).

Abstract Convexity of Functions with Respectto the Set of Lipschitz (Concave) Functions

The paper is devoted to the abstract $${\mathcal{H}}$$ -convexity of functions (where $${\mathcal{H}}$$ is a given set of elementary functions) and its realization in the cases when $${\mathcal{H}}$$ is the space of Lipschitz functions or the set of Lipschitz concave functions. The notion of regular $${\mathcal{H}}$$ -convex functions is introduced. These are functions representable as the upper envelopes of the set of their maximal (with respect to the pointwise order) $${\mathcal{H}}$$ -minorants. As a generalization of the global subdifferential of a convex function, we introduce the set of maximal support $${\mathcal{H}}$$ -minorants at a point and the set of lower $${\mathcal{H}}$$ -support points. Using these tools, we formulate both a necessary condition and a sufficient one for global minima of nonsmooth functions. In the second part of the paper, the abstract notions of $${\mathcal{H}}$$ -convexity are realized in the specific cases when functions are defined on a metric or normed space $$X$$ and the set of elementary functions is the space $${\mathcal{L}}(X,{\mathbb{R}})$$ of Lipschitz functions or the set $${\mathcal{L}}\widehat{C}(X,{\mathbb{R}})$$ of Lipschitz concave functions, respectively. An important result of this part of the paper is the proof of the fact that, for a lower semicontinuous function lower bounded by a Lipschitz function, the set of lower $${\mathcal{L}}$$ -support points and the set of lower $${\mathcal{L}}\widehat{C}$$ -support points coincide and are dense in the effective domain of the function. These results extend the known Brondsted–Rockafellar theorem on the existence of the subdifferential for convex lower semicontinuous functions to the wider class of lower semicontinuous functions and go back to the Bishop–Phelps theorem on the density of support points in the boundary of a closed convex set, which is one of the most important results of classical convex analysis.

Support points of lower semicontinuous functions with respect to the set of Lipschitz concave functions

For the functions defined on normed vector spaces, we introduce a new notion of the LC -convexity that generalizes the classical notion of convex functions. A function is called to be LC -convex if it can be represented as the upper envelope of some subset of Lipschitz concave functions. It is proved that the function is LC -convex if and only if it is lower semicontinuous and, in addition, it is bounded from below by a Lipschitz function. As a generalization of a global subdifferential of a classically convex function, we introduce the set of LC -minorants supported to a function at a given point and the set of LC -support points of a function that are then used to derive a criterion for global minimum points and a necessary condition for global maximum points of nonsmooth functions. An important result of the article is to prove that for a LC - convex function, the set of LC -support points is dense in its effective domain. This result extends the well-known Brondsted– Rockafellar theorem on the existence of the sub-differential for classically convex lower semicontinuous functions to a wider class of lower semicontinuous functions and goes back to the one of the most important results of the classical convex analysis – the Bishop–Phelps theorem on the density of support points in the boundary of a closed convex set.

Bishop-Phelps Theorem for Normed Cones

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

A domain-theoretic Bishop-Phelps theorem

A domain-theoretic Bishop-Phelps theorem

Bounded Holomorphic Functions Attaining their Norms in the Bidual

Under certain hypotheses on the Banach space X , we prove that the set of analytic functions in \mathcal A_u (X) (the algebra of all holomorphic and uniformly continuous functions in the ball of X ) whose Aron–Berner extensions attain their norms is dense in \mathcal A_u (X) . This Lindenstrauss type result also holds for functions with values in a dual space or in a Banach space with the so-called property ( \beta ). We show that the Bishop–Phelps theorem does not hold for \mathcal A_u (c_0, Z'') for a certain Banach space Z , while our Lindenstrauss theorem does. In order to obtain our results, we first handle their polynomial cases.

After the Bishop–Phelps theorem

We give an expository background to the Bishop–Phelps–Bollobás theorem and explore some progress and questions in recently developed areas of related research.

The domination property for efficiency and Bishop–Phelps theorem in locally convex spaces

We introduce the notion of Θ-closedness for any bounded convex set Θ and investigate the relationship between Θ-closedness and local closedness. Moreover we study some properties of σ-convex sets. By using Θ-closedness, local closedness and σ-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.

On the polynomial Lindenstrauss theorem

Under certain hypotheses on the Banach space X, we show that the set of N-homogeneous polynomials from X to any dual space, whose Aron–Berner extensions are norm attaining, is dense in the space of all continuous N-homogeneous polynomials. To this end we prove an integral formula for the duality between tensor products and polynomials. We also exhibit examples of Lorentz sequence spaces for which there is no polynomial Bishop–Phelps theorem, but our results apply. Finally we address quantitative versions, in the sense of Bollobás, of these results.

The Bishop–Phelps theorem in complete random normed modules endowed with the [formula omitted]-topology

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].

M-ideals and the Bishop-Phelps theorem

We give new proofs for the known important approximative properties of M-ideals, using only the definition of an M-ideal and the Bishop-Phelps theorem. Unlike the known proofs, these proofs do not use the 3-ball intersection property of M-ideals. Mathematics subject classification (2010): 46B20,41A50,41A65.

Ekelandʼs variational principle for an [formula omitted]-valued function on a complete random metric space

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.

AN ELEMENTARY PROOF OF JAMES’ CHARACTERIZATION OF WEAK COMPACTNESS

AbstractIn this paper we provide an elementary proof of James’ characterization of weak compactness in separable Banach spaces. The proof of the theorem does not rely upon either Simons’ inequality or any integral representation theorems. In fact the proof only requires the Krein–Milman theorem, Milman’s theorem and the Bishop–Phelps theorem.

Unified approach to some geometric results in variational analysis

Based on a study of a minimization problem, we present the following results applicable to possibly nonconvex sets in a Banach space: an approximate projection result, an extended extremal principle, a nonconvex separation theorem, a generalized Bishop–Phelps theorem and a separable point result. The classical result of Dieudonné (on separation of two convex sets in a finite-dimensional space) is also extended to a nonconvex setting.

A new proof of proximinality for $M$--ideals

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.

Existence of Efficient Points in Vector Optimization and Generalized Bishop–Phelps Theorem

In a set without linear structure equipped with a preorder, we give a general existence result for efficient points. In a topological vector space equipped with a partial order induced by a closed convex cone with a bounded base, we prove another kind of existence result for efficient points; this result does not depend on the Zorn lemma. As applications, we study a solution problem in vector optimization and generalize the Bishop--Phelps theorem to a topological vector space setting by showing that the B-support points of any sequentially complete closed subset A of a topological vector space E is dense in ∂A, where B is any bounded convex subset of E.

The Bishop–Phelps Theorem Fails for Uniform Non-selfadjoint Dual Operator Algebras

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.