Banach Space Theory Research Articles

On Convergence with respect to an Ideal and a Family of Matrices

P. Das et al. recently introduced and studied the notions of strong AI-summability with respect to an Orlicz function F and AI-statistical convergence, where A is a nonnegative regular matrix and I is an ideal on the set of natural numbers. In this paper, we will generalise these notions by replacing A with a family of matrices and F with a family of Orlicz functions or moduli and study the thus obtained convergence methods. We will also give an application in Banach space theory, presenting a generalisation of Simons' sup-limsup-theorem to the newly introduced convergence methods (for the case that the filter generated by the ideal I has a countable base), continuing some of the author's previous work.

On the Structure of Finite Level and ω-Decomposable Borel Functions

AbstractWe give a full description of the structure under inclusion of all finite level Borel classes of functions, and provide an elementary proof of the well-known fact that not every Borel function can be written as a countable union of Σα0-measurable functions (for every fixed 1 ≤ α < ω1). Moreover, we present some results concerning those Borel functions which are ω-decomposable into continuous functions (also called countably continuous functions in the literature): such results should be viewed as a contribution towards the goal of generalizing a remarkable theorem of Jayne and Rogers to all finite levels, and in fact they allow us to prove some restricted forms of such generalizations. We also analyze finite level Borel functions in terms of composition of simpler functions, and we finally present an application to Banach space theory.

On the numerical radius of Lipschitz operators in Banach spaces

We study the numerical radius of Lipschitz operators on Banach spaces via the Lipschitz numerical index, which is an analogue of the numerical index in Banach space theory. We give a characterization of the numerical radius and obtain a necessary and sufficient condition for Banach spaces to have Lipschitz numerical index 1. As an application, we show that real lush spaces and C-rich subspaces have Lipschitz numerical index 1. Moreover, using the Gâteaux differentiability of Lipschitz operators, we characterize the Lipschitz numerical index of separable Banach spaces with the RNP. Finally, we prove that the Lipschitz numerical index has the stability properties for the c0-, l1-, and l∞-sums of spaces and vector-valued function spaces. From this, we show that the C(K) spaces, L1(μ)-spaces and L∞(ν)-spaces have Lipschitz numerical index 1.

On statistical measure theory

The purpose of this paper is to unify various kinds of statistical convergence by statistical measure convergence and to present Jordan decomposition of finitely additive measures. It is done through dealing with the most generalized statistical convergence–ideal convergence by applying geometric functional analysis and Banach space theory. We first show that for each type of ideal I(⊂2N) convergence, there exists a set S of statistical measures such that the measure S-convergence is equivalent to the statistical convergence. To search for Jordan decomposition of measures of statistical type, we show that the subspace XI≡span¯{χA:A∈I} is an ideal of the space ℓ∞ in the sense of Banach lattice, hence the quotient space ℓ∞/XI is isometric to a C(K) space. We then prove that a statistical measure has a Jordan decomposition if and only if its corresponding functional is norm-attaining on ℓ∞, and which in turn induces an approximate null–ideal preserved Jordan decomposition theorem of finitely additive measures. Finally, we show this characterization and the approximate decomposition theorem are true for finitely additive measures defined on a general measurable space.

On Reduced Triangles in Normed Planes

A convex body is reduced if it does not properly contain a convex body of the same minimal width. In this paper we present new results on reduced triangles in normed (or Minkowski) planes, clearly showing how basic seemingly elementary notions from Euclidean geometry (like that of the regular triangle) spread when we extend them to arbitrary normed planes. Via the concept of anti-norms, we study the rich geometry of reduced triangles for arbitrary norms giving bounds on their side-lengths and on their vertex norms. We derive results on the existence and uniqueness of reduced triangles, and also we obtain characterizations of the Euclidean norm by means of reduced triangles. In the introductory part we discuss different topics from Banach Space Theory, Discrete Geometry, and Location Science which, unexpectedly, benefit from results on reduced triangles.

Geometry of Function Spaces

In recent years the use of the techniques from the geometric theory of Banach spaces in the study of function spaces has become an important tool and we are happy that this special issue has received the attention of both senior and young researchers in these areas. We have received papers from a wide cross-section of researchers in this area, covering existence theorems concerning the convolution of functions, distributions, and ultradistributions of Beurling type with supports satisfying suitable compatibility conditions and the Rayleigh equation of two deviating arguments. Application of the Leray-Schauder index theorem and the Leray-Schauder fixed point theorem towards the existence of nontrivial periodic solutions of this type of equations is covered. Also covered is smooth approximation of Lipschitz functions on Finsler manifolds, keeping control on the corresponding Lipschitz constants. An article that extends the usual notion ofKothe dual of a Banach function space to a broad class of Banach lattices is also in this collection. On abstract Banach space theory, there is an article that studies a weakening of the notion of central subspace of a Banach space, as well as an article that studies notions of generalized approximation properties.

The Hahn Sequence Space Defined by the Cesáro Mean

The -space of all sequences is given as such that converges and is a null sequence which is called the Hahn sequence space and is denoted by . Hahn (1922) defined the space and gave some general properties. G. Goes and S. Goes (1970) studied the functional analytic properties of this space. The study of Hahn sequence space was initiated by Chandrasekhara Rao (1990) with certain specific purpose in the Banach space theory. In this paper, the matrix domain of the Hahn sequence space determined by the Cesáro mean first order, denoted by , is obtained, and some inclusion relations and some topological properties of this space are investigated. Also dual spaces of this space are computed and, matrix transformations are characterized.

An introduction to the Ribe program

We survey problems, results, ideas, and recent progress in the Ribe program. The goal of this research program, which is motivated by a classical rigidity theorem of Martin Ribe, is to obtain structural results for metric spaces that are inspired by the local theory of Banach spaces. We also present examples of applications of the Ribe program to several areas, including group theory, theoretical computer science, and probability theory.

Some new results on decidability for elementary algebra and geometry

We carry out a systematic study of decidability for theories (a) of real vector spaces, inner product spaces, and Hilbert spaces and (b) of normed spaces, Banach spaces and metric spaces, all formalized using a 2-sorted first-order language. The theories for list (a) turn out to be decidable while the theories for list (b) are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic.We find that the purely universal and purely existential fragments of the theory of normed spaces are decidable, as is the ∀∃ fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbertʼs 10th problem show that the ∃∀ fragments for metric and normed spaces and the ∀∃ fragment for normed spaces are all undecidable.

Compact spaces of Szlenk index ω

The Szlenk index has found many applications in the isomorphic theory of Banach spaces. Its definition is based in some kind of interplay between a weak topology and the norm metric with not much care on the linear structure. There is no obstacle to consider the notion of Szlenk index in more general settings. In this paper we study the compact spaces of Szlenk index ω at most with respect to an associated metric. We include new applications to Banach spaces of the these methods, where the estimations of the growth speed of the finite Szlenk indices play a fundamental role.

The Features of a Class of Orthogonal Multiple Vector Ternary Wavelet Wraps and Applications in Physics

Wavelet analysishas been a powerful tool for exploring and solving many complicated problems in natural science and enginee-ring computation. In this work, the notion of orthogonal matrix-valued trivariate small-wave wraps and wavelet frame wraps, which are generalization of univariate small-wave wraps, is introduced. A new procedure for constructing these vector trivariate small-wave wraps is presented. Their characteristics are studied by using time-frequency analysis method, Banach space theory and finite group theory. Orthogonal formulas concerning the wavelet packs are established. The biorthogonality formulas concerning these wavelet wraps are established. Moreover, it is shown how to draw new Riesz bases of space from these wavelet wraps.

Bell violations through independent bases games

In a recent paper, Junge and Palazuelos presented two two-player games exhibiting interesting properties. In their first game, entangled players can perform notably better than classical players. The quantitative gap between the two cases is remarkably large, especially as a function of the number of inputs to the players. In their second game, entangled players can perform notably better than players that are restricted to using a maximally entangled state (of arbitrary dimension). This was the first game exhibiting such a behavior. The analysis of both games is heavily based on non-trivial results from Banach space theory and operator space theory. Here we provide alternative proofs of these two results. Our proofs are arguably simpler, use elementary probabilistic techniques and standard quantum information arguments, and also give better quantitative bounds.

Convergence results for function spaces over o-minimal structures

We begin the development of a theory of Banach spaces in the definable setting of o-minimal structures. We outline several results which develop the theory of compact embeddings for explicitly given function spaces. One aim is to explain the substantive underpinnings of an important observation used in the proof of the Reparameterization Theorem of Pila and Wilkie in (2). We place this observation in the broader context of our theory and demonstrate how it may be refined further. 2000 Mathematics Subject Classification 03C64 (primary); 46B99 (secondary)

Local complementation and the extension of bilinear mappings

AbstractWe study different aspects of the connections between local theory of Banach spaces and the problem of the extension of bilinear forms from subspaces of Banach spaces. Among other results, we prove that if X is not a Hilbert space then one may find a subspace of X for which there is no Aron–Berner extension. We also obtain that the extension of bilinear forms from all the subspaces of a given X forces such X to contain no uniform copies of ℓpn for p ∈ [1, 2). In particular, X must have type 2 − ϵ for every ϵ > 0. Also, we show that the bilinear version of the Lindenstrauss–Pełczyński and Johnson–Zippin theorems fail. We will then consider the notion of locally α-complemented subspace for a reasonable tensor norm α, and study the connections between α-local complementation and the extendability of α*-integral operators.

On strong solutions of a Robin problem modelling heat conduction in materials with corroded boundary

Motivated by a practical problem on a corrosion process, we shall study a third kind of BVP for a large class of elliptic equations in vector-valued L p spaces. Particularly we will determine optimal spaces for boundary data and get maximal regularity for inhomogeneous equations. Then based on these results we shall treat some nonlinear problems. Our approach will be based on the semigroup theory, the interpolation theory of Banach spaces, fractional powers of positive operators, operator-valued Fourier multiplier theorems and the Banach fixed point theorem.

Grothendieck’s Theorem, past and present

Probably the most famous of Grothendieck’s contributions to Banach space theory is the result that he himself described as “the fundamental theorem in the metric theory of tensor products”. That is now commonly referred to as “Grothendieck’s theorem” (GT in short), or sometimes as “Grothendieck’s inequality”. This had a major impact first in Banach space theory (roughly after 1968), then, later on, in C ¤ -algebra theory, (roughly after 1978). More recently, in this millennium, a new version of GT has been successfully developed in the framework of “operator spaces” or non-commutative Banach spaces. In addition, GT independently surfaced in several quite unrelated fields: in connection with Bell’s inequality in quantum mechanics, in graph theory where the Grothendieck constant of a graph has been introduced and in computer science where the Grothendieck inequality is invoked to replace certain NP hard problems by others that can be treated by “semidefinite programming’ and hence solved in polynomial time. In this expository paper, we present a review of all these topics, starting from the original GT. We concentrate on the more recent developments and merely outline those of the first Banach space period since detailed accounts of that are already available, for instance the author’s 1986 CBMS notes.

Geometry Where Direction Matters—Or Does It?

Minkowski Geometry, the geometric study of finite-dimensional real normed spaces, originated, as the name suggests, with HermannMinkowski in his Geometry of Numbers. We note that this term is also used for Minkowski’s spacetime (or Lorentz) geometry, a completely different field. Minkowski Geometry is closely related to Convex Geometry, Banach Space Theory, Finsler Geometry, and Distance Geometry. In recent times it was also enriched by applied disciplines, such as Operations Research, Optimization, Discrete Geometry, Computational Geometry, and Location Science. Wewould like to present a few interesting results from Minkowski Geometry, mostly from the viewpoint of Elementary Geometry.

Banach space theory: the basis for linear and nonlinear analysis

Preface.- Basic Concepts in Banach Spaces.- Hahn-Banach and Banach Open Mapping Theorems.- Weak Topologies and Banach Spaces.- Schauder Bases.- Structure of Banach Spaces.- Finite-Dimensional Spaces.- Optimization.- C^1 Smoothness in Separable Spaces.- Superreflexive Spaces.- Higher Order Smoothness.- Dentability and differentiability.- Basics in Nonlinear Geometric Analysis.- Weakly Compactly Generated Spaces.- Topics in Weak Topologies on Banach Spaces.- Compact Operators on Banach Spaces.- Tensor Products.- Appendix.- References.- Symbol Index.- Subject Index.- Author Index.

Complex extreme points and complex convexity in Orlicz–Bochner function spaces

It is well known that extreme points which are connected with strict convexity of the whole spaces, are the most basic and important geometric points in geometric theory of Banach spaces. In this paper, criteria for complex extreme points, complex strict convexity and complex uniform convexity in Orlicz–Bochner function spaces are given.

New Criteria of Some Bounded Approximation Properties

The bounded (resp. bounded compact) approximation property is well known in the theory of Banach spaces. This paper is concerned with some bounded approximation properties in the more general setting. We establish various new criteria of bounded approximation properties.