Marián Fabian

The Czech Academy of Sciences | AVCR · Department of Topology and Functional Analysis

Influenced by a recent note by M. Ivanov and N. Zlateva, we prove a statement in the style of Nash-Moser-Ekeland theorem for mappings from a Fréchet-Montel space with values in any Fréchet space (not necessarily standard). The mapping under consideration is supposed to be continuous and directionally differentiable (in particular Gateaux differenti...

We present a criterion for local surjectivity of mappings between graded Fréchet spaces in the spirit of a well-known criterion in Banach spaces. As applications, we get “hard inverse mapping theorem” in the flavor of Nash–Moser. The technology of proofs was strongly influenced by a recent paper of Ekeland.

We prove that the property of a set-valued mapping F: X ⇒ Y to be locally metrically regular (and consequently, the properties of the mapping to be linearly open or pseudo-Lipschitz) is separably reducible by rich families of separable subspaces of X ×Y. In fact, we prove that, moreover, this extends to computation of the functor reg F that associa...

There are two basic ways of weakening the definition of the well-known metric regularity property by fixing one of the points involved in the definition. The first resulting property is called metric subregularity and has attracted a lot of attention during the last decades. On the other hand, the latter property which we call semiregularity can be...

We show a way of constructing projectional skeleton using the concept of rich families in Banach spaces which admit a projectional generator. If a Banach space $X$ is Asplund and weakly compactly generated, then we show the existence of a commutative 1-projectional skeleton $(Q_\gamma:\ \gamma\in\Gamma)$ on $X$ such that $(Q_\gamma{}^*:\ \gamma\in\...

Asplund property of a Banach space X is characterized by the existence of a rich family, in the product , consisting of some carefully chosen separable subspaces. This structural result is then used to add a lot of precision and simplicity to the known separable reductions of Fréchet subdifferentials.

In [5], we presented the separable reduction for a general statement covering practically all important properties of Fréchet subdifferentials, in particular: the non-emptiness of subdifferentials, the non-zeroness of normal cones, the fuzzy calculus, and the extremal principle; all statements being considered in the Fréchet sense. As in earlier st...

Based on a primal regularity criterion we provide lower bounds for the regularity modulus of a nonlinear single-valued mapping F from a Banach space X into another Banach space Y . We focus on the case when F is defined on a proper (closed convex) subset of X only rather than on the whole of X. Three possible ways of approximating F around the refe...

In the framework of Asplund spaces, we use two equivalent instruments - rich families and suitable models from logic - for performing separable reductions of various statements on Frechet subdifferentiability of functions. This way, isometrical results are actually obtained and several existed proofs are substantially simplified. Everything is base...

R.Deville and J.Rodríguez proved that, for every Hilbert generated space X, every Pettis integrable function f: [0, 1] → X is McShane integrable. R.Avilés, G. Plebanek, and J.Rodríguez constructed a weakly compactly generated Banach space X and a scalarly null (hence Pettis integrable) function from [0, 1] into X, which was not McShane integrable....

G. Godefroy and the second author of this note proved in 1988 that in duals to Asplund spaces there always exists a projectional resolution of the identity. A few years later, Ch. Stegall succeeded to drop from the original proof a deep lemma of S. Simons. Here, we rewrite the condensed argument of Ch. Stegall in a more transparent and detailed way...

We show that if μ is a probability measure and X is a Banach space, then the Lebesgue- Bochner space L1(μ,X) admits an equivalent norm which is rotund (uniformly rotund in every direction, locally uniformly rotund, or midpoint locally uniformly rotund) if X does. We also prove that if X admits a uniformly rotund norm, then the space L1(μ,X) has an...

We prove a separable reduction theorem for the Fréchet subdifferential that contains all earlier results of that sort as particular cases.

A large supply of quasi-monotone multivalued mappings with values in a weak* fragmentable dual Banach space is shown to be generically single-directional. This is of some interest in analysis of quasi-convex functions. The paper extends/strengthens some results of a recent paper by Aussel and Eberhard (Math Program 30, 2012).

The aim of this note is twofold. First, we prove an analogue of the well-known Robinson-Ursescu Theorem on the relative openness with a linear rate (restrictive metric regularity) of a multivalued mapping. Second, we prove a generalization of Graves Open Mapping Theorem for a class of mappings which can be approximated at a reference point by a bun...

We show that, if μ is a probability measure and X is a Banach space, then the space L 1(μ;X) of Bochner integrable functions admits an equivalent Gâteaux (or uniformly Gâteaux) smooth norm provided that X has such a norm, and that if X admits an equivalent Fréchet (resp. uniformly Fréchet) smooth norm, then L 1(μ;X) has an equivalent renorming whos...

The paper presents a general primal space classification scheme of necessary and sufficient criteria for the error bound property incorporating the existing conditions. Several primal space derivative-like objects – slopes – are used to characterize the error bound property of extended-real-valued functions on metric sapces.

In this chapter, we begin by proving the Brouwer and the Schauder fixed-point theorems. Then we turn to results on homeomorphisms of convex sets and spaces. We prove Keller’s theorem on homeomorphism of infinite-dimensional compact convex sets in Banach spaces to \mathbb I\mathbb N{\mathbb I}^{{\mathbb N}} . We also prove the Kadec theorem on the...

In this chapter, we will first discuss the properties of smoothness in ℓ p spaces and in Hilbert spaces. Then we study spaces that have countable James boundary in connection with their higher order smoothness, and its applications. In particular, we study spaces of continuous functions on countable compact spaces.

An indispensable tool in the study of deeper structural properties of a Banach space X is its weak topology, i.e., the topology on X of the pointwise convergence on elements of the dual space X *, or the weak *topology on X *, i.e., the topology on X * of the pointwise convergence on elements of X. The topology on X * of the uniform convergence on...

In this chapter we study superreflexive Banach spaces. These spaces admit many characterizations by means of equivalent renormings, local properties, uniform smoothness, and dentability properties. We also discuss the structure of these spaces and basic sequences in them.

The main topic of the present chapter is the dentability of bounded sets and the closely related Radon–Nikodým property (RNP) of Banach spaces. This property has several equivalent characterizations and applications. In particular, Asplund spaces are characterized by the Radon–Nikodým property of their dual spaces. As another application, we show t...

The Hahn-Banach theorem, in the geometrical form, states that a closed and convex set can be separated from any external point by means of a hyperplane. This intuitively appealing principle underlines the role of convexity in the theory. It is the first, and most important, of the fundamental principles of functional analysis. The rich duality theo...

In this chapter we study the weak and weak* topologies of Banach spaces in more detail. We discuss several types of compacta (Eberlein, uniform Eberlein, scattered, Corson, and more), weakly Lindelöf determined spaces and properties of tightness in weak topologies. We discuss some applications in the structural properties of some Banach spaces.

In this chapter we study separable Asplund spaces, i.e., Banach spaces with a separable dual space. These spaces admit many equivalent characterizations, in particular by means of C 1-smooth renormings and differentiability properties of convex functions. Asplund spaces also play an important role in applications. We study basic results in smooth a...

In this chapter we study weakly compact operators and the related class of Banach spaces that are generated by weakly compact sets (i.e., weakly compactly generated spaces, in short WCG spaces). We focus on their decomposition properties, renormings, and on the topological properties of their dual spaces. We prove that WCG spaces are generated by r...

The interplay between the structure of an infinite-dimensional Banach space and properties of its finite-dimensional subspaces belongs to the subject of the local theory of Banach spaces. It is a vast and deep part of Banach space theory intimately related to probability and combinatorics. Our goal is to familiarize the reader with some of its basi...

This chapter is an introduction to the topological theory of tensor products of Banach spaces. The focus lies on the applications of tensors in the duality theory for spaces of operators, and their structure as Banach spaces. We discuss the role of the approximation property and Enflo’s example of a Banach space without the approximation property.

In this chapter we discuss some basic techniques in nonlinear analysis on Banach spaces that are frequently used in applications in related fields. The classical approach uses differentiability, and we discuss this concept in infinite-dimensional Banach spaces.

In this short chapter we collect, for the reader’s convenience, some basic definitions and results that are used in the book. A list of sources for them is provided at the end of each section here.

We collect facts about Gul’ko, descriptive, Gruenhage, and fragmentable compact spaces. In several instances we provide direct proofs of the results discussed. We show how they reflect the geometrical structure of corresponding Banach spaces C(K). In particular, we provide proofs, by a simple transfer of Day’s norm, of recent renorming results for...

Let X be a Banach space whose norm is simultaneously LUR and Gateaux (Fréchet) smooth. Under some assumptions, it is shown that the infimal convolution of a fairly general function on X and the square of the norm is generically strongly attained and hence is Gateaux (Fréchet) differentiable. This contains a result of S. Dutta on distance functions.

The paper presents a general classification scheme of necessary and sufficient criteria for the error bound property incorporating the existing conditions. Several derivative-like objects both from the primal as well as from the dual space 122 M.J. Fabian et al. are used to characterize the error bound property of extended-real-valued functions on...

We will use the concept of strong generating and a simple renorming theorem to give new proofs to slight generalizations of some results of Argyros and Rosenthal on weakly compact sets in L1(μ) spaces for finite measures μ.

We define asymptotically p-flat and innerly asymptotically p-flat sets in Banach spaces in terms of uniform weak* Kadec–Klee asymptotic smoothness, and use these concepts to characterize weakly compactly generated (Asplund) spaces that are c0(ω1)-generated or ℓp(ω1)-generated, where p(1,∞). In particular, we show that every subspace of c0(ω1) is c0...

The dual X∗ of a Banach space X admits a dual σ-LUR norm if (and only if) X∗ admits a σ-weak∗ Kadets norm if and only if X∗ admits a dual weak∗ LUR norm and moreover X is σ-Asplund generated.

A characterization of Banach spaces admitting uniformly Gâteaux smooth norms in terms of σ-finite dual dentability indices is given. Some applications in the area of weak compactness are discussed. We also study σ-locally uniformly rotund dual renormings in connection with σ-countable dual dentability indices.

Using separable projectional resolutions of the identity, we provide a different proof of a result of Argyros and Mercourakis on the behavior of fundamental biorthogonal systems in weakly compactly generated (in short, WCG) Banach spaces. This result is used to discuss the example given by Argyros of a non-WCG subspace of a WCG space of the form C(...

We prove a variational principle in reflexive Banach spaces X with Kadec-Klee norm, which asserts that any Lipschitz (or any proper lower semicontinuous bounded from below extended real-valued) function in X can be perturbed with a parabola in such a way that the perturbed function attains its infimum (even more can be said - the infimum is well-po...

Let (X,‖⋅‖) be a reflexive Banach space with Kadec–Klee norm. Let f:X→(−∞,+∞] be a function which is either Lipschitzian or is proper, bounded below, and lower semi-continuous. Then f is supported from below by residually many parabolas opening downward, that is, the infimal convolution of ‖⋅‖2 and f is attained at residually many points of X.

Assume that every coercive, continuous, and bounded below function defined on a real Banach space attains, after a suitable linear perturbation, a (not necessarily strong) minimum. Then the space is dentable.

We present YAGO, a light-weight and extensible ontology with high coverage and quality. YAGO builds on entities and relations and currently contains more than 1 million entities and 5 million facts. This includes the Is-A hierarchy as well as non-taxonomic relations between entities (such as HASONEPRIZE). The facts have been automatically extracted...

Asplund generated Banach spaces are used to give new characteriza- tions of subspaces of weakly compactly generated spaces and to prove some results on Radon-Nikodym compacta. We show, typically, that in the framework of weakly Lindelof determined Banach spaces, subspaces of weakly compactly generated spaces are the same as �-Asplund generated spac...

This paper shows that every continuous convex function defined on an Asplund generated space can be represented as the point-wise limit of a non-decreasing sequence of continuous convex functions which are locally affine at all points of dense open sets.

We extend the definition of the limiting Fréchet subdifferential and the limiting Fréchet normal cone from Asplund spaces to Asplund generated spaces. Then we prove a sum rule, a mean value theorem, and other statements for this concept.

It is shown that most of the well known classes of nonseparable Ba- nach spaces related to the weakly compact generating can be charac- terized by elementary properties of the closure of the coe-cient space of Markusevic bases for such spaces. In some cases, such property is then shared by all Markusevic bases in the space.

We study the ε-Fréchet differentiability of Lipschitz functions on Asplund generated Banach spaces. We prove a mean value theorem and its equivalent, a formula for Clarke’s subdifferential, in terms of this concept. We inspect proofs of several statements based on the deep Preiss’s theorem on Fréchet differentiability of Lipschitz functions and we...

This is a short survey on some recent as well as classical results and open problems in smoothness and renormings of Banach spaces. Applications in general topology and nonlinear analysis are considered. A few new results and new proofs are included. An effort has been made that a young researcher may enjoy going through it without any special pre-...

We study countable splitting of Markushevich bases in weakly Lindelöf Banach spaces in connection with the geometry of these spaces.

If X is an infinite-dimensional Banach space, with separable dual, and $M \subset X^{\ast}$ is an analytic set such that any point $x^{\ast} \in M$ can be reached from 0 by a continuous path contained (except for the point x*) in the interior of M, then M is the range of the derivative of a C1-smooth function on X with bounded nonempty support.

A quantitative version of Krein's Theorem on convex hulls of weak compact sets is proved. Some applications to weakly compactly generated Banach spaces are given.

We give characterizations of weakly compactly generated spaces, their subspaces, Vašák spaces, weakly Lindelöf determined spaces as well as several other classes of Banach spaces related to uniform Gâteaux smoothness, in terms of the presence of a total subset of the space with some additional properties. In addition, we describe geometrically, whe...

A close connection of the strict convexity of the Day norm to the concept of the Gruenhage compacta is shown. As a byproduct we give an elementary characterization of Gul'ko compacta in the sigma-product of lines and a more elementary proof of Mercourakis' renorming result for Vašák spaces. This note is a result of our effort to classify those Bana...

We prove that a Banach space X is a subspace of a weakly com-pactly generated Banach space if and only if, for every ε > 0, X can be covered by a countable collection of bounded closed convex symmetric sets the weak * closure in X ** of each of them lies within the distance ε from X. As a corollary, we give a new, short functional-analytic proof to...

We study a variational principle in which there is one common perturbation function Φ for every proper lower semicontinuous extended real-valued function f defined on a metric space X. Necessary and sufficient conditions are given in order for the perturbed function f+Φ to attain its minimum. In the case of a separable Banach space we obtain a spec...

Using Stegall’s variational principle, we present a simple proof of Pitt’s theorem that bounded linear operators from ℓ q into ℓ p are compact for 1≤p<q<+∞.

We study relationships between two normal compactness properties of sets in Banach spaces that play an essential role in many aspects of variational analysis and its applications, particularly in calculus rules for generalized differentiation, necessary optimality and suboptimality conditions for optimization problems, etc. Both properties automati...

We classify several classes of the subspaces of Banach spaces X for which there is a bounded linear operator from a Hilbert space onto a dense subset in X. Dually, we provide optimal affine homeomorphisms from weak star dual unit balls onto weakly compact sets in Hilbert spaces or in c0(Γ) spaces in their weak topology. The existence of such embedd...

We answer positively a question raised by S. Argyros: Given any coanalytic, nonalytic subset � ' of the irrationals, we construct, in Mer- courakis space c1(� ' ), an adequate compact which is Gul'ko and not Ta- lagrand. Further, given any Borel, non Fsubset � ' of the irrationals, we construct, in c1(� ' ), an adequate compact which is Talagrand a...

If a Banach space has a LipschitzC 1-smooth bump function, then it admits other bumps of the same smoothness whose gradients exactly fill the dual unit ball and other reasonable figures. This strengthens a result of Azagra and Deville who were able to cover the dual unit ball.

We study Banach spaces X such that given a norming subspace Y of X *, X admits an equivalent Gâteaux differentiable norm that is Y-lower semicontinuous. Applications are shown in the area of Corson compact sets and in questions on the separability of supports of measures on Corson compacts. Several open problems are discussed.

Two smoothness characterizations of weakly compact sets in Ba-nach spaces are given. One that involves pointwise lower semicontin-uous norms and one that involves projectional resolutions of identity. The Gâteaux smoothness of norms has a profound impact on the structure of nonseparable Banach spaces, especially, if the smoothness is accompanied by...

The extremal points of set systems for generalizing conventional notions of optimal solutions to constrained optimization problems were studied. A method of separable reduction in nonseparable Banach spaces applied to extremal principles was developed. The developments were used to obtain direct proofs of two versions of the extremal principle.

We establish su#cient conditions on the shape of a set A included in the space L n s (X, Y ) of the n-linear symmetric mappings between Banach spaces X and Y , to ensure the existence of a C n -smooth mapping f : X -# Y , with bounded support, and such that f (n) (X) = A, provided that X admits a C n - smooth bump with bounded n-th derivative and d...

It is shown that a Banach spaceX admits an equivalent uniformly Gâteaux smooth norm if and only if the dual ball ofX* in its weak star topology is a uniform Eberlein compact.

Los Espacios de Banach proporcionan un marco para el análisis funcional lineal y no lineal, teor'ia de operadores, análisis abstracto, probabilidad, optimización y otras ramas de la matemática. Este libro pretende ser una introducción al análisis funcional lineal y de algunas partes de la teor'ia de Espacios de Banach de dimensión infinita

In this chapter we shall introduce Schauder bases, an important concept in Banach space theory. Elements of a Banach space with a Schauder basis may be represented as infinite sequences of “coordinates,” which is very natural and useful for analytical work. Although not every separable Banach space admits a Schauder basis (this is a deep and diffi...

Most of the theory presented in this text is valid for both real and complex scalar fields. When the proofs are similar, we formulate the theorems without specifying the field over which we are working. When theorems are not valid in both fields or their proofs are different, we specify the scalar field in the formulation of a theorem. K denotes si...

The focus of the study in the present chapter is on Banach spaces containing subspaces isomorphic to c 0 or ℓ 1. We prove Sobczyk’s theorem on complementability of c 0 in separable overspaces, lifting property of ℓ 1 and Pełczyński’s characterization of separable Banach spaces containing ℓ 1. We present Rosenthal’s ℓ 1 theorem, Odell–Rosenthal theo...

In this chapter we study basic properties of compact operators on Banach spaces. We present the elementary spectral theory of compact operators in Banach spaces, including the spectral radius and properties of eigenvalues. Then we discus basic spectral properties of selfadjoint operators on Hilbert spaces, their spectral decomposition, and show so...

This paper shows that the product of a G^ ateaux dierentiability space and a separable Banach space is again a G^ ateaux dierentiability space. A Banach space is called a G^ ateaux dierentiability space if every convex con- tinuous function on it is G^ ateaux dierentiable at the points of a dense set. For more information about this class and its r...

Let f be a real-valued function on an open subset U of a Banach space X. Let x ∈ U. We say that f is Gâteaux differentiable at x if there is F ∈ X* such that $$\mathop {\lim }\limits_{t \to 0} \frac{{f(x + th) - f(x)}}{t} = F(h)

Let X be a Banach space, n э N.

Let (X, ‖ · ‖) be a Banach space. For every ε ∈ (0, 2], we define the modulus of convexity (or rotundity) of ‖ · ‖ by $${\delta _X}\left( \varepsilon \right) = \inf \left\{ {1 - \left\| {\frac{{x + y}}{2}} \right\|;\,x,y \in {B_X},\,\left\| {x - y} \right\| \geqslant \varepsilon } \right\}$$.

Let K be a compact set, and let L be a pointwise closed set in C (K).

Norms and bump functions on a Banach space X that locally depend on finitely many elements of the dual space X, (sometimes called coordinates) are useful in renorming theory (cf. e. g. [3, Ch. V], [16]) or in the area of poly-hedral spaces (cf. e. g. [11] and references therein). Such norms share some properties of norms on finite dimensional space...

Given a normed space (X, ‖ · ‖), by X** we denote the space (X*)* with the norm \( \left\| F \right\| = \mathop {\sup }\limits_{f \in {B_x}*} \left| {F(f)} \right| \). We define higher duals by induction as X*** = (X**)*, etc.

In this chapter, we restrict ourselves to the real scalar field.

Every separable Banach space with C (n)-smooth norm (Lipschitz bump function) admits an equivalent norm (a Lipschitz bump function) which is both uniformly Gteaux smooth and C (n)-smooth. If a Banach space admits a uniformly Gteaux smooth bump function, then it admits an equivalent uniformly Gteaux smooth norm.

We develop a method of separable reduction for Fréchet-like normals and ε-normals to arbitrary sets in general Banach spaces. This method allows us to reduce certain problems involving such normals in nonseparable spaces to the separable case. It is particularly helpful in Asplund spaces where every separable subspace admits a Fréchet smooth renorm...

We show that the Asplund property of Banach spaces is not only sufficient but also a necessary condition for the fulfillment of some basic results in nonsmooth analysis involving Frchet-like normals and subdifferentials as well as their sequential limits. In this way we obtain new characterizations of Asplund spaces within the framework of nonsmoot...

A family of compact spaces containing continuous images of Radon-Nikodym compacta is introduced and studied. A family of Banach spaces containing subspaces of Asplund generated (i.e., GSG) spaces is introduced and studied. Further, for a continu- ous image of a Radon-Nikodym compact K we prove: If K is totally disconnected, then it is Radon-Nikodym...

We show that if a continuous bump function on a Banach space X locally depends on finitely many elements of a set F in X*, then the norm closed linear span of F equals to X*. Some corollaries for Markuševič bases and Asplund spaces are derived.

It is shown that the dual unit ball B X * of a Banach space X * in its weak star topology is a uniform Eberlein compact if and only if X admits a uniformly Gâteaux smooth norm and X is a subspace of a weakly compactly generated space. The bidual unit ball B X ** of a Banach space X ** in its weak star topology is a uniform Eberlein compact if and o...

. Various properties of Banach spaces, including the reflexivity and the Schur property of a space, are characterized in terms of properties of corresponding classes of locally Lipschitz functions on those spaces. AMS Classification. 46A55, 46B20, 52A41. 1 Research partially supported by NSERC and the Shrum Endowment. 2 Research partially supported...

A new smooth variational principle for spaces admitting Fréchet differentiable bump functions is proved. Further it is shown that each proper lower semicontinuous bounded below function can be supported by a smooth function with locally Hölder derivative if and only if the space is superreflexive. Some geometrical refinements of the Borwein–Preiss...

We show that in every infinite dimensional Banach space there is a compact set C and x in C such that C is regular at x (in the sense of nonsmooth analysis) but such that the associated distance function dC is not regular at x and neither is −dC.

We study the relationships between Gateaux, Frechet and weak Hadamard differentiability of convex functions and of equivalent norms. As a consequence we provide related characterizations of infinite dimensional Banach spaces and of Banach spaces containing l 1 . Explicit examples are given. Some renormings of WCG Asplund spaces are made in this vei...