Stepanoff's theorem in separable Banach spaces (original) (raw)

Geometry and Gâteaux smoothness in separable Banach spaces

Operators and Matrices, 2012

It is a classical fact, due to Day, that every separable Banach space admits an equivalent Gâteaux smooth renorming. In fact, it admits an equivalent uniformly Gâteaux smooth norm, as was shown later byŠmulyan. It is therefore rather unexpected that the existence of Gâteaux smooth renormings satisfying various quantitative estimates on the directional derivative has rather strong structural and geometrical implications for the space. For example, by a result of Vanderwerff, if the directional derivatives satisfy a p-estimate, where p varies arbitrarily with respect to the point and the direction in question, then the Banach space must be an Asplund space. In the present survey paper, we discuss the interplay between various types of Gâteaux differentiability of norms and extreme points with the geometry of separable Banach spaces. In particular, we present various characterizations of Asplund, reflexive, superreflexive, and other classes of separable Banach spaces, via smooth as well as rotund renormings. We also include open problems of various levels of difficulty, with the hope of stimulating research in the area of smoothness and renormings of Banach spaces. In nonlinear analysis, the differentiability of norms plays an important role. The most important type of differentiability is that of Fréchet differentiability. However, in many instances it suffices to use weaker forms of differentiability, i.e., variants of the Gâteaux differentiability (that are more often accessible). This happens especially when some convexity arguments can be combined with Baire category techniques. The present paper surveys some of these results and discusses several ideas and constructions in their proofs. We focus on the interplay of these concepts with the geometry of separable spaces, for example with problems on containment of c 0 or 1 , with superreflexivity, the Radon-Nikodým property, etc. Several open problems in this area are discussed. We refer to, e.g., [Gode], [DGZb], [Fab], [AlKal06], [BoVa10], and [FHHMZ] for all unexplained notions and results used in this note.

On the near differentiability property of Banach spaces

Journal of Mathematical Analysis and Applications, 2006

Let μ be a scalar measure of bounded variation on a compact metrizable abelian group G. Suppose that μ has the property that for any measure σ whose Fourier-Stieltjes transformσ vanishes at ∞, the measure μ * σ has Radon-Nikodým derivative with respect to λ, the Haar measure on G. Then L. Pigno and S. Saeki showed that μ itself has Radon-Nikodým derivative. Such property is not shared by vector measures in general. We say that a Banach space X has the near differentiability property if every X-valued measure of bounded variation shares the above property. We prove that Banach spaces with the Radon-Nikodým property have the near differentiability property, while Banach spaces with the near differentiability property enjoy the near Radon-Nikodým property. We also show that the Banach spaces L 1 [0, 1] and L 1 /H 1 0 have the near differentiability property. Lastly, we show that Banach spaces with the near differentiability property have type II-Λ-Radon-Nikodým property, whenever Λ is a Riesz subset of type 0 of G.

On Ordinary and Standard "Lebesgue Measures" in Separable Banach Spaces

2013

By using results from a paper [G.R. Pantsulaia, On ordinary and standard Lebesgue measures on R ∞ , Bull. Pol. Acad. Sci. Math. 57 (3-4) (2009), 209-222] and an approach based in a paper [T. Gill, A.Kirtadze, G.Pantsulaia , A.Plichko, The existence and uniqueness of translation invariant measures in separable Banach spaces, Functiones et Approximatio, Commentarii Mathematici, 16 pages, to appear ], a new class of translation-invariant quasi-finite Borel measures (the so called, ordinary and standard "Lebesgue Measures") in an infinite-dimensional separable Banach space X is constructed and some their properties are studied in the present paper. Also, various interesting examples of generators of two-sided (left or right) shy sets with domain in non-locally compact Polish Groups are considered.

Existence and uniqueness of translation invariant measures in separable Banach spaces

Functiones et Approximatio Commentarii Mathematici, 2014

It is shown that for the vector space R N (equipped with the product topology and the Yamasaki-Kharazishvili measure) the group of linear measure preserving isomorphisms is quite rich. Using Kharazishvili's approach, we prove that every infinite-dimensional Polish linear space admits a σ-finite non-trivial Borel measure that is translation invariant with respect to a dense linear subspace. This extends a recent result of Gill, Pantsulaia and Zachary on the existence of such measures in Banach spaces with Schauder bases. It is shown that each σ-finite Borel measure defined on an infinite-dimensional Polish linear space, which assigns the value 1 to a fixed compact set and is translation invariant with respect to a linear subspace fails the uniqueness property. For Banach spaces with absolutely convergent Markushevich bases, a similar problem for the usual completion of the concrete σ-finite Borel measure is solved positively. The uniqueness problem for non-σ-finite semi-finite translation invariant Borel measures on a Banach space X which assign the value 1 to the standard rectangle (i.e., the rectangle generated by an absolutely convergent Markushevich basis) is solved negatively. In addition, it is constructed an example of such a measure µ 0 B on X, which possesses a strict uniqueness property in the class of all translation invariant measures which are defined on the domain of µ 0 B and whose values on non-degenerate rectangles coincide with their volumes.

Separable measures and the Dunford-Pettis property

Bulletin of the Australian Mathematical Society, 1991

Let X be a complete regular space. We denote by Cb(X) the Banach space of all real-valued bounded continuous functions on X endowed with the supremumnorm.In this paper we give a characterisation of weakly compact operators defined from Cb(X) into a Banach space E which are β∞-continuous, where β∞ is a locally convex topology on Cb(X) introduced by Wheeler. We also prove that (Cb(X), β∞) has the strict Dunford-Pettis property and, if X is a σ-compact space, (Cb(X), β∞), has the Dunford-Pettis property.

On the drop and weak drop properties for a Banach space

Bulletin of the Australian Mathematical Society, 1990

Rolewicz' drop property is a modification of a concept underlying Danes' drop theorem. We characterise the drop property by the upper semicontinuity and compact valued property of the duality mapping for the dual. The characterisation suggests that we define a weak drop property which we show characterises the reflexivity of the space. Consider the Banach space (X, ||-||) with open unit ball B(X) = {x 6 X: \\x\\ < 1} and closed unit ball B(X) = {x G X: \\x\\ < 1}. Given x £ B~(X), the set D(x, ~B(X)) = co{x, ~B(X)}, the convex hull of x and ~B(X), is called the drop generated by x. Danes [3] proved that in any Banach space (X, \\-\\), for every closed set C at positive distance from B(X), there exists an x G C such that D(x, -B(Z)) fl C = {x}. Rolewicz [9], modifying the Danes' drop theorem assumption, said that the norm ||-|| of X has the drop property if for every closed set C disjoint from B~{X), there exists an x G C such that D(x, ~B(XJ) ClC = {x}. He also introduced an associated sequential concept; a sequence {x n } in Jf \ B(X) such that x n +i £ D(x n , B(X)) for all n, is called a stream. Rolewicz proved that (i) the norm ||-|| has the drop property if and only if each stream in X\B(X) contains a convergent subsequence, [9, Proposition 2, p.29), and (ii) if the norm ||-|| has the drop property then X is reflexive, [9, Theorem 5, p.34].