Rotundity in transitive and 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 isometric reflexions in Banach spaces

We obtain the following characterization of Hilbert spaces. Let E be a Banach space whose unit sphere S has a hyperplane of symmetry. Then E is a Hilbert space iff any of the following two conditions is fulfilled: a) the isometry group Iso E of E has a dense orbit in S; b) the identity component G 0 of the group Iso E endowed with the strong operator topology acts topologically irreducible on E. Some related results on infinite dimentional Coxeter groups generated by isometric reflexions are given which allow to analyse the structure of isometry groups containing sufficiently many reflexions.

Review of the Banach-Stone Theorem

Journal of Development Review

This is a quick overview of the isomorphism between spaces of continuous functions, or C(X) type spaces, that depend on compact Hausdorff spaces outfitted with the uniform norm. When two compact metric spaces, X and Y, are homeomorphic, Banach assumed the problem in 1932. He came to the conclusion that if C(X) and C(Y) are isometric isomorphic, then X and Y are homeomorphic. Stone then generalized this outcome for a general compact Hausdorff space in 1937. Then it is frequently referred to as the Banach-Stone theorem. There are numerous variations of this classic result. We can derive the topological features of X and Y from Gelfand and Kolmogoroff's algebraic version, which was published in 1939.

On nicely smooth Banach spaces

1996

In this work, we obtain some necessary and some sufficient conditions for a space to be nicely smooth, and show that they are equivalent for separable or Asplund spaces. We obtain a sufficient condition for the Ball Generated Property (BGP), and conclude that Property (II)(II)(II) implies the BGP, which, in turn, implies the space is nicely smooth. We show that

A New Class of Banach Spaces and Its Relation with Some Geometric Properties of Banach Spaces

Abstract and Applied Analysis, 2012

By introducing the concept ofL-limited sets and thenL-limited Banach spaces, we obtain some characterizations of it with respect to some well-known geometric properties of Banach spaces, such as Grothendieck property, Gelfand-Phillips property, and reciprocal Dunford-Pettis property. Some complementability of operators on such Banach spaces are also investigated.

A metric interpretation of reflexivity for Banach spaces

Duke Mathematical Journal

We define two metrics d1,α and d∞,α on each Schreier family Sα, α < ω1, with which we prove the following metric characterization of reflexivity of a Banach space X: X is reflexive if and only if there is an α < ω1, so that there is no mapping Φ : Sα → X for which cd∞,α(A, B) ≤ Φ(A) − Φ(B) ≤ Cd1,α(A, B) for all A, B ∈ Sα. Secondly, we prove for separable and reflexive Banach spaces X, and certain countable ordinals α that max(Sz(X), Sz(X *)) ≤ α if and only if (Sα, d1,α) does not bi-Lipschitzly embed into X. Here Sz(Y) denotes the Szlenk index of a Banach space Y .