LOCAL AND UNIFORM NEAR SMOOTHNESS OF SOME BANACH SPACES

In this paper we give an estimate of the modulus of near smoothness of the space $c_o(E_i)$. In the case of the space $c_o(l_{p_i})$ the exact formula for this modulus is derived. Moreover, we show that the properties of near uniform smoothness and local near uniform smoothness are hereditary with respect to the product space $c_o(E_i)$.

The Modulus of Nearly Uniform Smoothness in Orlicz Sequence Spaces

It is well known that the modulus of nearly uniform smoothness related with the fixed point property is important in Banach spaces. In this paper, we prove that the modulus of nearly uniform smoothness in Köthe sequence spaces without absolutely continuous norm is ΓX(t)=t. Meanwhile, the formula of the modulus of nearly uniform smoothness in Orlicz sequence spaces equipped with the Luxemburg norm is given. As a corollary, we get a criterion for nearly uniform smoothness of Orlicz sequence spaces equipped with the Luxemburg norm. Finally, the equivalent conditions of R(a,l(Φ))<1+a and RW(a,l(Φ))<1+a are given.

ON ASYMPTOTIC UNIFORM SMOOTHNESS AND NONLINEAR GEOMETRY OF BANACH SPACES

These notes concern the nonlinear geometry of Banach spaces, asymptotic uniform smoothness and several Banach–Saks-like properties. We study the existence of certain concentration inequalities in asymptotically uniformly smooth Banach spaces as well as weakly sequentially continuous coarse (Lipschitz) embeddings into those spaces. Some results concerning the descriptive set theoretical complexity of those properties are also obtained. We finish the paper with a list of open problem.

Smoothness and uniform smoothness

The notions of smoothness and uniform smoothness of a space are discussed. The relation with differentiability of the norm is shown. The main tool, the modulus of smoothness of a space is studied.

More moduli and coefficients

The general construction of multi-dimensional Milman’s moduli is described. Two-dimensional moduli are related to uniform convexity and uniform smoothness. The James constant measuring nonsquareness of the ball is discussed. A universal modulus, called also the modulus of squareness, and related both to convexity and smoothness is studied.

The case of Banach lattices

Uniform monotonicity and order uniform smoothness for Banach lattices are discussed as counterparts of uniform convexity and uniform smoothness. Corresponding moduli are defined. Analogies and differences are presented.

Uniform smoothness vs uniform convexity

The aim of the chapter is to present duality between uniform convexity and uniform smoothness. Lindenstrauss formulas relating moduli of convexity and smoothness are discussed as the main tool. A section deals with the notion of noncreasy and uniformly noncreasy spaces.