Uniform Convexity and Uniform Smoothness

2018 ◽  
pp. 61-80
Keyword(s):  
Uniform Convexity ◽  
Uniform Smoothness

Nearly Uniform Convexity and Nearly Uniform Smoothness

1997 ◽  
pp. 85-108
Author(s):  
J. M. Ayerbe Toledano ◽  
T. Domínguez Benavides ◽  
G. López Acedo
Keyword(s):  
Uniform Convexity ◽  
Uniform Smoothness

scholarly journals On a Numerical Radius Preserving Onto Isometry onL(X)

2016 ◽  
Vol 2016 ◽  
pp. 1-3
Author(s):  
Sun Kwang Kim
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Numerical Range ◽  
Complex Banach Space ◽  
Uniform Convexity ◽  
Numerical Radius ◽  
Uniform Smoothness

We study a numerical radius preserving onto isometry onL(X). As a main result, whenXis a complex Banach space having both uniform smoothness and uniform convexity, we show that an onto isometryTonL(X)is numerical radius preserving if and only if there exists a scalarcTof modulus 1 such thatcTTis numerical range preserving. The examples of such spaces are Hilbert space andLpspaces for1<p<∞.

More moduli and coefficients

2018 ◽  
pp. 85-101
Author(s):  
Kazimierz Goebel ◽  
Stanisław Prus
Keyword(s):  
Uniform Convexity ◽  
General Construction ◽  
Two Dimensional ◽  
James Constant ◽  
Uniform Smoothness

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.

scholarly journals The modulus of near smoothness of the lp product of a sequence of Banach spaces

1997 ◽  
Vol 39 (2) ◽  
pp. 153-165
Author(s):  
Leszek Olszowy
Keyword(s):  
Functional Analysis ◽  
Banach Spaces ◽  
Measure Of Noncompactness ◽  
Uniform Convexity ◽  
Uniform Smoothness ◽  
Classical Geometry ◽  
New Concepts

In the classical geometry of Banach spaces the notions of smoothness, uniform smoothness, strict and uniform convexity introduced by Day [1] and Clarkson [2] play a very important role and are used in many branches of functional analysis ([3,4,5], for example). In recent years a lot of papers have appeared containing interesting generalizations of these notions in terms of a measure of noncompactness. These new concepts investigated in this paper as near uniform smoothness, local near uniform smoothness and modulus of near smoothness have been introduced by Stachura and Sekowski [6] and Banaś [7] (see also [8,9]).

The case of Banach lattices

2018 ◽  
pp. 147-156
Author(s):  
Kazimierz Goebel ◽  
Stanisław Prus
Keyword(s):  
Banach Lattices ◽  
Uniform Convexity ◽  
Uniform Smoothness ◽  
Uniform Monotonicity

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

2018 ◽  
pp. 59-69
Author(s):  
Kazimierz Goebel ◽  
Stanisław Prus
Keyword(s):  
Main Tool ◽  
Uniform Convexity ◽  
Uniform Smoothness

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.

scholarly journals Modulus of nearly uniform smoothness and Lindenstrauss formulae

1995 ◽  
Vol 37 (2) ◽  
pp. 143-153 ◽  
Author(s):  
Tomás Domínguez Benavides
Keyword(s):  
Banach Space ◽  
Strong Relationship ◽  
Uniform Convexity ◽  
Measures Of Noncompactness ◽  
Conjugate Space ◽  
Uniform Smoothness ◽  
Image Position ◽  
Smooth Space

AbstractThe Lindenstrauss formulawhich states a strong relationship between the (Clarkson) modulus of uniform convexity δx of a Banach space X and the modulus of uniform smoothness px* of the conjugate space X*, is well known. Following the idea of the definitions of nearly uniform smooth space by S. Prus and modulus of uniform smoothness we define a modulus of nearly uniform smoothness and prove some Lindenstrauss type formulae concerning this modulus and the modulus of nearly uniform convexity for some measures of noncompactness.

scholarly journals On Milman’s moduli for Banach spaces

2001 ◽  
Vol 6 (2) ◽  
pp. 115-129 ◽  
Author(s):  
Elisabetta Maluta ◽  
Stanislaw Prus ◽  
Mariusz Szczepanik
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Uniform Convexity ◽  
Infinite Dimensional ◽  
Uniform Smoothness

We show that infinite dimensional geometric moduli introduced by Milman are strongly related to nearly uniform convexity and nearly uniform smoothness. An application of those moduli to fixed point theory is given.

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