On nearly uniformly convex and k-uniformly convex spaces

1984 ◽  
Vol 95 (2) ◽  
pp. 325-327 ◽  
Author(s):  
V. I. Istrăt‚escu ◽  
J. R. Partington
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Normal Structure ◽  
Uniformly Convex ◽  
Uniformly Convex Space ◽  
Uniformly Convex Spaces ◽  
Convex Spaces

AbstractIn this note we prove that every nearly uniformly convex space has normal structure and that K-uniformly convex spaces are super-reflexive.We recall [1] that a Banach space is said to be Kadec–Klee if whenever xn → x weakly and ∥n∥ = ∥x∥ = 1 for all n then ∥xn −x∥ → 0. The stronger notions of nearly uniformly convex spaces and uniformly Kadec–Klee spaces were introduced by R. Huff in [1]. For the reader's convenience we recall them here.

Hadamard–Landau inequalities in uniformly convex spaces

1981 ◽  
Vol 90 (2) ◽  
pp. 259-264 ◽  
Author(s):  
J. R. Partington
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Uniformly Convex ◽  
Image Position ◽  
Uniformly Convex Spaces ◽  
Convex Spaces

The inequalityfor fεLp(− ∞, ∞)or Lp(0, ∞) (1≤p ≤ ∞), and its extensionfor T an Hermitian or dissipative linear operator, in general unbounded, on a Banach space X, for xεX, have been considered by many authors. In particular, forms of inequality (1) have been given by Hadamard(7), Landau(15), and Hardy and Little-wood(8),(9). The second inequality has been discussed by Kallman and Rota(11), Bollobás (2) and Kato (12), and numerous further references may be found in the recent papers of Kwong and Zettl(i4) and Bollobás and Partington(3).

Lower bounds for normal structure coefficients

1992 ◽  
Vol 121 (3-4) ◽  
pp. 245-252 ◽  
Author(s):  
T. Domínguez Benavides ◽  
G. López Acedo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Normal Structure ◽  
Uniformly Convex ◽  
Modulus Of Convexity ◽  
Image Position ◽  
Uniformly Convex Spaces ◽  
Lower Boundedness ◽  
Convergent Sequences ◽  
Convex Spaces

SynopsisUsing some new expressions for the weakly convergent sequences coefficient WCS(X) the lower boundednessis proved, where δ(-) is the (Clarkson) modulus of convexity. We also define a modulus of noncompact convexity concerning nearly uniformly convex spaces which is used to obtain another lower bound for WCS(X). The computation of this modulus in Ip-spaces shows that our second lower bound is the best possible in these spaces.

On Nonreflexive Banach Spaces Which Contain No c0 or lp

1980 ◽  
Vol 32 (6) ◽  
pp. 1382-1389 ◽  
Author(s):  
P. G. Casazza ◽  
Bor-Luh Lin ◽  
R. H. Lohman
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Reflexive Banach Space ◽  
Analytical Description ◽  
Basic Sequence ◽  
Isomorphic Copy ◽  
Uniformly Convex ◽  
Infinite Dimensional ◽  
Symmetric Basis ◽  
Uniformly Convex Space

The first infinite-dimensional reflexive Banach space X such that no subspace of X is isomorphic to c0 or lp, 1 ≦ p < ∞, was constructed by Tsirelson [8]. In fact, he showed that there exists a Banach space with an unconditional basis which contains no subsymmetric basic sequence and which contains no superreflexive subspace. Subsequently, Figiel and Johnson [4] gave an analytical description of the conjugate space T of Tsirelson's example and showed that there exists a reflexive Banach space with a symmetric basis which contains no superreflexive subspace; a uniformly convex space with a symmetric basis which contains no isomorphic copy of lp, 1 < p < ∞; and a uniformly convex space which contains no subsymmetric basic sequence and hence contains no isomorphic copy of lp, 1 < p < ∞. Recently, Altshuler [2] showed that there is a reflexive Banach space with a symmetric basis which has a unique symmetric basic sequence up to equivalence and which contains no isomorphic copy of lp, 1 < p < ∞.

On the Banach–Saks property

1977 ◽  
Vol 82 (3) ◽  
pp. 369-374 ◽  
Author(s):  
J. R. Partington
Keyword(s):  
Banach Space ◽  
Bounded Sequence ◽  
Uniformly Convex ◽  
Image Position ◽  
Uniformly Convex Spaces ◽  
Convex Spaces

A Banach space X is said to have the Banach–Saks property (BS) if every bounded sequence (xn) in X has a subsequence (), which is (C, 1) convergent in norm to a point x in X; that is,Kakutani (7) showed that all uniformly convex spaces are (BS); moreover, all (BS) spaces are reflexive. It is further known that both these implicationsare strict: see, for example, Baernstein (1) and Diestel (4).

A new class of Banach space with the drop property

2012 ◽  
Vol 142 (1) ◽  
pp. 215-224 ◽  
Author(s):  
Suyalatu Wulede ◽  
Wudunqiqige Ha
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Uniformly Convex ◽  
Strongly Convex ◽  
New Class ◽  
Uniformly Convex Spaces ◽  
Convex Spaces

We discuss a new class of Banach spaces which are wider than the strongly convex spaces introduced by Congxin Wu and Yongjin Li. We prove that the new class of Banach spaces lies strictly between either the class of uniformly convex spaces and strongly convex spaces or the class of fully k-convex spaces and strongly convex spaces. The new class of Banach spaces has inclusive relations with neither the class of locally uniformly convex spaces nor the class of nearly uniformly convex spaces. We obtain in addition some characterizations of this new class of Banach spaces.

A note on k-uniformly convex spaces

1985 ◽  
Vol 97 (3) ◽  
pp. 489-490
Author(s):  
Jong Sook Bae ◽  
Sung Kyu Choi
Keyword(s):  
Banach Space ◽  
Short Note ◽  
Uniform Convexity ◽  
Uniformly Convex ◽  
Uniformly Convex Spaces ◽  
Convex Spaces

AbstractIn this short note we prove that Istrǎƫescu's notion of k-uniform (k-locally uniform) convexity of a Banach space is actually equivalent to the notion of uniform (locally uniform) convexity. Thus theorem 2 in [3] and theorem 2·6·28 in [2] are trivially true.

A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi

1976 ◽  
Vol 19 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Joseph Bogin
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Compact Convex ◽  
Continuous Mapping ◽  
Normal Structure ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Image Position

In [7], Goebel, Kirk and Shimi proved the following:Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K:(1)where ai≥0 and Then F has a fixed point in K.In this paper we shall prove that this theorem remains true in any Banach space X, provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).

Sign in / Sign up

Export Citation Format

Share Document