A note on k-uniformly 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.

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On nearly uniformly convex and k-uniformly convex spaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100061594 ◽

1984 ◽

Vol 95 (2) ◽

pp. 325-327 ◽

Cited By ~ 3

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.

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Hadamard–Landau inequalities in uniformly convex spaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100058710 ◽

1981 ◽

Vol 90 (2) ◽

pp. 259-264 ◽

Cited By ~ 1

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).

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Strict and uniform convexity

Elements of Geometry of Balls in Banach Spaces ◽

10.1093/oso/9780198827351.003.0003 ◽

2018 ◽

pp. 28-47

Author(s):

Kazimierz Goebel ◽

Stanisław Prus

Keyword(s):

Main Tool ◽

Strict Convexity ◽

Uniform Convexity ◽

Uniformly Convex ◽

Modulus Of Convexity ◽

Nearest Point ◽

Uniformly Convex Spaces ◽

Convex Spaces

Ways to classify and measure convexity of balls are described. Properties like strict convexity, uniform convexity, and squareness are discussed. The main tool, the modulus of convexity of a space, is studied. In the case of uniformly convex spaces, nearest point projections and asymptotic centres of sequences are presented.

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On the Banach–Saks property

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100054025 ◽

1977 ◽

Vol 82 (3) ◽

pp. 369-374 ◽

Cited By ~ 14

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).

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A new class of Banach space with the drop property

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210510000545 ◽

2012 ◽

Vol 142 (1) ◽

pp. 215-224 ◽

Cited By ~ 3

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.

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