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).

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).

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.

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.

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.

scholarly journals A note on best approximation and invertibility of operators on uniformly convex Banach spaces

1991 ◽  
Vol 14 (3) ◽  
pp. 611-614 ◽  
Author(s):  
James R. Holub
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Best Approximation ◽  
Bounded Linear Operator ◽  
Convex Banach Space ◽  
Sufficient Condition ◽  
Uniformly Convex ◽  
Bounded Linear ◽  
Uniformly Convex Banach Spaces

It is shown that ifXis a uniformly convex Banach space andSa bounded linear operator onXfor which‖I−S‖=1, thenSis invertible if and only if‖I−12S‖<1. From this it follows that ifSis invertible onXthen either (i)dist(I,[S])<1, or (ii)0is the unique best approximation toIfrom[S], a natural (partial) converse to the well-known sufficient condition for invertibility thatdist(I,[S])<1.

The Range Sequence of an Operator

1974 ◽  
Vol 17 (1) ◽  
pp. 145-147
Author(s):  
F.-H. Vasilescu
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Linear Subspaces ◽  
Image Position

Let T be a linear operator on a Banach space X and consider the sequence of rangeswhere the inclusions are not necessarily proper. The linear subspaces Xn=TnX (n>0) are, in general, not closed but they have some remarkable properties [1], [2]. Let X0=X and denote by |x|0 (x∈X0) the norm of X0.

On Asymptotic Centers and Fixed Points of Nonexpansive Mappings

1980 ◽  
Vol 32 (2) ◽  
pp. 421-430 ◽  
Author(s):  
Teck-Cheong Lim
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Nonempty Subset ◽  
Nonexpansive Mappings ◽  
Chebyshev Center ◽  
Bounded Subset ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Asymptotic Centers ◽  
Image Position

Let X be a Banach space and B a bounded subset of X. For each x ∈ X, define R(x) = sup{‖x – y‖ : y ∈ B}. If C is a nonempty subset of X, we call the number R = inƒ{R(x) : x ∈ C} the Chebyshev radius of B in C and the set the Chebyshev center of B in C. It is well known that if C is weakly compact and convex, then and if, in addition, X is uniformly convex, then the Chebyshev center is unique; see e.g., [9].Let {Bα : α ∈ ∧} be a decreasing net of bounded subsets of X. For each x ∈ X and each α ∈ ∧, define

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