On the Support of a Measure in a Banach Space and Finite Representability

1992 ◽  
Vol 36 (2) ◽  
pp. 381-385 ◽  
Author(s):  
I. K. Matsak ◽  
A. N. Plichko
Keyword(s):  
Banach Space ◽  
Finite Representability ◽  
Support Of A Measure

COMPARING THE GENERALISED ROUNDNESS OF METRIC SPACES

2016 ◽  
Vol 95 (2) ◽  
pp. 299-314
Author(s):  
LUKIEL LEVY-MOORE ◽  
MARGARET NICHOLS ◽  
ANTHONY WESTON
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Metric Spaces ◽  
New Technique ◽  
Local Theory ◽  
Second Dual ◽  
Finite Representability ◽  
A New Technique

Motivated by the local theory of Banach spaces, we introduce a notion of finite representability for metric spaces. This allows us to develop a new technique for comparing the generalised roundness of metric spaces. We illustrate this technique by applying it to Banach spaces and metric trees. In the realm of Banach spaces we obtain results such as the following: (1) if${\mathcal{U}}$is any ultrafilter and$X$is any Banach space, then the second dual$X^{\ast \ast }$and the ultrapower$(X)_{{\mathcal{U}}}$have the same generalised roundness as$X$, and (2) no Banach space of positive generalised roundness is uniformly homeomorphic to$c_{0}$or$\ell _{p}$,$2<p<\infty$. For metric trees, we give the first examples of metric trees of generalised roundness one that have finite diameter. In addition, we show that metric trees of generalised roundness one possess special Euclidean embedding properties that distinguish them from all other metric trees.

scholarly journals SYMMETRIC FINITE REPRESENTABILITY OF ℓp IN ORLICZ SPACES

2021 ◽  
Vol 26 (4) ◽  
pp. 15-24
Author(s):  
S. V. Astashkin
Keyword(s):  
Banach Space ◽  
Orlicz Space ◽  
Orlicz Spaces ◽  
Large Dimension ◽  
Complete Proof ◽  
Finite Dimensional ◽  
Finite Representability

It is well known that a Banach space need not contain any subspace isomorphic to a space ℓp (1 6 p ) or c0 (it was shown by Tsirelson in 1974). At the same time, by the famous Krivines theorem, every Banach space X always contains at least one of these spaces locally, i.e., there exist finite-dimensional subspaces of X of arbitrarily large dimension n which are isomorphic (uniformly) to ℓnp for some 1 6 p or cn0 . In thiscase one says that ℓp (resp. c0) is finitely representable in X. The main purpose of this paper is to give a characterization (with a complete proof) of the set of p such that ℓp is symmetrically finitely representable in a separable Orlicz space.

On the domain of Riesz mean in the space Ls

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 925-940 ◽  
Author(s):  
Medine Yeşilkayagil ◽  
Feyzi Başar
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Double Sequence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Double Sequences ◽  
Riesz Mean ◽  
Double Sequence Space ◽  
Necessary And Sufficient ◽  
Barrelled Space

Let 0 < s < ?. In this study, we introduce the double sequence space Rqt(Ls) as the domain of four dimensional Riesz mean Rqt in the space Ls of absolutely s-summable double sequences. Furthermore, we show that Rqt(Ls) is a Banach space and a barrelled space for 1 ? s < 1 and is not a barrelled space for 0 < s < 1. We determine the ?- and ?(?)-duals of the space Ls for 0 < s ? 1 and ?(bp)-dual of the space Rqt(Ls) for 1 < s < 1, where ? ? {p, bp, r}. Finally, we characterize the classes (Ls:Mu), (Ls:Cbp), (Rqt(Ls) : Mu) and (Rqt(Ls):Cbp) of four dimensional matrices in the cases both 0 < s < 1 and 1 ? s < 1 together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.

Random iterative algorithms and almost sure stability in Banach spaces

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3611-3626 ◽  
Author(s):  
Abdul Khan ◽  
Vivek Kumar ◽  
Satish Narwal ◽  
Renu Chugh
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Iterative Algorithms ◽  
Random Operator ◽  
Almost Sure Stability ◽  
Random Fixed Point ◽  
Stochastic Version ◽  
Contractive Type ◽  
Bochner Integrability ◽  
Weakly Contractive

Many popular iterative algorithms have been used to approximate fixed point of contractive type operators. We define the concept of generalized ?-weakly contractive random operator T on a separable Banach space and establish Bochner integrability of random fixed point and almost sure stability of T with respect to several random Kirk type algorithms. Examples are included to support new results and show their validity. Our work generalizes, improves and provides stochastic version of several earlier results by a number of researchers.

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