Unconditional Convergence of Weakly Sub-Gaussian Series in Banach Spaces

2007 ◽  
Vol 51 (2) ◽  
pp. 305-324 ◽  
Author(s):  
N. N. Vakhania ◽  
V. V. Kvaratskhelia
Keyword(s):  
Banach Spaces ◽  
Unconditional Convergence ◽  
Gaussian Series

Complex convexity and the geometry of Banach spaces

1986 ◽  
Vol 99 (3) ◽  
pp. 495-506 ◽  
Author(s):  
S. J. Dilworth
Keyword(s):  
Banach Spaces ◽  
Present Article ◽  
Uniform Convexity ◽  
Unconditional Convergence

The notion of PL-convexity was introduced in [4]. In the present article several results are proved which related PL-convexity to various aspects of the geometry of Banach spaces. The first section introduces the moduli of comples convexity and makes a comparison with the more familiar modulus of uniform convexity. It is shown that unconditional convergence of implies convergence of . In the next section the moduli and are shown to be related. The method of proof gives rise to a theorem about strict c-convexity of Lp(X) and a result on the representability in Lp(X).

Absolute and Unconditional Convergence in Normed Linear Spaces

1962 ◽  
Vol 58 (4) ◽  
pp. 575-579 ◽  
Author(s):  
D. Rutovitz
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Normed Linear Space ◽  
Unconditional Convergence ◽  
Geometrical Approach ◽  
Linear Spaces ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Open Question ◽  
General Banach Space

In 1933 Orlicz proved various results concerning unconditional convergence in Banach spaces (4), which were noted by Banach ((l), p. 240) who remarked that absolute and unconditional convergence are equivalent in finite dimensional Banach spaces, but that whether or not the two are non-equivalent in all infinite dimensional spaces was still an open question. MacPhail (3) gave a criterion for the equivalence of the two notions of convergence in a general Banach space and used it to prove non-equivalence in the spaces l1 and L1. In 1950 Dvoretzky and Rogers demonstrated the non-equivalence of the two types of convergence in any infinite dimensional normed linear space, using an elegant and instructive geometrical approach (2). The result has also been proved by a different method by Grothendieck (5).

Unconditional Convergence of Random Series and the Geometry of Banach Spaces

2000 ◽  
Vol 7 (1) ◽  
pp. 85-96 ◽  
Author(s):  
V. Kvaratskhelia
Keyword(s):  
Banach Spaces ◽  
Unconditional Convergence ◽  
Random Series

Abstract The a.s. unconditionally convergent random series are investigated. The connection of the a.s. unconditionally convergence with the geometry of spaces is established.

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