scholarly journals Axiomatisations of the average and a further generalisation of monotonic sequences

1974 ◽  
Vol 15 (1) ◽  
pp. 63-65 ◽  
Author(s):  
John Bibby
Keyword(s):  
Bounded Sequence ◽  
Monotonic Sequence

A bounded monotonic sequence is convergent. This paper shows that a bounded sequence which is g-monotonic (to be defined) also converges. The proof generalises one attributed to Professor R. A. Rankin by Copson [1]. The theorem requires two definitions: the first axiomatises the notion of “average“ and the second generalises the concept of monotonicity.

On a generalisation of monotonic sequences

1970 ◽  
Vol 17 (2) ◽  
pp. 159-164 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Difference Equation ◽  
Bounded Sequence ◽  
Real Numbers ◽  
Monotonic Sequence ◽  
Image Position

A bounded monotonic sequence is convergent. Dr J. M. Whittaker recently suggested to me a generalisation of this result, that, if a bounded sequence {an} of real numbers satisfies the inequalitythen it is convergent. This I was able to prove by considering the corresponding difference equation

VMO Space Associated with Parabolic Sections and its Application

2015 ◽  
Vol 58 (3) ◽  
pp. 507-518
Author(s):  
Ming-Hsiu Hsu ◽  
Ming-Yi Lee
Keyword(s):  
Hardy Space ◽  
Weak Convergence ◽  
Bounded Sequence ◽  
Almost Everywhere Convergence ◽  
Almost Everywhere

AbstractIn this paper we define a space VMO𝒫 associated with a family 𝒫 of parabolic sections and show that the dual of VMO𝒫 is the Hardy space . As an application, we prove that almost everywhere convergence of a bounded sequence in implies weak* convergence

Center Points Of Nets

1975 ◽  
Vol 27 (2) ◽  
pp. 418-422 ◽  
Author(s):  
C. L. Anderson ◽  
W. H. Hyams ◽  
C. K. McKnight
Keyword(s):  
Metric Space ◽  
Limit Point ◽  
Cauchy Sequence ◽  
Bounded Sequence ◽  
Center Point

Suppose x = (x∝) is a net with values in a metric space X having metric ρ. If a point z in X can be found to minimizethen z is called a center point (c.p.) of x. The space X is (netwise) c.p. complete if every bounded net has at least one c.p.; it is sequentially c.p. complete if every bounded sequence has a c.p. Netwise c.p. completeness implies sequential c.p. completeness, and the latter implies completeness since any c.p. of a Cauchy sequence will necessarily be a limit point of that sequence.These notions are related to the set centers of Calder et al. [2].

scholarly journals Kadec–Pełczyński decomposition for Haagerup Lp-spaces

2002 ◽  
Vol 132 (1) ◽  
pp. 137-154 ◽  
Author(s):  
NARCISSE RANDRIANANTOANINA
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Bounded Sequence ◽  
Point Property ◽  
Weakly Compact ◽  
Von Neumann ◽  
Lp Spaces ◽  
Reflexive Subspace ◽  
Finite Von Neumann Algebras ◽  
Bounded Sequences

Let [Mscr ] be a von Neumann algebra (not necessarily semi-finite). We provide a generalization of the classical Kadec–Pełczyński subsequence decomposition of bounded sequences in Lp[0, 1] to the case of the Haagerup Lp-spaces (1 [les ] p < 1 ). In particular, we prove that if { φn}∞n=1 is a bounded sequence in the predual [Mscr ]∗ of [Mscr ], then there exist a subsequence {φnk}∞k=1 of {φn}∞n=1, a decomposition φnk = yk+zk such that {yk, k [ges ] 1} is relatively weakly compact and the support projections supp(zk) ↓k 0 (or similarly mutually disjoint). As an application, we prove that every non-reflexive subspace of the dual of any given C*-algebra (or Jordan triples) contains asymptotically isometric copies of [lscr ]1 and therefore fails the fixed point property for non-expansive mappings. These generalize earlier results for the case of preduals of semi-finite von Neumann algebras.

scholarly journals Statistical limit point theorems

2000 ◽  
Vol 23 (11) ◽  
pp. 741-752 ◽  
Author(s):  
Jeff Zeager
Keyword(s):  
Complex Number ◽  
Limit Point ◽  
Statistical Convergence ◽  
Nonnegative Matrix ◽  
Bounded Sequence ◽  
Regular Matrix ◽  
Number Sequence ◽  
Statistical Limit ◽  
Limit Points

It is known that given a regular matrixAand a bounded sequencexthere is a subsequence (respectively, rearrangement, stretching)yofxsuch that the set of limit points ofAyincludes the set of limit points ofx. Using the notion of a statistical limit point, we establish statistical convergence analogues to these results by proving that every complex number sequencexhas a subsequence (respectively, rearrangement, stretching)ysuch that every limit point ofxis a statistical limit point ofy. We then extend our results to the more generalA-statistical convergence, in whichAis an arbitrary nonnegative matrix.

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