scholarly journals A note on bump functions that locally depend on finitely many coordinates

1997 ◽  
Vol 56 (3) ◽  
pp. 447-451 ◽  
Author(s):  
M. Fabian ◽  
V. Zizler
Keyword(s):  
Banach Space ◽  
Linear Span ◽  
Bump Function ◽  
Closed Linear Span ◽  
Asplund Spaces

We show that if a continuous bump function on a Banach space X locally depends on finitely many elements of a set F in X*, then the norm closed linear span of F equals to X*. Some corollaries for Markuševič bases and Asplund spaces are derived.

scholarly journals Some results on the span of families of Banach valued independent, random variables

1991 ◽  
Vol 14 (2) ◽  
pp. 381-384
Author(s):  
Rohan Hemasinha
Keyword(s):  
Banach Space ◽  
Probability Space ◽  
Linear Span ◽  
Random Variables ◽  
Isomorphic Copy ◽  
Independent Random Variables ◽  
Closed Linear Span

LetEbe a Banach space, and let(Ω,ℱ,P)be a probability space. IfL1(Ω)contains an isomorphic copy ofL1[0,1]then inLEP(Ω)(1≤P<∞), the closed linear span of every sequence of independent,Evalued mean zero random variables has infinite codimension. IfEis reflexive orB-convex and1<P<∞then the closed(in LEP(Ω))linear span of any family of independent,Evalued, mean zero random variables is super-reflexive.

RUC-systems and Besselian systems in Banach spaces

1989 ◽  
Vol 106 (1) ◽  
pp. 163-168 ◽  
Author(s):  
D. J. H. Garling ◽  
N. Tomczak-Jaegermann
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Span ◽  
Random Variables ◽  
Biorthogonal System ◽  
Independent Random Variables ◽  
Closed Linear Span ◽  
Image Position

Let (rj) be a Rademacher sequence of random variables – that is, a sequence of independent random variables, with , for each j. A biorthogonal system in a Banach space X is called an RUC-system[l] if for every x in [ej] (the closed linear span of the vectors ej), the seriesconverges for almost every ω. A basis which, together with its coefficient functionals, forms an RUC-system is called an RUC-basis. A biorthogonal system is an RLTC-svstem if and only if there exists 1 ≤ K < ∞ such thatfor each x in [ej]: the RUC-constant of the system is the smallest constant K satisfying (1) (see [1], proposition 1.1).

A Note on Unconditional Bases

1972 ◽  
Vol 15 (3) ◽  
pp. 369-372 ◽  
Author(s):  
J. R. Holub ◽  
J. R. Retherford
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Linear Span ◽  
Unique Sequence ◽  
Schauder Basis ◽  
Norm Topology ◽  
Closed Linear Span ◽  
Unconditional Bases ◽  
Image Position

A sequence (xi) in a Banach space X is a Schauder basis for X provided for each x∊X there is a unique sequence of scalars (ai) such that1.1convergence in the norm topology. It is well known [1] that if (xi) is a (Schauder) basis for X and (fi) is defined by1.2where then fi(xj) = δij and fi∊X* for each positive integer i.A sequence (xi) is a éasic sequence in X if (xi) is a basis for [xi], where the bracketed expression denotes the closed linear span of (xi).

Monotone and E-Schauder Bases of Subspaces

1968 ◽  
Vol 20 ◽  
pp. 233-241 ◽  
Author(s):  
John P. Russo
Keyword(s):  
Normed Linear Space ◽  
Linear Span ◽  
Linear Topological Space ◽  
Principal Result ◽  
Schauder Basis ◽  
Topological Spaces ◽  
Dense Subspace ◽  
Closed Linear Span ◽  
Whole Space ◽  
Schauder Bases

The notions of monotone bases and bases of subspaces are well known in a normed linear space setting and have obvious extensions to pseudo-metrizable linear topological spaces. In this paper, these notions are extended to arbitrary linear topological spaces. The principal result gives a list of properties that are equivalent to a sequence (Mi) of complete subspaces being an e-Schauder basis of subspaces for the closed linear span of . A corollary of this theorem is the fact that an e-Schauder basis for a dense subspace of a linear topological space is an e-Schauder basis for the whole space.

Smooth Partitions of Unity on Banach Spaces

1998 ◽  
Vol 41 (2) ◽  
pp. 145-150
Author(s):  
R. Fry
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Bump Function

AbstractIt is shown that if a Banach space X admits a Ck-smooth bump function, and X* is Asplund, then X admits Ck-smooth partitions of unity.

scholarly journals A CONTINUITY CHARACTERIZATION OF ASPLUND SPACES

2011 ◽  
Vol 83 (3) ◽  
pp. 450-455
Author(s):  
J. R. GILES
Keyword(s):  
Banach Space ◽  
Asplund Space ◽  
Weakly Compact ◽  
Asplund Spaces ◽  
Subdifferential Mapping

AbstractA Banach space is an Asplund space if every continuous gauge has a point where the subdifferential mapping is Hausdorff weak upper semi-continuous with weakly compact image. This contributes towards the solution of a problem posed by Godefroy, Montesinos and Zizler.

Porosity, Γ‎N- and Γ‎-Null Sets

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Modulus Of Smoothness ◽  
Separable Space ◽  
Bump Function ◽  
Asplund Spaces ◽  
Porous Sets

This chapter introduces the notion of porosity “at infinity” (formally defined as porosity with respect to a family of subspaces) and discusses the main result, which shows that sets porous with respect to a family of subspaces are Γ‎ₙ-null provided X admits a continuous bump function whose modulus of smoothness (in the direction of this family) is controlled by tⁿ logⁿ⁻¹ (1/t). The first of these results characterizes Asplund spaces: it is shown that a separable space has separable dual if and only if all its porous sets are Γ‎₁-null. The chapter first describes porous and σ‎-porous sets as well as a criterion of Γ‎ₙ-nullness of porous sets. It then considers the link between directional porosity and Γ‎ₙ-nullness. Finally, it tackles the question in which spaces, and for what values of n, porous sets are Γ‎ₙ-null.

scholarly journals Separable determination of Fréchet differentiability of convex functions

1995 ◽  
Vol 52 (1) ◽  
pp. 161-167 ◽  
Author(s):  
J.R. Giles ◽  
Scott Sciffer
Keyword(s):  
Banach Space ◽  
Open Convex ◽  
Asplund Spaces ◽  
Fréchet Differentiability ◽  
Continuous Convex Function ◽  
Separability Condition ◽  
Frechet Differentiability ◽  
Open Convex Subset ◽  
Insight Into

For a continuous convex function on an open convex subset of any Banach space a separability condition on its image under the subdifferential mapping is sufficient to guarantee the generic Fréchet differentiability of the function. This gives a direct insight into the characterisation of Asplund spaces.

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