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).

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).

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.

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.

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 On the average distance property in finite dimensional real Banach spaces

1995 ◽  
Vol 51 (1) ◽  
pp. 87-101 ◽  
Author(s):  
Reinhard Wolf
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Positive Real Number ◽  
Real Banach Space ◽  
Average Distance ◽  
Positive Real ◽  
Finite Dimensional ◽  
Distance Property ◽  
Average Distance Property ◽  
Image Position

The average distance Theorem of Gross implies that for each N-dimensional real Banach space E (N ≥ 2) there is a unique positive real number r(E) with the following property: for each positive integer n and for all (not necessarily distinct) x1, x2, …, xn, in E with ‖x1‖ = ‖x2‖ = … = ‖xn‖ = 1, there exists an x in E with ‖x‖ = 1 such that.In this paper we prove that if E has a 1-unconditional basis then r(E)≤2−(l/N) and equality holds if and only if E is isometrically isomorphic to Rn equipped with the usual 1-norm.

scholarly journals 1-Complemented Subspaces of Spaces With 1-Unconditional Bases

1997 ◽  
Vol 49 (6) ◽  
pp. 1242-1264 ◽  
Author(s):  
Beata Randrianantoanina
Keyword(s):  
Sequence Space ◽  
Lorentz Space ◽  
Linear Span ◽  
Full Description ◽  
Monotone Sequence ◽  
Closed Linear Span ◽  
Complex Spaces ◽  
Complemented Subspaces ◽  
Unconditional Bases ◽  
Constant Coefficients

AbstractWe prove that if X is a complex strictly monotone sequence space with 1-unconditional basis, Y ⊆ X has no bands isometric to ℓ22 and Y is the range of norm-one projection from X, then Y is a closed linear span a family of mutually disjoint vectors in X.We completely characterize 1-complemented subspaces and norm-one projections in complex spaces ℓp(ℓq) for 1 ≤ p,q > ∞.Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are 1-complemented in (real or complex) Orlicz or Lorentz sequence spaces. In particular if an Orlicz or Lorentz space X is not isomorphic to ℓp for some 1 ≤ p,q > ∞ then the only subspaces of X which are 1-complemented and disjointly supported are the closed linear spans of block bases with constant coefficients.

Some Imbedding Theorems for Sobolev Spaces

1971 ◽  
Vol 23 (3) ◽  
pp. 517-530 ◽  
Author(s):  
R. A. Adams ◽  
John Fournier
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Euclidean Space ◽  
Sobolev Space ◽  
Dimensional Euclidean Space ◽  
Dense Subspace ◽  
Bounded Subset ◽  
Partial Derivatives ◽  
Image Position ◽  
Derivatives Of

We shall be concerned throughout this paper with the Sobolev space Wm,p(G) and the existence and compactness (or lack of it) of its imbeddings (i.e. continuous inclusions) into various LP spaces over G, where G is an open, not necessarily bounded subset of n-dimensional Euclidean space En. For each positive integer m and each real p ≧ 1 the space Wm,p(G) consists of all u in LP(G) whose distributional partial derivatives of all orders up to and including m are also in LP(G). With respect to the norm1.1Wm,p(G) is a Banach space. It has been shown by Meyers and Serrin [9] that the set of functions in Cm(G) which, together with their partial derivatives of orders up to and including m, are in LP(G) forms a dense subspace of Wm,p(G).

scholarly journals On decay rates of the solutions of parabolic Cauchy problems

2020 ◽  
pp. 1-19
Author(s):  
José Bonet ◽  
Wolfgang Lusky ◽  
Jari Taskinen
Keyword(s):  
Cauchy Problem ◽  
Positive Integer ◽  
Linear Span ◽  
Decay Rates ◽  
Polynomial Decay ◽  
Parabolic Partial Differential Equations ◽  
Cauchy Problems ◽  
Arbitrary Positive Integer ◽  
Closed Linear Span ◽  
The Cauchy Problem

We consider the Cauchy problem for a general class of parabolic partial differential equations in the Euclidean space ℝ N . We show that given a weighted L p -space $L_w^p({\mathbb {R}}^N)$ with 1 ⩽ p < ∞ and a fast growing weight w, there is a Schauder basis $(e_n)_{n=1}^\infty$ in $L_w^p({\mathbb {R}}^N)$ with the following property: given an arbitrary positive integer m there exists n m  > 0 such that, if the initial data f belongs to the closed linear span of e n with n ⩾ n m , then the decay rate of the solution of the problem is at least t−m for large times t. The result generalizes the recent study of the authors concerning the classical linear heat equation. We present variants of the result having different methods of proofs and also consider finite polynomial decay rates instead of unlimited m.

scholarly journals Pre-Schauder Bases in Topological Vector Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1026 ◽  
Author(s):  
Francisco Javier García-Pacheco ◽  
Francisco Javier Pérez-Fernández
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Banach Spaces ◽  
Topological Vector Space ◽  
Linear Span ◽  
Vector Spaces ◽  
Schauder Basis ◽  
Topological Vector Spaces ◽  
Hausdorff Topological Vector Space ◽  
Schauder Bases

A Schauder basis in a real or complex Banach space X is a sequence ( e n ) n ∈ N in X such that for every x ∈ X there exists a unique sequence of scalars ( λ n ) n ∈ N satisfying that x = ∑ n = 1 ∞ λ n e n . Schauder bases were first introduced in the setting of real or complex Banach spaces but they have been transported to the scope of real or complex Hausdorff locally convex topological vector spaces. In this manuscript, we extend them to the setting of topological vector spaces over an absolutely valued division ring by redefining them as pre-Schauder bases. We first prove that, if a topological vector space admits a pre-Schauder basis, then the linear span of the basis is Hausdorff and the series linear span of the basis minus the linear span contains the intersection of all neighborhoods of 0. As a consequence, we conclude that the coefficient functionals are continuous if and only if the canonical projections are also continuous (this is a trivial fact in normed spaces but not in topological vector spaces). We also prove that, if a Hausdorff topological vector space admits a pre-Schauder basis and is w * -strongly torsionless, then the biorthogonal system formed by the basis and its coefficient functionals is total. Finally, we focus on Schauder bases on Banach spaces proving that every Banach space with a normalized Schauder basis admits an equivalent norm closer to the original norm than the typical bimonotone renorming and that still makes the basis binormalized and monotone. We also construct an increasing family of left-comparable norms making the normalized Schauder basis binormalized and show that the limit of this family is a right-comparable norm that also makes the normalized Schauder basis binormalized.

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Dongni Tan ◽  
Xujian Huang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Phase Function ◽  
Linear Isometry ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

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