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.

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).

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 Hidden regular variation of moving average processes with heavy-tailed innovations

2014 ◽  
Vol 51 (A) ◽  
pp. 267-279 ◽  
Author(s):  
Sidney I. Resnick ◽  
Joyjit Roy
Keyword(s):  
Sequence Space ◽  
Regular Variation ◽  
Moving Average ◽  
Random Variables ◽  
Continuity Properties ◽  
Hidden Regular Variation ◽  
Constant Coefficients ◽  
Moving Average Processes ◽  
Regularly Varying ◽  
Heavy Tailed

We look at joint regular variation properties of MA(∞) processes of the form X = (Xk, k ∈ Z), where Xk = ∑j=0∞ψjZk-j and the sequence of random variables (Zi, i ∈ Z) are independent and identically distributed with regularly varying tails. We use the setup of MO-convergence and obtain hidden regular variation properties for X under summability conditions on the constant coefficients (ψj: j ≥ 0). Our approach emphasizes continuity properties of mappings and produces regular variation in sequence space.

scholarly journals Character Density in Central Subalgebras of Compact Quantum Groups

2017 ◽  
Vol 60 (3) ◽  
pp. 449-461 ◽  
Author(s):  
Mahmood Alaghmandan ◽  
Jason Crann
Keyword(s):  
Fourier Analysis ◽  
Quantum Group ◽  
Quantum Groups ◽  
Linear Span ◽  
Compact Groups ◽  
Closed Linear Span ◽  
Weak Density ◽  
Compact Quantum Groups ◽  
Open Question ◽  
Density Results

AbstractWe investigate quantum group generalizations of various density results from Fourier analysis on compact groups. In particular, we establish the density of characters in the space of fixed points of the conjugation action on L2(𝔾) and use this result to show the weak* density and normdensity of characters in ZL∞(G) and ZC(G), respectively. As a corollary, we partially answer an open question of Woronowicz. At the level of L1(G), we show that the center Z(L1(G)) is precisely the closed linear span of the quantum characters for a large class of compact quantum groups, including arbitrary compact Kac algebras. In the latter setting, we show, in addition, that Z(L1(G)) is a completely complemented Z(L1(G))-submodule of L1(G).

scholarly journals Hidden regular variation of moving average processes with heavy-tailed innovations

2014 ◽  
Vol 51 (A) ◽  
pp. 267-279
Author(s):  
Sidney I. Resnick ◽  
Joyjit Roy
Keyword(s):  
Sequence Space ◽  
Regular Variation ◽  
Moving Average ◽  
Random Variables ◽  
Continuity Properties ◽  
Hidden Regular Variation ◽  
Constant Coefficients ◽  
Moving Average Processes ◽  
Regularly Varying ◽  
Heavy Tailed

We look at joint regular variation properties of MA(∞) processes of the form X = (X k , k ∈ Z), where X k = ∑ j=0 ∞ψ j Z k-j and the sequence of random variables (Z i , i ∈ Z) are independent and identically distributed with regularly varying tails. We use the setup of M O -convergence and obtain hidden regular variation properties for X under summability conditions on the constant coefficients (ψ j : j ≥ 0). Our approach emphasizes continuity properties of mappings and produces regular variation in sequence space.

The Measure Algebra as an Operator Algebra

1968 ◽  
Vol 20 ◽  
pp. 1391-1396 ◽  
Author(s):  
Donald E. Ramirez
Keyword(s):  
Operator Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Linear Span ◽  
Measure Algebra ◽  
Linear Functionals ◽  
Bounded Operators ◽  
Bounded Linear ◽  
Closed Linear Span ◽  
Arens Multiplication

In § I, it is shown that M(G)*, the space of bounded linear functionals on M(G), can be represented as a semigroup of bounded operators on M(G).Let △ denote the non-zero multiplicative linear functionals on M(G) and let P be the norm closed linear span of △ in M(G)*. In § II, it is shown that P, with the Arens multiplication, is a commutative B*-algebra with identity. Thus P = C(B), where B is a compact, Hausdorff space.

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).

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