scholarly journals Approximation on Boundary Sets

1979 ◽  
Vol 22 (3) ◽  
pp. 377-379
Author(s):  
James Li-Ming Wang
Keyword(s):  
Complex Plane ◽  
Open Subset ◽  
Linear Span ◽  
Infinite Length ◽  
Closed Linear Span ◽  
Bounded Open Subset ◽  
Boundary Sets ◽  
Uniformly Closed ◽  
Dirichlet Algebra

AbstractLet U be a bounded open subset of the complex plane. By a well known result of A. M. Davie, C(bU) is the uniformly-closed linear span of A(U) and the powers (z-zi)-n, n = 1, 2, 3, … with zi a point in each component of U. We show that if A(U) is a Dirichlet algebra and bU is of infinite length, then one power of (z - zi) is superfluous.

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 Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians

2020 ◽  
Vol 18 (1) ◽  
pp. 1-9
Author(s):  
Carlo Mariconda ◽  
Giulia Treu
Keyword(s):  
Convex Function ◽  
Calculus Of Variations ◽  
Open Subset ◽  
Lipschitz Function ◽  
Lavrentiev Phenomenon ◽  
Smooth Functions ◽  
Bounded Open Subset ◽  
Admissible Functions

Abstract We consider the classical functional of the Calculus of Variations of the form $$\begin{array}{} \displaystyle I(u)=\int\limits_{{\it\Omega}}F(x, u(x), \nabla u(x))\,dx, \end{array}$$ where Ω is a bounded open subset of ℝn and F : Ω × ℝ × ℝn → ℝ is a Carathéodory convex function; the admissible functions u coincide with a prescribed Lipschitz function ϕ on ∂Ω. We formulate some conditions under which a given function in ϕ + $\begin{array}{} \displaystyle W^{1,p}_0 \end{array}$(Ω) with I(u) < +∞ can be approximated in the W1,p norm and in energy by a sequence of smooth functions that coincide with ϕ on ∂Ω. As a particular case we obtain that the Lavrentiev phenomenon does not occur when F(x, u, ξ) = f(x, u) + h(x, ξ) is convex and x ↦ F(x, 0, 0) is sufficiently smooth.

The universal multiplicity theory for analytic operator-valued functions

1995 ◽  
Vol 118 (2) ◽  
pp. 315-320 ◽  
Author(s):  
Jón Arason ◽  
Robert Magnus
Keyword(s):  
Banach Space ◽  
Complex Plane ◽  
Open Subset ◽  
Closed Subset ◽  
Complex Banach Space ◽  
Singular Set ◽  
Analytic Operator ◽  
Set Of Points

An analytic operator-valued function A is an analytic map A: D → L(E, E), where D = D(A) is an open subset of the complex plane C and E = E(A) is a complex Banach space. For such a function A the singular set σ(A) of A is defined as the set of points z ∈ D such that A(z) is not invertible. It is a relatively closed subset of D.

scholarly journals THE CLOSED RANGE PROPERTY FOR BANACH SPACE OPERATORS

2008 ◽  
Vol 50 (1) ◽  
pp. 17-26 ◽  
Author(s):  
THOMAS L. MILLER ◽  
VLADIMIR MÜLLER
Keyword(s):  
Banach Space ◽  
Complex Plane ◽  
Open Subset ◽  
Weak Topology ◽  
Bounded Operator ◽  
Complex Banach Space ◽  
Fréchet Space ◽  
Closed Range ◽  
Open Set ◽  
Banach Space Operators

AbstractLetTbe a bounded operator on a complex Banach spaceX. LetVbe an open subset of the complex plane. We give a condition sufficient for the mappingf(z)↦ (T−z)f(z) to have closed range in the Fréchet spaceH(V,X) of analyticX-valued functions onV. Moreover, we show that there is a largest open setUfor which the mapf(z)↦ (T−z)f(z) has closed range inH(V,X) for allV⊆U. Finally, we establish analogous results in the setting of the weak–* topology onH(V, X*).

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 Effective energy integral functionals for thin films in the Orlicz–Sobolev space setting

2013 ◽  
Vol 46 (3) ◽  
Author(s):  
Włodzimierz Laskowski ◽  
Hong Thai Nguyen
Keyword(s):  
Thin Film ◽  
Thin Films ◽  
Sobolev Space ◽  
Open Subset ◽  
Integral Functionals ◽  
Effective Energy ◽  
Energy Integral ◽  
Space Setting ◽  
Bounded Open Subset

AbstractWe consider an elastic thin film as a bounded open subset

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 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.

Homogenization of the Torsion Problem and the Neumann Problem in Nonregular Periodically Perforated Domains

1997 ◽  
Vol 07 (06) ◽  
pp. 847-870 ◽  
Author(s):  
Marc Briane
Keyword(s):  
Asymptotic Behavior ◽  
Neumann Problem ◽  
Open Subset ◽  
Linear Operators ◽  
Periodic Case ◽  
Torsion Problem ◽  
Stiff Problem ◽  
Perforated Domains ◽  
Bounded Open Subset ◽  
The Neumann Problem

This paper is devoted to the homogenization of the torsion problem (or stiff problem) and the Neumann problem (or soft problem) for second-order elliptic but not necessarily symmetric linear operators set in a bounded open subset Ω of ℝN. More precisely, we study the asymptotic behavior of the equations [Formula: see text] in Ω with ε → 0, where Sε is a closed subset of Ω, which represents the set of the inclusions for the stiff problem or the holes for the soft one, and Ωε = Ω \ Sε. The stiff problem corresponds to δ → + ∞ and ν = 0, the soft one to δ → 0 and ν = 1. We prove a homogenization result in the periodic case without assuming any regularity on the set Sε and thus generalizing the result of Cioranescu and Saint Jean Paulin.7

Sign in / Sign up

Export Citation Format

Share Document