Eberlein compacts and spaces of continuous functions

1979 ◽  
Vol 85 (2) ◽  
pp. 305-313
Author(s):  
Richard J. Hunter ◽  
J. W. Lloyd
Keyword(s):  
Topological Space ◽  
Weak Topology ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Locally Convex Spaces ◽  
Weakly Compact ◽  
Spaces Of Continuous Functions ◽  
Eberlein Compact ◽  
Necessary And Sufficient ◽  
Locally Convex

AbstractLet X be a Hausdorff topological space. We consider various locally convex spaces of continuous real valued functions on X and give necessary and sufficient conditions in order that (i) they contain an absolutely convex weakly compact total subset and (ii) they contain an absolutely convex total subset which is an Eberlein compact, when given the weak topology.

Embeddings of weighted Sobolev spaces into spaces of continuous functions

1992 ◽  
Vol 439 (1906) ◽  
pp. 279-296 ◽  
Keyword(s):  
Weighted Sobolev Spaces ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Hölder Continuous ◽  
Compact Embeddings ◽  
Spaces Of Continuous Functions ◽  
Hölder Continuous Functions ◽  
Continuous And Compact Embeddings ◽  
Necessary And Sufficient ◽  
Theoretical Results

We give sufficient conditions and necessary conditions (which in some cases are both necessary and sufficient) for continuous and compact embeddings of the weighted Sobolev space W 1,p ( Ω ;v 0 v 1 )into spaces of weighted continuous and Holder continuous functions. The theoretical results are illustrated by several examples.

Projections and embeddings of locally convex operator spaces and their duals

1984 ◽  
Vol 96 (2) ◽  
pp. 321-323 ◽  
Author(s):  
Jan H. Fourie ◽  
William H. Ruckle
Keyword(s):  
Dual Space ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Operator Spaces ◽  
Complemented Subspace ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Continuous Linear Operators

AbstractLet E, F be Hausdorff locally convex spaces. In this note we consider conditions on E and F such that the dual space of the space Kb (E, F) (of quasi-compact operators) is a complemented subspace of the dual space of Lb (E, F) (of continuous linear operators). We obtain necessary and sufficient conditions for Lb(E, F) to be semi-reflexive.

XI.—On Unconditional Convergence in Topological Vector Spaces

1969 ◽  
Vol 68 (2) ◽  
pp. 145-157 ◽  
Author(s):  
A. P. Robertson
Keyword(s):  
Sufficient Conditions ◽  
Direct Proof ◽  
Vector Spaces ◽  
Unconditional Convergence ◽  
Partial Sums ◽  
Locally Convex Spaces ◽  
Topological Groups ◽  
Topological Vector Spaces ◽  
Necessary And Sufficient ◽  
Locally Convex

SynopsisFor a series of elements of a topological vector space, necessary and sufficient conditions are found, in terms of the set of finite partial sums, for unconditional convergence and for the corresponding Cauchy condition. The extent to which these results remain valid for topological groups is investigated. A new and direct proof, for locally convex spaces, is given of the theorem of Orlicz.

scholarly journals On Homomorphisms, Open Operators and their Adjoints

2001 ◽  
Vol 8 (4) ◽  
pp. 823-844
Author(s):  
D. Zarnadze
Keyword(s):  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Locally Convex Spaces ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Convex Spaces

Abstract The well-known A. Grothendieck's theorem on a homomorphism between locally convex spaces is generalized to the case of topologies which are incompatible with dualities. On the basis of this theorem, necessary and sufficient conditions are obtained for a weak homomorphism (resp. its adjoint operator, resp. its double adjoint operator) to be again a homomorphism in various topologies of the initial (resp. dual, resp. bidual) spaces. Some new classes of pairs of locally convex spaces satisfying these conditions are established. The results obtained have enabled us to reveal new properties of frequently encountered homomorphisms and weakly open operators, as well as to strengthen and generalize some well-known results.

scholarly journals Multiplication operators on weighted spaces in the non-locally convex framework

1997 ◽  
Vol 20 (1) ◽  
pp. 75-79 ◽  
Author(s):  
L. A. Khan ◽  
A. B. Thaheem
Keyword(s):  
Topological Space ◽  
Hausdorff Space ◽  
Weighted Space ◽  
Topological Algebra ◽  
Sufficient Conditions ◽  
Multiplication Operators ◽  
Completely Regular ◽  
General Topological Space ◽  
Necessary And Sufficient ◽  
Locally Convex

LetXbe a completely regular Hausdorff space,Ea topological vector space,Va Nachbin family of weights onX, andCV0(X,E)the weighted space of continuousE-valued functions onX. Letθ:X→Cbe a mapping,f∈CV0(X,E)and defineMθ(f)=θf(pointwise). In caseEis a topological algebra,ψ:X→Eis a mapping then defineMψ(f)=ψf(pointwise). The main purpose of this paper is to give necessary and sufficient conditions forMθandMψto be the multiplication operators onCV0(X,E)whereEis a general topological space (or a suitable topological algebra) which is not necessarily locally convex. These results generalize recent work of Singh and Manhas based on the assumption thatEis locally convex.

scholarly journals Necessary and sufficient conditions for precompact sets to be metrisable

2006 ◽  
Vol 74 (1) ◽  
pp. 7-13 ◽  
Author(s):  
J.C. Ferrando ◽  
J. Kasakol ◽  
M. López Pellicer
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Locally Convex Spaces ◽  
Compact Sets ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Convex Spaces ◽  
Compact Subsets

This self-contained paper characterises those locally convex spaces whose (weakly) precompact (respectively, compact) subsets are metrisable. Applications and examples are provided. Our approach also applies to get Cascales-Orihuela's, Valdivia's and Robertson's metrisation theorems for (pre)compact sets.

scholarly journals Note on pointwise contractive projections

1993 ◽  
Vol 16 (4) ◽  
pp. 817-818
Author(s):  
L. M. Sanchez Ruiz ◽  
J. R. Ferrer Villanueva
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Compact Open Topology ◽  
Completely Regular ◽  
Contractive Projection ◽  
Necessary And Sufficient ◽  
Contractive Projections

LetC(X)be the space of real-valued continuous functions on a Hausdorff completely regular topological spaceX. endowed with the compact-open topology. In this paper necessary and sufficient conditions are given for a subspace ofC(X)to be the range of a pointwise contractive projection inC(X).

scholarly journals A variant of a fixed point theorem of Browder-Fan and Reich

1985 ◽  
Vol 8 (4) ◽  
pp. 693-696
Author(s):  
V. M. Sehgal
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Compact Subset ◽  
Weak Topology ◽  
Fixed Point Theorems ◽  
Hausdorff Space ◽  
Sufficient Conditions ◽  
Weakly Compact ◽  
Continuous Seminorm ◽  
Locally Convex

LetSbe a convex, weakly compact subset of a locally convex Hausdorff space(E,τ)andf:S→Ebe a continuous multifunction from its weak topologyωtoτ. letρbe a continuous seminorm on(E,τ)and for subsetsA,BofEletp(A,B)=inf{p(x−y):x ϵ A, y ϵ B}. In this paper, sufficient conditions are developed for the existence of anx ϵ Ssatisfyingp(x,fx)=p(fx,S). The result is then used to prove several fixed point theorems.

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