A note on the weak topology of spaces $$C_k(X)$$ of continuous functions

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Jerzy Ka̧kol ◽  
Santiago Moll-López
Keyword(s):  
Weak Topology ◽  
Continuous Functions

On Additive Operators

1971 ◽  
Vol 23 (3) ◽  
pp. 468-480 ◽  
Author(s):  
N. A. Friedman ◽  
A. E. Tong
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Weak Topology ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Additive Functional ◽  
Representation Theorems ◽  
Image Position ◽  
Vector Valued ◽  
Kernel Representation

Representation theorems for additive functional have been obtained in [2, 4; 6-8; 10-13]. Our aim in this paper is to study the representation of additive operators.Let S be a compact Hausdorff space and let C(S) be the space of real-valued continuous functions defined on S. Let X be an arbitrary Banach space and let T be an additive operator (see § 2) mapping C(S) into X. We will show (see Lemma 3.4) that additive operators may be represented in terms of a family of “measures” {μh} which take their values in X**. If X is weakly sequentially complete, then {μh} can be shown to take their values in X and are vector-valued measures (i.e., countably additive in the norm) (see Lemma 3.7). And, if X* is separable in the weak-* topology, T may be represented in terms of a kernel representation satisfying the Carathéordory conditions (see [9; 11; §4]):

Spaces of Continuous Vector Functions as Duals

1988 ◽  
Vol 31 (1) ◽  
pp. 70-78 ◽  
Author(s):  
Michael Cambern ◽  
Peter Greim
Keyword(s):  
Hilbert Space ◽  
Weak Topology ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Continuous Vector ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractA well known result due to Dixmier and Grothendieck for spaces of continuous scalar-valued functions C(X), X compact Hausdorff, is that C(X) is a Banach dual if, and only if, Xis hyperstonean. Moreover, for hyperstonean X, the predual of C(X) is strongly unique. Here we obtain a formulation of this result for spaces of continuous vector-valued functions. It is shown that if E is a Hilbert space and C(X, (E, σ *) ) denotes the space of continuous functions on X to E when E is provided with its weak * ( = weak) topology, then C(X, (E, σ *) ) is a Banach dual if, and only if, X is hyperstonean. Moreover, for hyperstonean X, the predual of C(X, (E, σ *) ) is strongly unique.

Uniqueness of Preduals for Spaces of Continuous Vector Functions

1989 ◽  
Vol 32 (1) ◽  
pp. 98-104 ◽  
Author(s):  
Michael Cambern ◽  
Peter Greim
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Weak Topology ◽  
Reflexive Banach Space ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Continuous Vector

AbstractA. Grothendieck has shown that if the space C(X) is a Banach dual then X is hyperstonean; moreover, the predual of C(X) is strongly unique. In this article we give a vector analogue of Grothendieck's result. We show that if E* is a reflexive Banach space and C(X, (E*, σ*)) denotes the space of continuous functions on X to E* when E* is provided with its weak* (= weak) topology then the full content of Grothendieck's theorem for C(X) can be established for C(X,(E*,σ*)). This improves a result previously obtained for the case in which E* is Hilbert space.

Weak Continuity of a Composition Map Between Spaces of Compact Operators and Banach Valued Continuous Functions

1991 ◽  
Vol 34 (2) ◽  
pp. 145-146 ◽  
Author(s):  
Rajappa K. Asthagiri
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Weak Topology ◽  
Hausdorff Space ◽  
Compact Operators ◽  
Continuous Functions ◽  
Linear Operators ◽  
Weak Continuity ◽  
Sequential Continuity ◽  
Bounded Sets

AbstractThis paper characterizes the Banach space E for the sequential continuity and the continuity on bounded sets of the composition map m: C(S, E)wk x K{E,F)wk —> C(S,F)wk. Here, K(E,F) denotes the Banach space of compact linear operators on the Banach space E to the Banach space F with the usual operator norm, and for any Banach space E, Ewk denote the Banach space E with the weak topology. Also we denote by C(S, E) the Banach space of E valued continuous functions on a nonvoid compact Hausdorff space S with sup norm.

Approximation of Young measures by functions and application to a problem of optimal design for plates with variable thickness

1994 ◽  
Vol 124 (3) ◽  
pp. 399-422 ◽  
Author(s):  
E. Bonnetier ◽  
C. Conca
Keyword(s):  
Optimal Design ◽  
Variable Thickness ◽  
Weak Topology ◽  
Fundamental Theorem ◽  
Design Problem ◽  
Continuous Functions ◽  
Young Measures ◽  
Optimal Design Problem ◽  
Sequence Of Functions

Given a parametrised measure and a family of continuous functions (φn), we construct a sequence of functions (uk) such that, as k→∞, the functions φn(uk) converge to the corresponding moments of the measure,in the weak * topology. Using the sequence (uk) corresponding to a dense family of continuous functions, a proof of the fundamental theorem for Young measures is given.We apply these techniques to an optimal design problem for plates with variable thickness. The relaxation of the compliance functional involves three continuous functions of the thickness. We characterise a set of admissible generalised thicknesses, on which the relaxed functional attains its minimum.

scholarly journals Function spaces and the Mosco topology

1990 ◽  
Vol 42 (1) ◽  
pp. 7-19 ◽  
Author(s):  
Gerald Beer ◽  
Robert Tamaki
Keyword(s):  
Banach Spaces ◽  
Function Space ◽  
Weak Topology ◽  
Continuous Functions ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Compact Open Topology ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Weakly Continuous

Let X and Y be Banach spaces and let C(X, Y) be the functions from X to Y continuous with respect to the weak topology on X and the strong topology on Y. By the Mosco topology τM on C(X, Y) we mean the supremum of the Fell topologies determined by the weak and strong topologies on X × Y, where functions are identified with their graphs. The function space is Hausdorff if and only if both X and Y are reflexive. Moreover, τM coincides with the stronger compact-open topology on C(X, Y) provided X is reflexive and Y is finite dimensional. We also show convergence in either sense is properly weaker than continuous convergence, even for continuous linear functionals, whenever X is infinite dimensional. For real-valued weakly continuous functions, τM is the supremum of the Mosco epitopology and the Mosco hypotopology if and only if X is reflexive.

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.

Approximation in statistical sense to B -continuous functions by positive linear operators

2010 ◽  
Vol 47 (3) ◽  
pp. 289-298 ◽  
Author(s):  
Fadime Dirik ◽  
Oktay Duman ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Statistical Approximation ◽  
Classical Version ◽  
Statistical Sense

In the present work, using the concept of A -statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued B -continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.

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