Extensions of Continuous and Lipschitz Functions

2000 ◽  
Vol 43 (2) ◽  
pp. 208-217 ◽  
Author(s):  
Eva Matoušková
Keyword(s):  
Closed Subset ◽  
Hausdorff Space ◽  
Lipschitz Constant ◽  
Continuous Extension ◽  
Supremum Norm ◽  
Entire Space ◽  
Continuous Norm ◽  
Weakly Continuous ◽  
Lower Semi ◽  
Weakly Closed

AbstractWe show a result slightly more general than the following. Let K be a compact Hausdorff space, F a closed subset of K, and d a lower semi-continuous metric on K. Then each continuous function ƒ on F which is Lipschitz in d admits a continuous extension on K which is Lipschitz in d. The extension has the same supremum norm and the same Lipschitz constant.As a corollary we get that a Banach space X is reflexive if and only if each bounded, weakly continuous and norm Lipschitz function defined on a weakly closed subset of X admits a weakly continuous, norm Lipschitz extension defined on the entire space X.

scholarly journals Compactifications of totally bounded quasi-uniform spaces

1986 ◽  
Vol 28 (1) ◽  
pp. 31-36 ◽  
Author(s):  
P. Fletcher ◽  
W. F. Lindgren
Keyword(s):  
Hausdorff Space ◽  
Uniform Space ◽  
Continuous Extension ◽  
Sufficient Condition ◽  
Uniform Spaces ◽  
Completely Regular ◽  
Totally Bounded ◽  
Continuous Map ◽  
Total Boundedness ◽  
Uniformly Continuous

The notation and terminology of this paper coincide with that of reference [4], except that here the term, compactification, refers to a T1-space. It is known that a completely regular totally bounded Hausdorff quasi-uniform space (X, ) has a Hausdorff compactification if and only if contains a uniformity compatible with ℱ() [4, Theorem 3.47]. The use of regular filters by E. M. Alfsen and J. E. Fenstad [1] and O. Njåstad [5], suggests a construction of a compactification, which differs markedly from the construction obtained in [4]. We use this construction to show that a totally bounded T1 quasi-uniform space has a compactification if and only if it is point symmetric. While it is pleasant to have a characterization that obtains for all T1-spaces, the present construction has several further attributes. Unlike the compactification obtained in [4], the compactification given here preserves both total boundedness and uniform weight, and coincides with the uniform completion when the quasi-uniformity under consideration is a uniformity. Moreover, any quasi-uniformly continuous map from the underlying quasi-uniform space of the compactification onto any totally bounded compact T1-space has a quasi-uniformly continuous extension to the compactification. If is the Pervin quasi-uniformity of a T1-space X, the compactification we obtain is the Wallman compactification of (X, ℱ ()). It follows that our construction need not provide a Hausdorff compactification, even when such a compactification exists; but we obtain a sufficient condition in order that our compactification be a Hausdorff space and note that this condition is satisfied by all uniform spaces and all normal equinormal quasi-uniform spaces. Finally, we note that our construction is reminiscent of the completion obtained by Á. Császár for an arbitrary quasi-uniform space [2, Section 3]; in particular our Theorem 3.7 is comparable with the result of [2, Theorem 3.5].

scholarly journals On the theorem of Kishi for a continuous function-kernel

1974 ◽  
Vol 53 ◽  
pp. 127-135 ◽  
Author(s):  
Isao Higuchi ◽  
Masayuki Itô
Keyword(s):  
Continuous Function ◽  
Potential Theory ◽  
Compact Hausdorff Space ◽  
Symmetric Function ◽  
Hausdorff Space ◽  
Locally Compact ◽  
Continuity Principle ◽  
Lower Semi ◽  
Locally Compact Hausdorff Space

In the potential theory with respect to a non-symmetric function-kernel, the following theorem is obtained by M. Kishi ([3]).Let X be a locally compact Hausdorff space and G be a lower semi-continuous function-kernel on X. Assume that G(x, x)>0 for any x in X and that G and the adjoint kernel Ğ of G satisfy “the continuity principle”.

A note on elementary equivalence of C(K) spaces

1987 ◽  
Vol 52 (2) ◽  
pp. 368-373 ◽  
Author(s):  
S. Heinrich ◽  
C. Ward Henson ◽  
L. C. Moore
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Supremum Norm ◽  
Closed Unit Ball ◽  
Elementary Equivalence ◽  
Bounded Number ◽  
Totally Disconnected

In this paper we give a closer analysis of the elementary properties of the Banach spaces C(K), where K is a totally disconnected, compact Hausdorff space, in terms of the Boolean algebra B(K) of clopen subsets of K. In particular we sharpen a result in [4] by showing that if B(K1) and B(K2) satisfy the same sentences with ≤ n alternations of quantifiers, then the same is true of C(K1) and C(K2). As a consequence we show that for each n there exist C(K) spaces which are elementarily equivalent for sentences with ≤ n quantifier alternations, but which are not elementary equivalent in the full sense. Thus the elementary properties of Banach spaces cannot be determined by looking at sentences with a bounded number of quantifier alternations.The notion of elementary equivalence for Banach spaces which is studied here was introduced by the second author [4] and is expressed using the language of positive bounded formulas in a first-order language for Banach spaces. As was shown in [4], two Banach spaces are elementarily equivalent in this sense if and only if they have isometrically isomorphic Banach space ultrapowers (or, equivalently, isometrically isomorphic nonstandard hulls.)We consider Banach spaces over the field of real numbers. If X is such a space, Bx will denote the closed unit ball of X, Bx = {x ϵ X∣ ∣∣x∣∣ ≤ 1}. Given a compact Hausdorff space K, we let C(K) denote the Banach space of all continuous real-valued functions on K, under the supremum norm. We will especially be concerned with such spaces when K is a totally disconnected compact Hausdorff space. In that case B(K) will denote the Boolean algebra of all clopen subsets of K. We adopt the standard notation from model theory and Banach space theory.

scholarly journals STRONGLY BOUNDED REPRESENTING MEASURES AND CONVERGENCE THEOREMS

2010 ◽  
Vol 52 (3) ◽  
pp. 435-445 ◽  
Author(s):  
IOANA GHENCIU ◽  
PAUL LEWIS
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Convergence Theorems

AbstractLet K be a compact Hausdorff space, X a Banach space and C(K, X) the Banach space of all continuous functions f: K → X endowed with the supremum norm. In this paper we study weakly precompact operators defined on C(K, X).

Characterization of some classes of operators on spaces of vector-valued continuous functions

1985 ◽  
Vol 97 (1) ◽  
pp. 137-146 ◽  
Author(s):  
Fernando Bombal ◽  
Pilar Cembranos
Keyword(s):  
Banach Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Representing Measure ◽  
Bounded Linear ◽  
Second Dual ◽  
Image Position ◽  
Vector Valued

Let K be a compact Hausdorff space and E, F Banach spaces. We denote by C(K, E) the Banach space of all continuous. E-valued functions defined on K, with the supremum norm. It is well known ([6], [7]) that every operator (= bounded linear operator) T from C(K, E) to F has a finitely additive representing measure m of bounded semi-variation, defined on the Borel σ-field Σ of K and with values in L(E, F″) (the space of all operators from E into the second dual of F), in such a way thatwhere the integral is considered in Dinculeanu's sense.

Formal Power Series Over Commutative N-Algebras

1978 ◽  
Vol 30 (01) ◽  
pp. 66-84 ◽  
Author(s):  
Ernst August Behrens
Keyword(s):  
Power Series ◽  
Banach Algebra ◽  
Formal Power Series ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Identity Element ◽  
Closed Ideal ◽  
Supremum Norm ◽  
Maximal Ideals ◽  
Formal Power

A Banach algebra P over C with identity element is called an N-algebra if any closed ideal in P is the intersection of maximal ideals. An example is given by the algebra of the continuous C-valued functions on a compact Hausdorff space X under the supremum norm; two others are discussed in § 3.

scholarly journals Compact and extremally disconnected spaces

2004 ◽  
Vol 2004 (20) ◽  
pp. 1047-1056
Author(s):  
Bhamini M. P. Nayar
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Closed Subset ◽  
Hausdorff Space ◽  
Hausdorff Spaces ◽  
Compact Spaces ◽  
Hausdorff Topological Space ◽  
Extremally Disconnected

Viglino defined a Hausdorff topological space to beC-compact if each closed subset of the space is anH-set in the sense of Veličko. In this paper, we study the class of Hausdorff spaces characterized by the property that each closed subset is anS-set in the sense of Dickman and Krystock. Such spaces are calledC-s-compact. Recently, the notion of strongly subclosed relation, introduced by Joseph, has been utilized to characterizeC-compact spaces as those with the property that each function from the space to a Hausdorff space with a strongly subclosed inverse is closed. Here, it is shown thatC-s-compact spaces are characterized by the property that each function from the space to a Hausdorff space with a strongly sub-semiclosed inverse is a closed function. It is established that this class of spaces is the same as the class of Hausdorff, compact, and extremally disconnected spaces. The class ofC-s-compact spaces is properly contained in the class ofC-compact spaces as well as in the class ofS-closed spaces of Thompson. In general, a compact space need not beC-s-compact. The product of twoC-s-compact spaces need not beC-s-compact.

scholarly journals Varieties of topological groups II

1970 ◽  
Vol 2 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Sidney A. Morris
Keyword(s):  
Free Group ◽  
Compact Hausdorff Space ◽  
Closed Subset ◽  
Hausdorff Space ◽  
Projective Group ◽  
Necessary And Sufficient Condition ◽  
Topological Groups ◽  
Finitely Generated ◽  
Tychonoff Space ◽  
Necessary And Sufficient

This paper is a sequel to one entitled “Varieties of topological groups”. The variety V of topological groups is said to be full if it contains every group which is algebraically isomorphic to a group in V For any Tychonoff space X, the free group F of V on X exists, is Hausdorff and disconnected, and has X as a closed subset. Any subgroup of F which is algebraically fully invariant is a closed subset of F. If X is a compact Hausdorff space, then F is normal. Let V be a full Schreier variety and X a Tychonoff space, then all finitely generated subgroups of F are free in V.A β-variety V is one for which the free group of V on each compact Hausdorff space exists and is Hausdorff. For any β-variety V and Tychonoff space X, the free group of V exists, is Hausdorff and has X as a closed subset. A necessary and sufficient condition for V to be a β-variety is given.The concept of a projective (topological) group of a variety V is introduced. The projective groups of V are shown to be precisely the summands of the free groups of V. A finitely generated Hausdorff projective group of a Schreier variety V is free in V.

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