On linear operators and functors extending pseudometrics
| Autor: | C. Bessaga |
|---|---|
| Rok vydání: | 1993 |
| Předmět: | |
| Zdroj: | Fundamenta Mathematicae. 142:101-122 |
| ISSN: | 1730-6329 0016-2736 |
| Popis: | For some pairs (X,A), where X is a metrizable topological space and A its closed subset, continuous, linear (i.e., additive and positive-homogeneous) operators extending metrics for A to metrics for X are constructed. They are defined by explicit analytic formulas, and also regarded as functors between certain categories. An essential role is played by “squeezed cones” related to the classical cone construction. The main result: if A is a nondegenerate absolute neighborhood retract for metric spaces, then continuous linear operators extending metrics always exist. There are two roots for the present study. The first is the theorem of Hausdorff [H] on extending metrics for a closed subset A of a metrizable topological space X to metrics on the whole space X. The second is the Borsuk–Dugundji theorem ([Bo], [Du]) on the existence of continuous linear operators extending continuous functions on A to continuous functions on X. The author tried to prove the existence of continuous, linear (i.e., additive and positive-homogeneous) operators extending admissible metrics from A to X, and has only succeeded in special situations, e.g., when A is a nondegenerate ANR for metric spaces (Corollary 2.3). Very stimulating and helpful for our discussion was the paper [KN] of Nguyen Van Khue and Nguyen To Nhu: they were the first to construct continuous (but merely sublinear) operators extending metrics. The main part of the paper are §§ 1 and 2 which lead to the abovementioned Corollary 2.3. A crucial role is played by an extension of spaces and pseudometrics called the squeezed cone construction (in symbols: sc) which is related to the classical cone construction in topology. §3 contains examples of spaces with an “absolute” property of linear extending of metrics but lacking the ANR property. §4 is devoted to discussion of the linear extension constructions viewed as functors between certain categories. Finally, in §5 we ask some questions, mostly functional-analytic, related to the results, proofs and the philosophy of the paper; also some comments are included. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|