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The Attouch-Wets topology and a characterisation of normable linear spaces. (English) Zbl 0716.54013
Summary: Let X and Y be metric spaces and C(X,Y) be the space of all continuous functions from X to Y. If X is a locally connected space, the compact- open topology on C(X,Y) is weaker than the Attouch-Wets topology on C(X,Y). The result is applied on the space of continuous linear functions. Let X be a locally convex topological linear space, metrisable with an invariant metric, and \(X^*\) be a continuous dual. X is normable if and only if the strong topology on \(X^*\) and the Attouch-Wets topology coincide.

MSC:
54C35 Function spaces in general topology
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
46E10 Topological linear spaces of continuous, differentiable or analytic functions
46B99 Normed linear spaces and Banach spaces; Banach lattices
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References:
[1] Azé, Operations on convergent families of sets and functions (1987)
[2] Attouch, Quantiiative stability of variational systems: I (1988)
[3] DOI: 10.2307/1994611 · Zbl 0146.18204
[4] Robertson, Topological vector spaces (1964) · Zbl 0123.30202
[5] DOI: 10.1016/0022-247X(71)90200-9 · Zbl 0253.46086
[6] Azé, Functional analysis and approximation (1989)
[7] Kuratowski, Topology 1 (1966)
[8] Hamlett, The closed graph and p-closed graph properties in general topology: AMS Contemporary Series 3 (1981) · Zbl 0503.54014
[9] Beer, Bull. Austral. Math. Soc. 31 pp 421– (1985)
[10] Castaing, Convex analysis and measurable multifunctions: Lecture notes in mathematics 580 (1975)
[11] Beer, Bull. Austral. Math. Soc. 38 pp 239– (1988)
[12] DOI: 10.1016/0001-8708(69)90009-7 · Zbl 0192.49101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.