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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
##### Keywords:
normable linear space; Attouch-Wets topology
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##### References:
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