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Topology of compact convergence

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One of the topologies on a space of continuous functions; the same as the compact-open topology. For the space of linear mappings from a locally convex space into a locally convex space , the topology of compact convergence is one of the -topologies, i.e. a topology of uniform convergence on sets belonging to a family of bounded sets in ; it is compatible with the vector space structure of and it is locally convex.


Comments

Thus, the topology of compact convergence on is defined by the family of all compact sets, [a1].

The topology of pre-compact convergence is the -topology defined by the family of all pre-compact sets, [a2].

The topology of compact convergence in all derivatives in the space of all times differentiable real- or complex-valued functions on is defined by the family of pseudo-norms

The resulting space of functions is locally convex and metrizable, [a3].

References

[a1] F. Trèves, "Topological vectorspaces, distributions and kernels" , Acad. Press (1967) pp. 198
[a2] G. Köthe, "Topological vector spaces" , 1 , Springer (1969) pp. 263ff
[a3] J.L. Kelley, I. Namioka, "Linear topological spaces" , v. Nostrand (1963) pp. 82
How to Cite This Entry:
Topology of compact convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topology_of_compact_convergence&oldid=48991
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article