Topology of compact convergence

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

Comments

Thus, the topology of compact convergence on $ L( E, F ) $ is defined by the family of all compact sets, [a1].

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

The topology of compact convergence in all derivatives in the space $ C ^ {m} ( \mathbf R ^ {n} ) $ of all $ m $ times differentiable real- or complex-valued functions on $ \mathbf R ^ {n} $ is defined by the family of pseudo-norms

$$ \{ {q _ {K} ^ {(} m) } : {K \subset \mathbf R ^ {n} \textrm{ compact } } \} , $$

$$ q _ {K} ^ {(} m) ( f ) = \sup \{ | D ^ {p} f( x) | : x \in K, | p | \leq m \} . $$

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