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One of the topologies on a space of continuous functions; the same as the [[Compact-open topology|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, [[#References|[a1]]].

The topology of pre-compact convergence is the  $  \sigma $-
topology defined by the family of all pre-compact sets, [[#References|[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, [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Trèves,   "Topological vectorspaces, distributions and kernels" , Acad. Press  (1967)  pp. 198</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Köthe,   "Topological vector spaces" , '''1''' , Springer  (1969)  pp. 263ff</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.L. Kelley,   I. Namioka,   "Linear topological spaces" , v. Nostrand  (1963)  pp. 82</TD></TR></table>