View source for Topology of compact convergence

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093210/t0932101.png" /> from a locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093210/t0932102.png" /> into a locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093210/t0932103.png" />, the topology of compact convergence is one of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093210/t0932105.png" />-topologies, i.e. a topology of uniform convergence on sets belonging to a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093210/t0932106.png" /> of bounded sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093210/t0932107.png" />; it is compatible with the vector space structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093210/t0932108.png" /> and it is locally convex.



====Comments====
Thus, the topology of compact convergence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093210/t0932109.png" /> is defined by the family of all compact sets, [[#References|[a1]]].

The topology of pre-compact convergence is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093210/t09321010.png" />-topology defined by the family of all pre-compact sets, [[#References|[a2]]].

The topology of compact convergence in all derivatives in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093210/t09321011.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093210/t09321012.png" /> times differentiable real- or complex-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093210/t09321013.png" /> is defined by the family of pseudo-norms

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093210/t09321014.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093210/t09321015.png" /></td> </tr></table>

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>