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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 <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.
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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  ) $
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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 $;
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it is compatible with the vector space structure of  $  L( E, F  ) $
 +
and it is locally convex.
  
 
====Comments====
 
====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]]].
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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 <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]]].
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The topology of pre-compact convergence is the $  \sigma $-
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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
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The topology of compact convergence in all derivatives in the space $  C  ^ {m} ( \mathbf R  ^ {n} ) $
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of all $  m $
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times differentiable real- or complex-valued functions on $  \mathbf R  ^ {n} $
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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>
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$$
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\{ {q _ {K}  ^ {(} m) } : {K \subset  \mathbf R  ^ {n}  \textrm{ compact } } \}
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,
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$$
  
<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>
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$$
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q _ {K}  ^ {(} m) ( f  )  = \sup \{ | D  ^ {p} f( x) | : x \in K, | p | \leq  m \} .
 +
$$
  
 
The resulting space of functions is locally convex and metrizable, [[#References|[a3]]].
 
The resulting space of functions is locally convex and metrizable, [[#References|[a3]]].

Latest revision as of 08:26, 6 June 2020


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
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