Compact convergence

In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology.

DefinitionEdit

Let   be a topological space and   be a metric space. A sequence of functions

 ,  

is said to converge compactly as   to some function   if, for every compact set  ,

 

converges uniformly on   as  . This means that for all compact  ,

 

ExamplesEdit

  • If   and   with their usual topologies, with  , then   converges compactly to the constant function with value 0, but not uniformly.
  • If  ,   and  , then   converges pointwise to the function that is zero on   and one at  , but the sequence does not converge compactly.
  • A very powerful tool for showing compact convergence is the Arzelà–Ascoli theorem. There are several versions of this theorem, roughly speaking it states that every sequence of equicontinuous and uniformly bounded maps has a subsequence that converges compactly to some continuous map.

PropertiesEdit

  • If   uniformly, then   compactly.
  • If   is a compact space and   compactly, then   uniformly.
  • If   is locally compact, then   compactly if and only if   locally uniformly.
  • If   is a compactly generated space,   compactly, and each   is continuous, then   is continuous.

See alsoEdit

ReferencesEdit

  • R. Remmert Theory of complex functions (1991 Springer) p. 95