Compact convergence

From formulasearchengine
Jump to navigation Jump to search

{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence which generalizes the idea of uniform convergence. It is associated with the compact-open topology.

Definition

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 ,

Examples

  • 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 which converges compactly to some continuous map.

Properties

See also

References

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