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.

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

  • 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.

Properties

  • 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 also

References

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