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

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

is said to converge compactly if, for every compact set,

uniformly,

Examples

• converges compactly to the constant function with value 0, but uniform convergence does not hold.

• converges pointwise, 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

• compactly.

• is a compact space uniformly.

• is a locally compact space locally uniformly.

• is a compactly generated space is continuous is continuous.

See also

• Modes of convergence (annotated index)

• Montel's theorem

References

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