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