Get Compact Convergence essential facts below. View Videos
or join the Compact Convergence discussion
. Add Compact Convergence
to your PopFlock.com topic list for future reference or share
this resource on social media.
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.
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 ,
uniformly on as . This means that for all compact ,
- 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.
- 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.
- R. Remmert Theory of complex functions (1991 Springer) p. 95