- Compact convergence
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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.
Contents
Definition
Let be a topological space and (Y,dY) 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 K as . This means that for all compact ,
Examples
- If and with their usual topologies, with fn(x): = xn, then fn converges compactly to the constant function with value 0, but not uniformly.
- If X = (0,1], and fn(x) = xn, then fn converges pointwise to the function that is zero on (0,1) and one at 1, 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 which 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 fn is continuous, then f is continuous.
See also
References
- R. Remmert Theory of complex functions (1991 Springer) p. 95
Categories:- Functional analysis
- Convergence (mathematics)
- Topology of function spaces
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