Search results
Results from the WOW.Com Content Network
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 .
If the codomain of the functions under consideration has a uniform structure or a metric structure then the compact-open topology is the "topology of uniform convergence on compact sets." That is to say, a sequence of functions converges in the compact-open topology precisely when it converges uniformly on every compact subset of the domain. [2]
If is a compact interval (or in general a compact topological space), and () is a monotone increasing sequence (meaning () + for all n and x) of continuous functions with a pointwise limit which is also continuous, then the convergence is necessarily uniform (Dini's theorem).
If the domain of the functions is a topological space and the codomain is a uniform space, local uniform convergence (i.e. uniform convergence on a neighborhood of each point) and compact (uniform) convergence (i.e. uniform convergence on all compact subsets) may be defined. "Compact convergence" is always short for "compact uniform convergence ...
However, this name is frequently changed according to the types of sets that make up (e.g. the "topology of uniform convergence on compact sets" or the "topology of compact convergence", see the footnote for more details [3]).
For any open set, the set A(Ω) of all analytic functions : is a Fréchet space with respect to the uniform convergence on compact sets. The fact that uniform limits on compact sets of analytic functions are analytic is an easy consequence of Morera's theorem.
This criterion for uniform convergence is often useful in real and complex analysis. Suppose we are given a sequence of continuous functions that converges pointwise on some open subset G of R n. As noted above, it actually converges uniformly on a compact subset of G if it is equicontinuous on the compact set. In practice, showing the ...
However, since {} is bounded in operator norm, and the limit operator is continuous, a standard "" estimate shows that converges to uniformly on compact sets. Proof Essentially the same as that of the proof that a pointwise convergent sequence of equicontinuous functions on a compact set converges to a continuous function.