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 ...
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 ...
Hurwitz's theorem is used in the proof of the Riemann mapping theorem, [2] and also has the following two corollaries as an immediate consequence: . Let G be a connected, open set and {f n} a sequence of holomorphic functions which converge uniformly on compact subsets of G to a holomorphic function f.
If is a compact topological space, and () is a monotonically increasing sequence (meaning () + for all and ) of continuous real-valued functions on which converges pointwise to a continuous function :, then the convergence is uniform.
A discrete convergence action of a group on a compact metrizable space is called uniform (in which case is called a uniform convergence group) if the action of on () is co-compact. Thus Γ {\displaystyle \Gamma } is a uniform convergence group if and only if its action on Θ ( M ) {\displaystyle \Theta (M)} is both properly discontinuous and co ...