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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 .
A sequence of functions () converges uniformly to when for arbitrary small there is an index such that the graph of is in the -tube around f whenever . The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions () = (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).
In words, the Harnack inequality is a tool which directly propagates the Cauchy property of a sequence of harmonic functions at a single point to the Cauchy property at all points. Having established uniform convergence on compact sets, the harmonicity of the limit is an immediate corollary of the fact that the mean value property ...
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 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 ...
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.
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.