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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 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]
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).
Uniform convergence implies both local uniform convergence and compact convergence, since both are local notions while uniform convergence is global. If X is locally compact (even in the weakest sense: every point has compact neighborhood), then local uniform convergence is equivalent to compact (uniform) convergence.
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.
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 ...
The group operation on the dual group is given by pointwise multiplication of characters, the inverse of a character is its complex conjugate and the topology on the space of characters is that of uniform convergence on compact sets (i.e., the compact-open topology, viewing ^ as a subset of the space of all continuous functions from to .). This ...
Hence the sequence f n is uniformly bounded on compact sets. If two subsequences converge to holomorphic limits f and g, then f(0) = g(0) and with f'(0), g'(0) ≥ 0. By the first part and the assumptions it follows that f(D) = g(D). Uniqueness in the Riemann mapping theorem forces f = g, so the original sequence f n is uniformly convergent on ...