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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).
If u n (y) is a Cauchy sequence for any particular value of y, then the Harnack inequality applied to the harmonic function u m − u n implies, for an arbitrary compact set D containing y, that sup D |u m − u n | is arbitrarily small for sufficiently large m and n. This is exactly the definition of uniform convergence on compact sets.
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 ...
The Mackey topology or Arens-Mackey topology is the strongest locally convex topology on B(H) such that the dual is B(H) *, and is also the uniform convergence topology on Bσ(B(H) *, B(H)-compact convex subsets of B(H) *. It is stronger than all topologies below.
Local uniform convergence (i.e. uniform convergence on a neighborhood of each point) Compact (uniform) convergence (i.e. uniform convergence on all compact subsets) further instances of this pattern below. Implications: - "Global" modes of convergence imply the corresponding "local" and "compact" modes of convergence. E.g.: