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In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence which generalizes the idea of uniform convergence. It is associated with the compact-open topology.

Contents

Definition

Let (X_{1}, \mathcal{T}_{1}) be a topological space and (X2,d2) be a metric space. A sequence of functions

f_{n} : X_{1} \to X_{2}, n \in \mathbb{N},

is said to converge compactly as n \to \infty to some function f : X_{1} \to X_{2} if, for every compact set K \subseteq X_{1},

(f_{n})|_{K} \to f|_{K}

converges uniformly on K as n \to \infty. This means that for all compact K \subseteq X_{1},

\lim_{n \to \infty} \sup_{x \in K} d_{2} \left( f_{n} (x), f(x) \right) = 0.

Examples

  • If X_{1} = (0, 1) \subset \mathbb{R} and X_{2} = \mathbb{R} with their usual topologies, with fn(x): = xn, then fn converges compactly to the constant function with value 0, but not uniformly.
  • If X1 = (0,1], X_2=\R and fn(x) = xn, then fn converges pointwise to the function that is zero on (0,1) and one at 1, but the sequence does not converge compactly.

Properties

  • If f_{n} \to f uniformly, then f_{n} \to f compactly.
  • If f_{n} \to f compactly and (X_{1}, \mathcal{T}_{1}) is itself a compact space, then f_{n} \to f uniformly.
  • If f_n\to f compactly and each fn is continuous, then f is continuous.

See also


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