Compact convergence

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In mathematics compact convergence is a type of convergence which generalizes the idea of uniform convergence. It is also known as uniform convergence on compact sets or topology of compact convergence.

[edit] Definition

Let $(X_{1}, \mathcal{T}_{1})$ be a topological space and $(X 2, d 2)$ 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 $K \subseteq X_{1}$ ,

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

[edit] Examples

If $X_{1} = (0, 1) \subsetneq \mathbb{R}$ and $X_{2} = \mathbb{R}$ with their usual topologies, with $f n (x): = x n$ , then $f n$ converges compactly to the constant function with value 0, but not uniformly.

If $X 1 = (0,1]$ , $X_2=\R$ and $f n (x) = x n$ , then $f n$ converges pointwise to the function that is zero on $(0,1)$ and one at $1$ , but the sequence does not converge compactly.

[edit] 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 $X 1$ is locally compact, $f_n\to f$ compactly and each $f n$ is continuous, then $f$ is continuous.