Compact convergence

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.

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 $K\subseteq X_{1}$,

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

Examples

• If $X_{1}=(0,1)\subset \mathbb {R}$ 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 $X_{1}=(0,1]$, $X_{2}=\mathbb {R}$ 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.

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.

See also

• Modes of convergence (annotated index)