Compact convergence

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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.

Definition

Let ${\displaystyle (X,{\mathcal {T}})}$ be a topological space and ${\displaystyle (Y,d_{Y})}$ be a metric space. A sequence of functions

${\displaystyle f_{n}:X\to Y}$, ${\displaystyle n\in \mathbb {N} ,}$

is said to converge compactly as ${\displaystyle n\to \infty }$ to some function ${\displaystyle f:X\to Y}$ if, for every compact set ${\displaystyle K\subseteq X}$,

${\displaystyle f_{n}|_{K}\to f|_{K}}$

uniformly on ${\displaystyle K}$ as ${\displaystyle n\to \infty }$. This means that for all compact ${\displaystyle K\subseteq X}$,

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

Examples

• If ${\displaystyle X=(0,1)\subseteq \mathbb {R} }$ and ${\displaystyle Y=\mathbb {R} }$ with their usual topologies, with ${\displaystyle f_{n}(x):=x^{n}}$, then ${\displaystyle f_{n}}$ converges compactly to the constant function with value 0, but uniform convergence does not hold.
• If ${\displaystyle X=(0,1]}$, ${\displaystyle Y=\mathbb {R} }$ and ${\displaystyle f_{n}(x)=x^{n}}$, then ${\displaystyle f_{n}}$ converges pointwise to the function that is zero on ${\displaystyle (0,1)}$ and one at ${\displaystyle 1}$, but the sequence does not converge compactly.
• 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.

Properties

• If ${\displaystyle f_{n}\to f}$ uniformly, then ${\displaystyle f_{n}\to f}$ compactly.
• If ${\displaystyle (X,{\mathcal {T}})}$ is a compact space and ${\displaystyle f_{n}\to f}$ compactly, then ${\displaystyle f_{n}\to f}$ uniformly.
• If ${\displaystyle (X,{\mathcal {T}})}$ is a locally compact space, then ${\displaystyle f_{n}\to f}$ compactly if and only if ${\displaystyle f_{n}\to f}$ locally uniformly.
• If ${\displaystyle (X,{\mathcal {T}})}$ is a compactly generated space, ${\displaystyle f_{n}\to f}$ compactly, and each ${\displaystyle f_{n}}$ is continuous, then ${\displaystyle f}$ is continuous.

See also

• Modes of convergence (annotated index)
• Montel's theorem

References

• Reinhold Remmert Theory of complex functions (1991 Springer) p. 95