TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
compact convergence (or uniform convergence on compact sets) is a type of
convergence Convergence may refer to: Arts and media Literature *Convergence (book series), ''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-par ...
that generalizes the idea of
uniform convergenceIn the mathematical field of analysis, uniform convergence is a mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Language * Grammatical mode or grammatical mood, a category of verbal inflections t ...
. It is associated with the
compact-open topologyIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
.

# Definition

Let $\left(X, \mathcal\right)$ be a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
and $\left(Y,d_\right)$ be a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. A sequence of functions :$f_ : X \to Y$, $n \in \mathbb,$ is said to converge compactly as $n \to \infty$ to some function $f : X \to Y$ if, for every
compact set In mathematics, more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed set, closed (i.e., containing all its limit points) and bounded set, bounded (i.e., having all ...
$K \subseteq X$, :$f_, _ \to f, _$ uniformly on $K$ as $n \to \infty$. This means that for all compact $K \subseteq X$, :$\lim_ \sup_ d_ \left\left( f_ \left(x\right), f\left(x\right) \right\right) = 0.$

# Examples

* If $X = \left(0, 1\right) \subseteq \mathbb$ and $Y = \mathbb$ with their usual topologies, with $f_ \left(x\right) := x^$, then $f_$ converges compactly to the constant function with value 0, but not uniformly. * If $X=\left(0,1\right]$, $Y=\R$ and $f_n\left(x\right)=x^n$, then $f_n$ converges pointwise convergence, pointwise to the function that is zero on $\left(0,1\right)$ and one at $1$, but the sequence does not converge compactly. * A very powerful tool for showing compact convergence is the
Arzelà–Ascoli theorem The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, in ...
. There are several versions of this theorem, roughly speaking it states that every sequence of
equicontinuous In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood A neighbourhood ( British English, Australian English Australian English (Au ...
and
uniformly bounded In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
maps has a subsequence that converges compactly to some continuous map.

# Properties

* If $f_ \to f$ uniformly, then $f_ \to f$ compactly. * If $\left(X, \mathcal\right)$ is a
compact space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and $f_ \to f$ compactly, then $f_ \to f$ uniformly. * If $\left(X, \mathcal\right)$ is a
locally compact space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and ...
, then $f_ \to f$ compactly if and only if $f_ \to f$ locally uniformly. * If $\left(X, \mathcal\right)$ is a
compactly generated spaceIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
, $f_n\to f$ compactly, and each $f_n$ is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
, then $f$ is continuous.

# See also

*
Modes of convergence (annotated index) The purpose of this article is to serve as an Annotation, annotated Index (publishing), index of various modes of convergence and their logical relationships. For an expository article, see Modes of convergence. Simple logical relationships betwee ...
*
Montel's theorem In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after French mathematician Paul Montel, and give conditions under which a family of holomorphic fun ...

# References

*R. Remmert ''Theory of complex functions'' (1991 Springer) p. 95 {{DEFAULTSORT:Compact Convergence Functional analysis Convergence (mathematics) Topology of function spaces Topological spaces