In
mathematics compact convergence (or uniform convergence on compact sets) is a type of
convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
* "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that united the four Wei ...
that generalizes the idea of
uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
. It is associated with the
compact-open topology.
Definition
Let
be a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
and
be a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
. A sequence of functions
:
,
is said to converge compactly as
to some function
if, for every
compact set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
,
:
uniformly on
as
. This means that for all compact
,
:
Examples
* If
and
with their usual topologies, with
, then
converges compactly to the constant function with value 0, but not uniformly.
* If
,
and
, then
converges
pointwise convergence, pointwise to the function that is zero on
and one at
, 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
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein.
In particular, the concept applies to countable f ...
and
uniformly bounded
In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than or equal to the absolute value of any value of any of the functions in the family. ...
maps has a subsequence that converges compactly to some continuous map.
Properties
* If
uniformly, then
compactly.
* If
is a
compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
and
compactly, then
uniformly.
* If
is a
locally compact space In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
, then
compactly if and only if
locally uniformly.
* If
is a
compactly generated space
In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space ''X'' is compactly generated if it satisfies the following condition:
:A subsp ...
,
compactly, and each
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 g ...
, then
is continuous.
See also
*
Modes of convergence (annotated index)
The purpose of this article is to serve as an annotated index of various modes of convergence and their logical relationships. For an expository article, see Modes of convergence. Simple logical relationships between different modes of converg ...
*
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 f ...
References
*R. Remmert ''Theory of complex functions'' (1991 Springer) p. 95
{{DEFAULTSORT:Compact Convergence
Functional analysis
Convergence (mathematics)
Topology of function spaces
Topological spaces