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In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
compact convergence (or
uniform convergence on compact sets) is a type of convergence which generalizes the idea of
uniform convergenceIn the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. A sequence { f
n } of functions converges uniformly to a limiting function f if the speed of convergence of f
n to f does not depend on x.The concept is...
. It is associated with the
compact-open topologyIn mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly-used topologies on function spaces, and is applied in homotopy theory and functional analysis.- Definition :Let X and Y be...
.
Definition
Let be a
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
and be a
metric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
. A sequence of functions
,
is said to
converge compactly as to some function if, for every compact set ,
converges uniformlyIn the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. A sequence { f
n } of functions converges uniformly to a limiting function f if the speed of convergence of f
n to f does not depend on x.The concept is...
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
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.-Definition:...
to the function that is zero on and one at , but the sequence does not converge compactly.
Properties
- If uniformly, then compactly.
- If compactly and is itself a compact space
In mathematics, more specifically general topology and metric topology, a compact space is an abstract mathematical space in which, intuitively, whenever one takes an infinite number of "steps" in the space, eventually one must get arbitrarily close to some other point of the space...
, then uniformly.
- If is locally compact, compactly and each is continuous
In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous. A continuous function with a continuous inverse function is called bicontinuous...
, then is continuous.