Compact convergence

In mathematics it is called a sequence or series of functions on a topological space X with values ​​in a normed space E compactly convergent if it converges uniformly on every compact subset of X.

Its significance from the notion of compact convergence from the fact that from the locally uniform convergence of a sequence or series of functions follows the compact convergence and reversal for locally compact spaces is considered.

The topology of compact convergence

The special case of normed spaces

It is the space of functions from X into the normed vector space, which are restricted to any compact subset of X (in the sense of the norm on E). By definition of exists for two images f and g from B to K, the restricted distance

For each ( non-empty ) compact subset K of X. For the restrictions on K, this is a metric for B only a pseudo- metric, as can match the restrictions of two different functions on K. The compact convergence is the convergence with respect to this pseudo- metrics, ie a network converges compactly in, if for all compact.

If the space X is locally compact and it can be as countable union of many compact sets, ie in the form, represent, then you can this pseudo- metrics to the metric

To put together. Thus (b, d ) is added to a metric space.

In general cases, if no such representation of X is available or known can be achieved by any system compact sets that covers X filter to the respective pseudo- metrics is a family of pseudo metrics B, defining a uniform structure on B. Here, too, are the technical details in the article pseudometric explained.

Verallgemeinung on uniform spaces

Now, it is a uniform space, the uniform structure is given by a system of pseudo metrics. Again let the space of all functions that are limited on all compact sets, ie for each and each is finite. An important subspace is the space of all continuous functions.

A network of functions converges compactly to a function when compact for all and all. On obtained by the system of pseudo metrics, with and compact, a uniform structure.

Specifically, is a normed space, the uniform structure on is given by the norm, and one obtains the above presented a special case.

Examples

Completeness

Important figure spaces form a complete uniform structure with the topology of compact convergence. Two examples: the rooms or the continuous or holomorphic on a domain G of the complex plane functions form with respect to the uniform structure of compact convergence complete uniform space. In classical formulation, ie without topological terms, this can be expressed as follows:

• In an area are the functions, all continuous ( or holomorphic ), and is the result compactly convergent to a limit function f, then the limit function f is continuous (resp. holomorphic ) in G.
• The same is true for series and infinite products, when viewed as a function of consequences.