Compact convergence

In mathematics , a sequence or series of functions on a topological space with values ​​in a normalized space is called compactly convergent if it converges uniformly on every compact subset of . ${\displaystyle X}$ ${\displaystyle E}$ ${\displaystyle X}$

The concept of compact convergence derives its meaning from the fact that compact convergence follows from locally uniform convergence of a sequence or series of functions, and the converse applies to locally compact spaces. In general, however, this reversal does not apply, as explained in the article on the Arens Fort room .

The topology of compact convergence

The special case of standardized spaces

Let it be the space of the functions of in the normalized vector space which are restricted to every compact subset of (in the sense of the norm on ). According to the definition of exists for two mappings and from the on a restricted distance ${\displaystyle B=B_{k}(X,E)}$ ${\displaystyle X}$ ${\displaystyle (E,\|\cdot \|_{E})}$ ${\displaystyle X}$ ${\displaystyle E}$ ${\displaystyle B}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle B}$ ${\displaystyle K}$

${\displaystyle d_{K}(f,g):=\|\left.f\right|_{K}-\left.g\right|_{K}\|_{\infty }:=\sup _{x\in K}\|f(x)-g(x)\|_{E}}$

for every (non-empty) compact subset . For the restrictions on this is a metric , for only one pseudometric , since the restrictions of two different functions on can be the same. The compact convergence is the convergence with regard to these pseudometrics, that is, a network converges compactly to in if and only if for all compact ones . ${\displaystyle K\subset X}$ ${\displaystyle K}$ ${\displaystyle B}$ ${\displaystyle K}$ ${\displaystyle (f_{\alpha })_{\alpha \in A}}$ ${\displaystyle f}$ ${\displaystyle B}$ ${\displaystyle d_{K}(f_{\alpha },f)\rightarrow 0}$ ${\displaystyle K\subset X}$

If the space is locally compact and it can be represented as a union of countably many compact sets , i.e. in the form , then these pseudometrics can be used for the metric ${\displaystyle X}$ ${\displaystyle (K_{j})_{j\in \mathbb {N} }}$ ${\displaystyle \textstyle X=\bigcup _{j\in \mathbb {N} }(K_{j})}$ ${\displaystyle d_{j}=d_{K_{j}}}$

${\displaystyle d(f,g)=\sum \limits _{j=0}^{\infty }2^{-j}{\frac {d_{j}(f,g)}{1+d_{j}(f,g)}}}$

to compose it. This becomes a metric space . ${\displaystyle B}$ ${\displaystyle (B,d)}$

In general cases, when no such presentation is known or possible, can be explained by an arbitrary system of compact sets , which covers, with the respective pseudo metrics a family of pseudo-metrics to select a uniform structure to define. The technical details are also explained in the article pseudometrics . ${\displaystyle X}$ ${\displaystyle (K_{j})_{j\in J}}$ ${\displaystyle X}$ ${\displaystyle d_{j}=d_{K_{j}}}$ ${\displaystyle (d_{j})_{j\in J}}$ ${\displaystyle B}$ ${\displaystyle B}$

Generalization to uniform spaces

Now let us be a uniform room whose uniform structure is given by a system of pseudometrics . Let again be the space of all functions that are restricted to all compact sets, that is, for which is finite for each and every one . An important subspace is the space of all continuous functions . ${\displaystyle E}$ ${\displaystyle (d_{i})_{i\in I}}$ ${\displaystyle B_{k}(X,E)}$ ${\displaystyle f\colon X\to E}$ ${\displaystyle \textstyle \sup _{x\in K}d_{i}(f(x),y)}$ ${\displaystyle y\in E}$ ${\displaystyle i\in I}$ ${\displaystyle X\to E}$

A network of functions in converges compactly to a function if and only if ${\displaystyle (f_{\alpha })_{\alpha \in A}}$ ${\displaystyle B_{k}(X,E)}$ ${\displaystyle f\in B_{k}(X,E)}$

${\displaystyle \sup _{x\in K}d_{i}(f_{\alpha }(x),f(x)){\stackrel {\alpha \in A}{\longrightarrow }}0}$

for everyone and everyone compact. On the system of pseudometrics , where and compact and , a uniform structure is obtained. ${\displaystyle i\in I}$ ${\displaystyle K\subset X}$ ${\displaystyle B_{k}(X,E)}$ ${\displaystyle d_{i,K}}$ ${\displaystyle i\in I}$ ${\displaystyle K\subset X}$ ${\displaystyle \textstyle d_{i,K}(f,g):=\sup _{x\in K}d_{i}(f(x),g(x))}$

If a standardized space is specifically , the uniform structure is given by the standard, and the special case presented above is obtained. ${\displaystyle E}$ ${\displaystyle E}$

Locally compact and compact spaces

In locally compact , uniform spaces, the topology of compact convergence corresponds to the compact-open topology .

On compact , uniform spaces, the topology of compact convergence is called the topology of uniform convergence .

Examples

1. Power series of analytical functions on or converge compactly within their convergence interval or circle. ${\displaystyle X=\mathbb {R} }$ ${\displaystyle X=\mathbb {C} }$
2. Is , the system forms a countable system of compact sets that cover. With this, a metric of compact convergence can be introduced on the mapping set . ${\displaystyle X=\mathbb {R} }$ ${\displaystyle \{K_{j}:=[-j;j]\,|\,j\in \mathbb {N} \setminus \{0\}\}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle B_{k}(\mathbb {R} ,\mathbb {R} )}$
3. Correspondingly, the set of compactly restricted mappings from a -dimensional to a -dimensional real vector space can be given a metric. As a cover of the original image space here z. B. cube (the edge length with focus at the origin) or spheres (with a radius around the origin) can be selected. ${\displaystyle B_{k}(\mathbb {R} ^{n},\mathbb {R} ^{m})}$ ${\displaystyle n}$ ${\displaystyle m}$ ${\displaystyle 2j}$ ${\displaystyle j}$
4. If there is a limited, simply connected area of ​​the complex number plane, then it can be covered by the sets ( measures the distance from the edge in the sense of the Hausdorff metric , results in the empty set for smaller ones , then these must be from the family of pseudometrics in the definition taken out of the metric). Here, too, the topology of compact convergence proves to be metrizable. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle K_{j}:=\left\{x\in X\,|\,d_{H}(x,\partial X)\geq {\tfrac {1}{j}}\right\}}$ ${\displaystyle d_{H}}$ ${\displaystyle j\in \mathbb {N} }$

completeness

With the topology of compact convergence, important mapping spaces form a complete, uniform structure . Two examples: The spaces or the continuous or holomorphic functions in a field of the complex number plane form complete uniform spaces with regard to the uniform structure of the compact convergence. In classic formulation, i. H. Without topological terms, this can be expressed as follows: ${\displaystyle {\mathcal {C}}(G)}$ ${\displaystyle {\mathcal {O}}(G)}$ ${\displaystyle G}$

• Are in an area the functions , all continuous (or holomorphic ), and is the result of compact converges to a limit function , then the limit function is continuous (or holomorphic) in . ${\displaystyle G\subset \mathbb {C} }$ ${\displaystyle f_{n}:G\rightarrow \mathbb {C} }$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle (f_{n})}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle G}$
• The same applies to series and infinite products when viewed as sequences of functions. ${\displaystyle \textstyle \sum _{n=1}^{\infty }g_{n}}$ ${\displaystyle \textstyle \prod _{n=1}^{\infty }g_{n}}$

literature

• Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 .
• Reinhold Remmert : Function theory (= basic knowledge of mathematics. Vol. 5). 1st volume. 2nd, revised and expanded edition. Springer, Berlin et al. 1989, ISBN 3-540-51238-1 .
• Reinhold Remmert: Function theory (= basic knowledge of mathematics. Vol. 6). 2nd volume. Springer, Berlin et al. 1991, ISBN 3-540-12783-6 .