Arzelà-Ascoli's theorem

The Arzelà-Ascoli theorem , named after Cesare Arzelà (1847–1912) as an extension of a theorem by Giulio Ascoli (1843–1896), is an important sentence in functional analysis . It answers the question which subsets are ( relatively ) compact in certain function spaces .

Statement (scalar-valued case)

Let be a compact topological space and a subset of continuous real or complex valued functions . Then the following applies: The subset is relatively compact in the Banach space , provided with the supremum norm , if and only if uniformly continuous and is point-wise bounded, i.e. H. for each the set of function values ​​in is bounded in or . ${\displaystyle X}$ ${\displaystyle F\subseteq C(X)}$ ${\displaystyle f\colon X\to \mathbb {K} }$ ${\displaystyle F}$ ${\displaystyle C(X)}$ ${\displaystyle F}$ ${\displaystyle x\in X}$ ${\displaystyle \{f(x):f\in F\}}$ ${\displaystyle x}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

The importance of Arzelà-Ascoli's theorem shows itself in comparison to Riesz's compactness theorem , which states that spheres are not relatively compact in infinite-dimensional Banach spaces . Nevertheless, there are many compact subsets in infinite-dimensional Banach spaces and the Arzelà-Ascoli theorem characterizes them, at least in the special case that the Banach space is of the form . ${\displaystyle C(X)}$

Evidence sketch (in case X is a metric space)

The proof uses Cantor's diagonal method , in which partially convergent partial sequences are constructed in a recursive manner, in order to then obtain an everywhere convergent partial sequence across all partial sequences.

Let be any function sequence in the function family . It has to be shown that this contains a partial sequence that converges. ${\displaystyle \{f_{n}\}_{n\in \mathbb {N} }\subset F}$ ${\displaystyle F}$ ${\displaystyle C(X)}$

To do this, one chooses an ascending sequence of finite subsets , which “converges” to a subset that is dense in the compact set of points . ${\displaystyle A_{N}\subset A_{N+1}\subset X}$ ${\displaystyle \textstyle A_{\infty }:=\bigcup _{N=1}^{\infty }A_{N}}$ ${\displaystyle X}$

The function sequence , restricted to such a point set, contains, according to the assumption, a partial sequence that converges, because a finite Cartesian product of relatively compact sets is again relatively compact. ${\displaystyle \{f_{n,k}\}_{n\in \mathbb {N} }:=\{f_{n}{}_{|A_{k}}\}_{n\in \mathbb {N} }}$ ${\displaystyle A_{k}}$

Let be the zeroth given sequence. Then, starting with , a sub-sequence can be recursively selected in the function sequence which converges on the enlarged point set . Finally, according to Cantor's diagonal “trick”, the diagonal sequence on the dense subset converges to a function . ${\displaystyle (f_{n,0}=f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle N=1,2,\ldots }$ ${\displaystyle \{f_{n_{l},N-1}\}_{l\in \mathbb {N} }}$ ${\displaystyle \{f_{n_{l},N}\}_{l\in \mathbb {N} }}$ ${\displaystyle A_{N}}$ ${\displaystyle \{f_{n_{N},N}\}_{N\in \mathbb {N} }}$ ${\displaystyle A_{\infty }\subset X}$ ${\displaystyle f:A_{\infty }\to \mathbb {K} }$

From the uniform continuity it follows that the limit function obtained in this way can be continued to and it also follows that the diagonal sequence also converges in the supreme norm to the function constructed in this way: in , that is ${\displaystyle X}$ ${\displaystyle {\bar {f}}\colon X\to \mathbb {K} }$ ${\displaystyle \lim _{N\to \infty }f_{N,N}={\bar {f}}}$ ${\displaystyle C(X)}$

${\displaystyle \lim _{N\to \infty }\left(\sup _{x\in X}\|f_{N,N}(x)-{\bar {f}}(x)\|_{Y}\right)=0}$.

Applications

Functional analysis: compactness of operators

Arzelà-Ascoli's theorem can be used to prove that an operator is compact . For example, let the space be square integrable functions , then is defined by ${\displaystyle L^{2}([0,1])}$ ${\displaystyle F\colon L^{2}([0,1])\to C([0,1])}$

${\displaystyle F(x)(t):=\int _{0}^{1}(t^{2}+s^{2})(x(s))^{2}\mathrm {d} s}$

a nonlinear compact operator. For everyone and everyone is of the form and therefore continuous. Furthermore applies . So the subset relation holds for bounded and is therefore bounded and uniformly continuous. Hence, one can apply the Arzelà-Ascoli theorem and obtain that the set is relatively compact in terms of the supremum norm. Therefore, it maps bounded sets to relatively compact sets and is therefore a compact operator. ${\displaystyle t\in [0,1]}$ ${\displaystyle x\in L^{2}([0,1])}$ ${\displaystyle F(x)(t)}$ ${\displaystyle c_{1}t^{2}+c_{2}}$ ${\displaystyle |c_{1}|,|c_{2}|<\left\|x\right\|_{L^{2}(0,1)}}$ ${\displaystyle U\subset L^{2}([0,1])}$ ${\displaystyle F(U)\subset C([0,1])}$ ${\displaystyle F(U)}$ ${\displaystyle F(U)}$ ${\displaystyle C([0,1])}$ ${\displaystyle F}$

Ordinary differential equations

The set of Peano uses the Arzelà-Ascoli theorem to show that the operators used in the proof are relatively compact.

Generalizations

More general value ranges

Instead of scalar-valued functions, one can also consider functions with values ​​in , which can optionally be a standardized vector space , a topological vector space , a metric space or, more generally, a uniform space . The function space is still provided with the topology of uniform convergence. However, it is then no longer sufficient to require point-wise restriction, but the set of functions must be relatively compact (in ) point-wise . The following applies more precisely: ${\displaystyle Y}$ ${\displaystyle Y}$ ${\displaystyle C(X,Y)}$ ${\displaystyle Y}$

A subset is relatively compact with respect to the topology of uniform convergence if it is uniformly continuous and if it applies to each that it is relatively compact in space . ${\displaystyle F\subseteq C(X,Y)}$ ${\displaystyle x\in X}$ ${\displaystyle \{f(x):f\in F\}}$ ${\displaystyle Y}$

More general domains of definition

There are also generalizations in which the compact space is replaced by a more general topological space. In this case, however, the function space is then to be provided with the compact, open topology , i.e. the topology of uniform convergence on compact subsets . ${\displaystyle X}$

Application in differential geometry: compactness of the space of geodesics

Arzelà-Ascoli's theorem can be generalized to families of uniformly continuous functions with values ​​in a compact manifold . ${\displaystyle Y}$

In particular, you can apply it to families of pictures one interval in a compact Riemannian manifold and maintains that for fixed each family of - Quasigeodäten has a convergent subsequence. The convergence is uniform if is a finite interval and locally uniform if is. One can show that for a convergent sequence of geodesics the limit value is again a geodesic. ${\displaystyle F\subset C(I,Y)}$ ${\displaystyle I\subset \mathbb {R} }$ ${\displaystyle Y}$ ${\displaystyle (L,A)}$ ${\displaystyle (L,A)}$ ${\displaystyle I}$ ${\displaystyle I=\mathbb {R} }$

For a compact manifold , the space of all geodesics is compact with respect to the compact-open topology. ${\displaystyle Y}$

literature

• Cesare Arzelà: Un 'osservazione intorno alle serie di funzioni . Rend. dell 'Accad. R. delle Sci. dell'Istituto di Bologna, pp. 142-159 (1882-1883).
• Cesare Arzelà: Sulle funzioni di linee . Mem. Accad. Sci. Is. Bologna Cl. Sci. F sharp. Mat. Vol. 5 No 5, pp. 55-74 (1895).
• Giulio Ascoli: Le curve limiti di una varietà data di curve . Atti della R. Accad. dei Lincei Memorie della Cl. Sci. F sharp. Mat. Nat. Vol 18 No 3, pp. 521-586 (1883-1884).
• Johann Cigler , Hans-Christian Reichel : Topology . A basic lecture, ISBN 3-411-05121-3 .
• Harry Poppe: Compactness in General Function Spaces . Berlin 1974.