Arzelà–Ascoli theorem

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

Statement ( skalarwertiger case )

Be a compact topological space and a subset of continuous real - or complex-valued functions. Then: The subset is relatively compact in the Banach space if and only provided with the supremum norm, if equi is continuous and is pointwise limits, ie for each the set of function values ​​in restricted, respectively.

The meaning of the sentence of Arzelà - Ascoli shows in comparison to the compactness theorem of Riesz, which states that balls are not relatively compact in infinite-dimensional Banach spaces. Nevertheless, there are also infinite-dimensional Banach spaces in many compact subsets and the set of Arzelà - Ascoli characterizes this, at least in the special case that the Banach space of the form.

Sketch of proof (in the case that X is a metric space )

The proof uses the Cantor's diagonal method, in which a recursive type partially convergent sequences are constructed, only to get across all subsequences an everywhere convergent subsequence.

Be an arbitrary sequence of functions in the function family. We show that this one contains convergent subsequence.

You simply select an increasing sequence of finite subsets, " converges " to a subset which, which is dense in the compact point set.

The sequence of functions is restricted to such a set of points, by assumption, contains a convergent subsequence, since a finite Cartesian product of relatively compact sets is again relatively compact.

Be the zeroth, trivial subsequence. Then recursively, starting, in the sequence of functions to be selected is a subsequence that converges on the magnified set of points. Finally converges after Cantor's diagonal " trick", the diagonal sequence on the dense subset to a function.

From gleichgradigen continuity follows that the limit function thus obtained can be continued in a very steadily and it also follows that the diagonal sequence in the supremum norm converges to the so- constructed function: in, ie

Example

The set of Arzelà - Ascoli can be used to prove that an operator is compact. Be the space square integrable functions, then is defined by

A non-linear compact operator. For all and all is of the form, and thus steadily. Furthermore applies. So apply the subset relation for limited and is therefore limited and equicontinuous. Therefore, one can apply the theorem of Ascoli - Arzelà and receives that amount is relatively compact in relation to the supremum norm. So that's why is bounded sets onto relatively compact sets and thus is a compact operator.

Generalizations

• Instead of scalar-valued functions can also consider functions with values ​​in, optionally using a normed vector space is a topological vector space is a metric space or, more generally, can be a uniform space. The function room is still equipped with the topology of uniform convergence. Then, however, it is no longer sufficient to require pointwise boundedness, but the function set must pointwise relatively compact ( in ) to be. More precisely:

• There are also generalizations in which the compact space is replaced by a more general topological space. But this is then the function room to be provided with the compact - open topology, ie the topology of uniform convergence on compact subsets.