Ascoli-Arzela Theorem

Arzela's theorem is a statement that is a criterion for the precompactness of a set in a complete metric space in the special case when the space in question is the space of continuous functions on a segment of a real line. Named after the author, Cesare Arzela .

The Arzel – Ascoli theorem (or Ascoli – Arzela) is a generalization of the Arzel theorem to the case when families of mappings of metric compact sets are considered ( generalized Arzel theorem ).

The application of the Arzel theorem is associated with special properties of the families in question, namely, with uniform boundedness and equidistant continuity .

Content

1 Introduction
2 Definitions
- 2.1 Uniform Boundedness
- 2.2 Equal continuity
3 Wording
4 Proof
- 4.1 Necessity
- 4.2 Sufficiency
5 Applications
6 See also
7 Literature
8 Notes

Introduction

In mathematical analysis (and then in functional analysis ), various families of continuous functions defined on special sets ( metric compact sets ) are considered and the question of the “completeness” of such families is investigated. In particular, the question arises of the existence of a limit , for example, for a sequence of continuous numerical functions defined on a segment $[a,b]$ ${\ displaystyle [a, b]}$ , as well as the properties of this limit. According to the Cauchy criterion , the uniform limit of continuous functions is also a continuous function, which means the completeness of space $C[a,b]$ ${\ displaystyle C [a, b]}$ . What is significant here is that the domain of definition of functions is a compact subset of the real line (segment), and the functions take on a value in full metric space. We get a similar result if we take the class of continuous mappings of an arbitrary metric compactum into a complete metric space.

Class completeness $C[a,b]$ ${\ displaystyle C [a, b]}$ allows us to approximate any continuous function by a sequence of approximations, each of which is a function in a sense “simpler” than the original. This is evidenced by the Weierstrass theorem : each continuous function on a segment can be approximated arbitrarily accurately by polynomials.

Arzel's theorem refers to the case when a family of continuous functions is considered $F\subset C(K,Y)$ ${\ displaystyle F \ subset C (K, Y)}$ where $K$ ${\ displaystyle K}$ Is a metric compact, and $Y$ ${\ displaystyle Y}$ Is a complete metric space, and the question is examined whether it is possible to distinguish a convergent subsequence from this family. Since space $C(K,Y)$ ${\ displaystyle C (K, Y)}$ complete, the existence of a limit point essentially means the precompactness of the family $F$ ${\ displaystyle F}$ at $C(K,Y)$ ${\ displaystyle C (K, Y)}$ . Therefore, the theorem can be formulated in a general form, speaking precisely of precompactness.

Thus, the Arzel Theorem is a criterion for the precompactness of a family of continuous functions defined on a compact set and acting in a complete metric space.

The existing criterion for the precompactness of a set in full space requires verification of the completely boundedness of a given set. In practice, such a criterion is not effective. Therefore, it seems appropriate to somehow use the properties of the functions themselves that make up the family in order to obtain a precompact criterion suitable for practical use.

In the course of research, it turned out that such properties are the properties of uniform boundedness and equal continuity of the considered family.

Equal continuity was mentioned simultaneously by Giulio Ascoli (1883–1884) ^[1] and Cesare Arzela (1882–1883) ^[2] . A weak form of the theorem was proved by Ascoli in 1883–1884 ^[1] , who established a sufficient compactness condition, and by Arcela in 1895 ^[3] , who introduced the necessary condition and gave the first clear interpretation of the result. A further generalization of the theorem was proved by Frechet (1906) ^[4] for spaces in which the concept of a limit makes sense, for example, a metric space or Hausdorff Danford, Schwartz (1958) ^[5] . Modern formulations of the theorem allow a region and a range to be metric spaces. The most general statement of the theorem provides the necessary and sufficient conditions for the family of functions from a compact Hausdorff space to a uniform space to be compact in the topology of uniform convergence of Bourbaki (1998, § 2.5) ^[6] .

Definitions

Consider space $C[a,b]$ ${\ displaystyle C [a, b]}$ continuous functions defined on a segment $[a,b]$ ${\ displaystyle [a, b]}$ , together with the metric of uniform convergence. This is a complete metric space. It is known that:

For a subset of a complete metric space to be precompact, it is necessary and sufficient that it be completely bounded.

In case of space $C[a,b]$ ${\ displaystyle C [a, b]}$ However, a more effective criterion for precompactness can be used, but for this one will have to introduce the following two concepts.

We assume that $F$ ${\ displaystyle F}$ Is a family of continuous functions defined on the interval $[a,b]$ ${\ displaystyle [a, b]}$ .

Uniform Boundedness

Family $F$ ${\ displaystyle F}$ is called uniformly bounded if there is a constant constant for all elements of the family $K$ ${\ displaystyle K}$ , which limits all the functions of the family:

\forall f\in F\quad \forall x\in [a,b]\quad |f(x)|<K

{\ displaystyle \ forall f \ in F \ quad \ forall x \ in [a, b] \ quad | f (x) | <K}

.

Equal Continuity

Family $F$ ${\ displaystyle F}$ is called equidistant continuous if for any $\varepsilon >0$ ${\ displaystyle \ varepsilon> 0}$ exist $\delta >0$ ${\ displaystyle \ delta> 0}$ such that for every element $f\in F$ ${\ displaystyle f \ in F}$ and for any points $x_{1}$ ${\ displaystyle x_ {1}}$ and $x_{2}$ ${\ displaystyle x_ {2}}$ such that $|x_{1}-x_{2}|<\delta$ ${\ displaystyle | x_ {1} -x_ {2} | <\ delta}$ , the strict inequality holds $|f(x_{1})-f(x_{2})|<\varepsilon$ ${\ displaystyle | f (x_ {1}) - f (x_ {2}) | <\ varepsilon}$ .

Wording

Theorem.

Functional family $F$ ${\ displaystyle F}$ is precompact in full metric space $C[a,b]$ ${\ displaystyle C [a, b]}$ if and only if this family is

uniformly bounded
equally continuous.

Proof

In fact, it is necessary to show that both of the indicated properties of a family of functions are equivalent to the completely boundedness of this family.

Necessity

So let the family $F$ ${\ displaystyle F}$ - quite limited .

Fix $\varepsilon >0$ ${\ displaystyle \ varepsilon> 0}$ and build the final $(\varepsilon /3)$ ${\ displaystyle (\ varepsilon / 3)}$ -net view: $\{\varphi _{i}\}_{i=1}^{n}$ ${\ displaystyle \ {\ varphi _ {i} \} _ {i = 1} ^ {n}}$ .

Since each function of this system is continuous and, therefore, bounded, then for each such function there is a constant $K_{i}$ ${\ displaystyle K_ {i}}$ such that $|\varphi _{i}(x)|<K_{i}$ ${\ displaystyle | \ varphi _ {i} (x) | <K_ {i}}$ for everyone $x\in [a,b]$ ${\ displaystyle x \ in [a, b]}$ .

Since there are a finite set of such functions, we can take $K=\max _{i}K_{i}+\varepsilon /3$ ${\ displaystyle K = \ max _ {i} K_ {i} + \ varepsilon / 3}$ .

Now, if we take an arbitrary function $f\in F$ ${\ displaystyle f \ in F}$ , then for this function there is such an element $\varphi _{i}$ ${\ displaystyle \ varphi _ {i}}$ $(\varepsilon /3)$ ${\ displaystyle (\ varepsilon / 3)}$ networks that $|f(x)-\varphi _{i}(x)|<\varepsilon /3$ ${\ displaystyle | f (x) - \ varphi _ {i} (x) | <\ varepsilon / 3}$ for everyone $x\in [a,b]$ ${\ displaystyle x \ in [a, b]}$ . Obviously, in this case, the function $f$ ${\ displaystyle f}$ will be limited by constant $K$ ${\ displaystyle K}$ .

Thus, the family $F$ ${\ displaystyle F}$ is uniformly bounded .

Again, due to the continuity of each element $(\varepsilon /3)$ ${\ displaystyle (\ varepsilon / 3)}$ -nets, this element also turns out to be uniformly continuous and, therefore, $(\varepsilon /3)$ ${\ displaystyle (\ varepsilon / 3)}$ you can pick one $\delta _{i}$ ${\ displaystyle \ delta _ {i}}$ such that $|\varphi _{i}(x_{1})-\varphi _{i}(x_{2})|<\varepsilon /3$ ${\ displaystyle | \ varphi _ {i} (x_ {1}) - \ varphi _ {i} (x_ {2}) | <\ varepsilon / 3}$ for any points $x_{1},x_{2}\in [a,b]$ ${\ displaystyle x_ {1}, x_ {2} \ in [a, b]}$ such that $|x_{1}-x_{2}|<\delta _{i}$ ${\ displaystyle | x_ {1} -x_ {2} | <\ delta _ {i}}$ .

Put $\delta =\min _{i}\delta _{i}$ ${\ displaystyle \ delta = \ min _ {i} \ delta _ {i}}$ .

If we now consider an arbitrary function $f\in F$ ${\ displaystyle f \ in F}$ then for a given $\varepsilon >0$ ${\ displaystyle \ varepsilon> 0}$ strict inequality will occur $|f(x_{1})-f(x_{2})|<\varepsilon$ ${\ displaystyle | f (x_ {1}) - f (x_ {2}) | <\ varepsilon}$ for any points $x_{1},x_{2}\in [a,b]$ ${\ displaystyle x_ {1}, x_ {2} \ in [a, b]}$ such that $|x_{1}-x_{2}|<\delta$ ${\ displaystyle | x_ {1} -x_ {2} | <\ delta}$ .

Really, $|f(x_{1})-f(x_{2})|\leqslant |f(x_{1})-\varphi _{i}(x_{1})|+|\varphi _{i}(x_{1})-\varphi _{i}(x_{2})|+|\varphi _{i}(x_{2})-f(x_{2})|<\varepsilon /3+\varepsilon /3+\varepsilon /3=\varepsilon$ ${\ displaystyle | f (x_ {1}) - f (x_ {2}) | \ leqslant | f (x_ {1}) - \ varphi _ {i} (x_ {1}) | + | \ varphi _ { i} (x_ {1}) - \ varphi _ {i} (x_ {2}) | + | \ varphi _ {i} (x_ {2}) - f (x_ {2}) | <\ varepsilon / 3 + \ varepsilon / 3 + \ varepsilon / 3 = \ varepsilon}$ where $\varphi _{i}$ ${\ displaystyle \ varphi _ {i}}$ - suitable item $(\varepsilon /3)$ ${\ displaystyle (\ varepsilon / 3)}$ networks.

Thus, the family $F$ ${\ displaystyle F}$ is equally continuous .

In other words, complete boundedness implies uniform boundedness and equidistant continuity.

Sufficiency

Now it is necessary to prove that the uniform boundedness and uniform degree of continuity of the family $F$ ${\ displaystyle F}$ implies the existence of a finite $\varepsilon$ ${\ displaystyle \ varepsilon}$ networks for every end $\varepsilon >0$ ${\ displaystyle \ varepsilon> 0}$ .

Fix $\varepsilon >0$ ${\ displaystyle \ varepsilon> 0}$ .

Let be $K$ ${\ displaystyle K}$ Is a constant that appears in the definition of uniform boundedness.

Choose this $\delta >0$ ${\ displaystyle \ delta> 0}$ , which appears in the definition of uniform continuity and corresponds to $\varepsilon /5$ ${\ displaystyle \ varepsilon / 5}$ .

Consider a rectangle $[a,b]\times [-K,K]$ ${\ displaystyle [a, b] \ times [-K, K]}$ and break it into vertical and horizontal lines into rectangular cells smaller than $\delta$ ${\ displaystyle \ delta}$ horizontally and $\varepsilon /5$ ${\ displaystyle \ varepsilon / 5}$ vertically. Let be $x_{1}$ ${\ displaystyle x_ {1}}$ , $x_{2}$ ${\ displaystyle x_ {2}}$ , $\dots$ ${\ displaystyle \ dots}$ , $x_{N}$ ${\ displaystyle x_ {N}}$ - nodes of this lattice (abscissa).

If we now consider an arbitrary function $f\in F$ ${\ displaystyle f \ in F}$ , then for each node $x_{i}$ ${\ displaystyle x_ {i}}$ the lattice is sure to find such a point $(x_{i},y_{j})$ ${\ displaystyle (x_ {i}, y_ {j})}$ lattices that $|f(x_{i})-y_{j}|<\varepsilon /5$ ${\ displaystyle | f (x_ {i}) - y_ {j} | <\ varepsilon / 5}$ . If we now consider the broken function $\varphi$ ${\ displaystyle \ varphi}$ , which takes the corresponding values at the nodes, deviating from the function by no more than $\varepsilon /5$ ${\ displaystyle \ varepsilon / 5}$ , due to the fact that the function itself deviates on each interval by no more than $\varepsilon /5$ ${\ displaystyle \ varepsilon / 5}$ , the broken line on each such segment deviates by no more than $3\varepsilon /5$ ${\ displaystyle 3 \ varepsilon / 5}$ .

Since every point $x$ ${\ displaystyle x}$ segment $[a,b]$ ${\ displaystyle [a, b]}$ turns out to be on one of such segments, say, $[x_{k},x_{k}+1]$ ${\ displaystyle [x_ {k}, x_ {k} +1]}$ , it turns out that the deviation of the function from the constructed in this way broken does not exceed $\varepsilon$ ${\ displaystyle \ varepsilon}$ :

|f(x)-\varphi (x)|\leqslant |f(x)-f(x_{k})|+|f(x_{k})-\varphi (x_{k})|+|\varphi (x_{k})-\varphi (x)|<\varepsilon /5+\varepsilon /5+3\varepsilon /5=\varepsilon

{\ displaystyle | f (x) - \ varphi (x) | \ leqslant | f (x) -f (x_ {k}) | + | f (x_ {k}) - \ varphi (x_ {k}) | + | \ varphi (x_ {k}) - \ varphi (x) | <\ varepsilon / 5 + \ varepsilon / 5 + 3 \ varepsilon / 5 = \ varepsilon}

.

Thus, it is shown that a finite (!) System of broken functions of the indicated form is $\varepsilon$ ${\ displaystyle \ varepsilon}$ network for a given $\varepsilon >0$ ${\ displaystyle \ varepsilon> 0}$ .

Applications

Arzela's theorem finds its application in the theory of differential equations .

In Peano 's theorem (on the existence of a solution to the Cauchy problem ), a system of functions is constructed, which in the theory of differential equations is called Euler broken lines . This system turns out to be a uniformly bounded and equally continuous family of functions, from which, according to the Arzel theorem, we can distinguish a uniformly converging sequence of functions, the limit of which will be the desired solution to the Cauchy problem.

Literature

Kolmogorov A.N. , Fomin S.V. Elements of function theory and functional analysis. - ed. third, recycled. - M .: Science , 1972 . - 496 p.

Notes

↑ ¹ ² Ascoli, G. (1883-1884), “Le curve limiti di una varietà data di curve”, Atti della R. Accad. Dei Lincei Memorie della Cl. Sci. Fis. Mat. Nat. 18 (3): 521-586.
↑ Arzelà, Cesare (1882–1883), Un'osservazione intorno alle serie di funzioni, Rend. Dell 'Accad. R. Delle Sci. Dell'Istituto di Bologna: 142-159.
↑ Arzelà, Cesare (1895), Sulle funzioni di linee, Mem. Accad Sci. Isst. Bologna Cl. Sci. Fis. Mat. 5 (5): 55-74.
↑ Fréchet, Maurice (1906), Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo 22: 1-74, doi: 10.1007 / BF03018603.
↑ Dunford, Nelson; Schwartz, Jacob T. (1958), Linear operators, volume 1, Wiley-Interscience.
↑ Bourbaki, Nicolas (1998), General topology. Chapters 5-10, Elements of Mathematics, Berlin, New York: Springer-Verlag, MR1726872, ISBN 978-3-540-64563-4 .

[Ascoli18813-1] ¹ ² Ascoli, G. (1883-1884), “Le curve limiti di una varietà data di curve”, Atti della R. Accad. Dei Lincei Memorie della Cl. Sci. Fis. Mat. Nat. 18 (3): 521-586.

[2] Arzelà, Cesare (1882–1883), Un'osservazione intorno alle serie di funzioni, Rend. Dell 'Accad. R. Delle Sci. Dell'Istituto di Bologna: 142-159.

[3] Arzelà, Cesare (1895), Sulle funzioni di linee, Mem. Accad Sci. Isst. Bologna Cl. Sci. Fis. Mat. 5 (5): 55-74.

[4] Fréchet, Maurice (1906), Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo 22: 1-74, doi: 10.1007 / BF03018603.

[5] Dunford, Nelson; Schwartz, Jacob T. (1958), Linear operators, volume 1, Wiley-Interscience.

[6] Bourbaki, Nicolas (1998), General topology. Chapters 5-10, Elements of Mathematics, Berlin, New York: Springer-Verlag, MR1726872, ISBN 978-3-540-64563-4 .