5. Критерии компактности в пространствах С[0, 1], lp. Теорема Арцела

We give criteria for compactness in specific metric spaces.

Definition 12. The set M of functions continuous on the segment [0, 1] is called uniformly bounded if $ C: | x (t) | £ C, " t Î [0, 1], " x Î M.

Definition 13. The set M of functions continuous on the segment [0, 1] is called equicontinuous if for " e > 0 $ d ( e )> 0: | t 1 - t 2 | < d , t 1 , t 2 Î [0 , 1] Þ | x (t 1 ) - x (t 2 ) | < e , " x Î M.

Theorem 5 (Arzela) . The set M Ì C [0, 1] is relatively compact Û 1) M is uniformly bounded, 2) M is equicontinuous.

Necessity. We set e = 1 and for this e we construct a finite e -net x 1 (t), ..., x n (t) Î C [0, 1] for the set M. Then S (x k (t), 1) Q M. For any y Î М Ì S (x k (t), 1) there is m , 1 £ m £ n such that d (y, x m (t)) <1. Therefore, | y (t) - x m (t) | <1 and | y (t) | £ | x m (t) | + 1. Since there is a C such that | x k ( t ) | £ C for any k = 1, 2, 3, ..., n , then | y (t) | £ C + 1. Since y Î M in this inequality is arbitrary and since the right-hand side of the last inequality does not depend on the choice of this y, we obtain the uniform boundedness of the set of functions from M.

Now we take e > 0 arbitrarily and also construct a finite e -net, {x k ( t )}, k = 1, 2, ..., n . For a finite set of functions {x k ( t )}, due to its finiteness and uniform continuity of each of the functions, one can specify d > 0 such that | t 1 - t 2 | < d , t 1 , t 2 Î [0, 1] Þ | x k (t 2 ) - x k (t 2 ) | < e for any k = 1, 2, ...,n . Take an arbitrary x Î M. Then $ m is such that | x (t) - x m (t) | < e for " t Î [0, 1]. By the inequalities

| x (t 1 ) - x (t 2 ) | £ | x (t 1 ) - x m (t 1 ) | + | x m (t 1 ) - x m (t 2 ) | + | x m (t 2 ) - x (t 2 ) | < e + e + e = 3 e

for | t 1 - t 2 | < d , t 1 , t 2 Î [0, 1], it follows that | x (t 1 ) - x (t 2 ) | £ 3 e if | t 1 - t 2 | < d , t 1 , t 2 Î [0, 1]. This shows the equicontinuity of functions from the set M.

Adequacy. Let the set of functions M Ì C [0, 1] be uniformly bounded and equicontinuous. Let us construct a compact e -net for M. By the assumption of equicontinuity of the set M for " e > 0 $ d > 0: from | t 1 - t 2 | < d Þ | x ( t 1 ) - x ( t 2 ) | < e for " x Î M. natural number n so that 1 / n < d and divide the segment [0, 1] into n equal parts. For each function x Î M, we assign to it a set of numbers (x (0), x (1 / n), x (2 / n), ..., x (1)). This has constructed a mapping of the functions of the set M into the vector (x 1 , x 2 , ..., x n + 1 ) Î R n + 1 . Consider the set M n + 1 = {(x 1 , ..., x n + 1 ) Î R n + 1 : $ x Î M: (x (0), x (1 / n ), x (2 / n ), ..., x (1)) = (x 1 , x 2 , x 3 , ..., x n + 1 )}. Since | x ( t ) | £ M, " t Î [0, 1], " x Î M, then | x k | £ C for k = 1, 2, ..., n + 1, that is, the set M n +1 is bounded in R n + 1 , and therefore relatively compact in R n + 1 .

Let us construct a set of piecewise-linear functions M cl from the set M n +1 . Namely, for (x 1 , x 2 , x 3 , ..., x n + 1 ) Î M n +1, we set x cl ( t ) = n ( t - k / n ) ( x k +2 - x k +1 ) + x k +1 , for t Î [ k / n , (k + 1) / n ], k = 0, 1, 2, ..., n - 1. Geometrically, the latter means that we connect the points ( k / n , x k +1 ) and (( k +1) / n , x k + 2 ) by a straight line segment. Let us calculate the distance between two functions from M cl in the metric of the space С [0, 1]. We have

In this case, the second equality is fulfilled, since the difference of linear functions on the segment reaches its smaller and larger values ​​at the ends of the segment. This established an isometric isomorphism between metric spaces M n +1 with metric d ¥ ( x , y ) = | x k - y k | and M cl with the metric of the space C [0, 1].

Let x ( n ) Î M and x ( n ) cl Î M cl be piecewise linear functions constructed from x ( n ) in the above way. Since the set М n +1 is bounded in R n +1 and therefore relatively compact, and the convergence in the metric in R n +1 is equivalent to the convergence in the metric d ¥ ( x , y ) = max k | x k - y k | (show this), then the set Mcl is also relatively compact in C [0, 1]. To complete the proof we show that M cells compact e -net for the set M.

Due to the equicontinuity and the choice of n from t 1 , t 2 Î [( k –1) / n , k / n ], it follows that | x ( t 1 ) - x ( t 2 ) | < e for " х Î M. Let, for definiteness, at the ends of the segment x (( k –1) / n ) £ x ( k / n ). The latter means that the function x cl ( t ) increases on the interval [( k –1) / n , k / n ]. Then - e < x ( t ) - x ( k / n ) £ x ( t ) - x cl ( t ) £ x ( t ) - x (( k –1) / n ) < e for any t Î [( k –1) / n , k / n ]. Thus , d ( x ( t ), x cl ( t )) < e and M cl is a compact e -network for M. The theorem is proved.

Theorem 6. The set M Ì l p (1 £ p < ¥ ) is relatively compact if and only if 1) the set M is bounded, 2) for " e > 0 $ N ( e ): < e for " n ³ N, " x Î M.

Necessity. Necessity 1) of the condition is obvious. Let us prove the second condition. Let y (1) , y (2) , ..., y ( r ) be a finite e / 2 - network for the set M. Since this set is finite for " e > 0, $ N ( e ): < e / 2 for " n ³ N, m = 1, 2, ..., r . Then for arbitrary х Î M we choose у ( m ) so that d ( x , y ( m )) < e / 2. As a result, we have: £ £ d ( x , y ( m ) ) + e / 2 < e . We obtain the required inequality.

Adequacy. Let x = (x 1 , x 2 , ..., x m , x m +1 , x m +2 , ..) and P m x = (x 1 , x 2 , ..., x m , 0 , 0, ...), Q m x = x - P m x. According to the hypotheses of Theorem " e > 0 $ m : d (Q n x , 0) < e , n ³ m , " x Î M . The set M m = {P m x , x Î M} is isometrically isomorphic to a bounded set in R m ; therefore, it is relatively compact. Then for x Î M, P m x Î M m and d (x, P m x ) = d (Q m x , 0) < e . Hence M m is a compact e -net for M, hence M is compact. The theorem is proved.