Lecture
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.
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Functional analysis
Terms: Functional analysis