Eberlein Smulian Theorem The Krein-Smulianˇ Theorem

SMA 5878 Functional Analysis II

Alexandre do Nascimento Javier Cubas

Departamento de Matem´atica Instituto de Ciˆencias Matem´aticas e de Computa¸c˜ao Universidade de S˜ao Paulo

April 7, 2018

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem

Appendix B

Weakly Compactness

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem

In this lecture we present the proofs of Eberlein-Smulianˇ and Krein-Smulianˇ theorems.

Before we begin, remeber the very usefull resut: Theorem If K is a convex subset of a Banach space X , then the closure of K in the weak and strong topology coincide.

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem Eberlein-Smulianˇ Theorem

Compactness in general topological spaces cannot be caracterized with sequences as we do in metric spaces. Howhever in the weak topology is possible to do it as follows: Theorem (Eberlein-Smulian)ˇ Let W be a subset of a Banach space X . The following proprierties are equivalent: (A) The closure of W in the weak topology is compact in the weak topology. (B) Every sequence of elements of W has a weakly convergent subsequence in X . (C) Every sequence of elements of W has a limit point in X .

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem Lemma B.1.1

Lemma (B.1.1)

If X is a separable Banach space, then X ⇤ has a countable total subset A⇤ = a⇤ : n N with the proprierty { n 2 }

x X =sup x, an⇤ X ,X ⇤ . k k n N|h i | 2

Proof: Let a a sequence of unitary vectors which are dense in the { n}N unitary sphere of X .Foreachn N,leta⇤ such hat 2 n a , a = a = a = 1. h n n⇤iX ,X ⇤ k nkX k n⇤kX ⇤ We show that A⇤ = a⇤ : n N is total. { n 2 }

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem Lemma B.1.1

If y X 0 ,witha⇤(y)=0foralln N,takex = y/ y X , 2 { } n 2 k k then x =1anda (x) = 0. k kX n⇤ k !1 Let ank k N a subsequence of an n N such that ank x,then { } 2 { } 2 ! the sequences

a , a⇤ and x, a⇤ h nk nk iX ,X ⇤ h nk iX ,X ⇤ have the same limit when k goes to infinity and they are constant, i.e.,

x, a⇤ =0and a , a⇤ =1, h nk iX ,X ⇤ h nk nk iX ,X ⇤ which is a contradiction, q.e.d.

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem Lemma B.1.1

Now, we prove the second claim. For any x X with x X = 1, 2 k k k !1 there is a subsequence ank k N of an n N such that ank x { } 2 { } 2 ! and

1= x X =sup x, x⇤ X ,X ⇤ sup x, an⇤ X ,X ⇤ k k x =1|h i | n N|h i | k ⇤kX ⇤ 2

=limsup ank , an⇤ X ,X ⇤ lim ank , an⇤k X ,X ⇤ =1 k n N|h i | k |h i | !1 2 !1 and consequently, for all x X , 2

x X =sup x, an⇤ X ,X ⇤ . k k n N|h i | 2

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem Lemma B.1.2

Lemma (B.1.2)

Let X be a Banach space over K such that X ⇤ contains an countable total set. Then a weakly compact subset of X is metrizable.

Proof: Let A⇤ = an⇤ : n N be a total set of X ⇤ with { 2 } + an⇤ X ⇤ =1foralln N and define d : X X R the metric k k 2 n ⇥ ! define by d(x, y)= 1 2 x y, a .IfW X is n=0 |h n⇤iX ,X ⇤ | ⇢ weakly compact, note that W , x⇤ X ,X is a compact subset of K P h i ⇤ and, of the Uniform Boundedness Principle, W is a bounded set of X ( W X := supw W w X < ). k k 2 k k 1

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem Lemma B.1.2

If Ww and Wd denote the set W with weak topology and metric topology of d,respectively,letI : W W the identity operator. w ! d If I : W W is continuous then it is homeomorphism (since w ! d that W is weakly compact) and the result follows. Lack to prove that I : Ww Wd is continuous. Indeed, given ✏ > 0 let N N ! 2 such that

1 n 1 n ✏ 2 x y, a⇤ 2 W < |h niX ,X ⇤ |  k kX 2 n=XN+1 n=XN+1 and V = y W : x y, a < ✏ , n =0, 1, , N be { 2 |h n⇤iX ,X ⇤ | 2(N+1) ··· } a neighborhood of x in Ww such that

d(x, y) < ✏, y V , 8 2 This concludes the proof.

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem Corollary B.1.1

Corollary (B.1.1) In the same hypothesis of the Eberlein Smulian ‘s Theorem, we have that (A) implies (B).

Proof: Let wn n N asequenceinW and Y := span wn : n N . { } 2 { 2 } Since Y is also closed in the weak topology, see Theorem 1 and the weak topology is Hausdor↵,thesubsetW Y has compact \ closure in the weak topology in the Banach space Y .

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem Corollary B.1.1

Now, Y is separable and by the Lemmas B.1.1andB.1.2, we have that W Y with the weak topology is metrizable, and therefore \ wn n N has a weakly convergent subsequence to a element of Y , { } 2 i.e., w * y Y , in Y , nk 2 but X Y ,thenw * y X , in the weak topology of X . ⇤ ⇢ ⇤ nk 2

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem Lemma B.1.3

Lemma (B.1.3) (C) (A). ) Proof: If every sequence of W has a limit point in X , for a given x X ⇤ 2 ⇤ the subset W , x⇤ X ,X of K has the same proprierty in K. h i ⇤ It follows that W , x⇤ X ,X is a bounded subset of K and by h i ⇤ Uniform boundedness Principle W is bounded.

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem Lemma B.1.3

Let J : X X be the canonical aplication. ! ⇤⇤ Since J(W ) is bounded, the closure w ⇤(J(W )) of J(W )inthe weak⇤ topology of X ⇤⇤ is compact, Banach-Alaoglu ‘s Theorem. We claimed its enough to show w (J(W )) J(X ). ⇤ ⇢ (Then, of course, the case when X is reflexive is far more easier).

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem Lemma B.1.3

Let x w (JW )andx X , x = 1, w W with ⇤⇤ 2 ⇤ 1⇤ 2 ⇤ k 1⇤kX ⇤ 1 2

x⇤, x⇤⇤ Jw < 1. |h 1 1iX ⇤,X ⇤⇤ | Before we proceed, let F a finite dimensional subspace of X ⇤⇤. The unitary sphere of F is compact and therefore has a 1 net 4 x , , x . { 1⇤⇤ ··· n⇤⇤}

Choose xp⇤ on the unitary sphere of X ⇤ such that 3 xp⇤, xp⇤⇤ X ,X > ,1 p n.then,foranyx⇤⇤ F we have h i ⇤ ⇤⇤ 4   2 that 1 max x⇤, x⇤⇤ :1 p n x⇤⇤ . {|h p iX ⇤,X ⇤⇤ |   } 2k kX ⇤⇤

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem Lemma B.1.3

Now choose x , , x in X , x =1and 2⇤ ··· n⇤2 ⇤ k m⇤ kX ⇤ 1 max x⇤ , y ⇤⇤ :2 m n y ⇤⇤ {|h m iX ⇤,X ⇤⇤ |   2} 2k kX ⇤⇤ for all y span x , x Jw . Using again that ⇤⇤ 2 { ⇤⇤ ⇤⇤ 1} x w (J(W )), choose w W such that ⇤⇤ 2 ⇤ 2 2 1 max x⇤ , x⇤⇤ Jw :1 m n < . {|h m 2iX ⇤,X ⇤⇤ |   2} 2

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem Lemma B.1.3

Choose x⇤ , , x⇤ in the unitary sphere of X ⇤ such that n2+1 ··· n3 1 max x⇤ , y ⇤⇤ : n < m n y ⇤⇤ {|h m iX ⇤,X ⇤⇤ | 2  3} 2k kX ⇤⇤ for all y span x , x Jw , x Jw and, using again that ⇤⇤ 2 { ⇤⇤ ⇤⇤ 1 ⇤⇤ 2} x w (J(W )) choose w W such that ⇤⇤ 2 ⇤ 3 2 1 max x⇤ , x⇤⇤ Jw :1 m n < . {|h m 3iX ⇤,X ⇤⇤ |   3} 3

This process can be continued indefinitely. Let wn n N be the { } 2 sequence resulting from this construction.

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem Lemma B.1.3

By hypothesis, there exist a point x X which is a limit point of 2 the sequence wn n N in the weak topology of X . { } 2 Since Z = span wn : n N is weakly closed, x Z and take { 2 } 2 R = span x , x Jw , x Jw , . ⇤⇤ { ⇤⇤ ⇤⇤ 1 ⇤⇤ 2 ···} We have that, for all y R , ⇤⇤ 2 ⇤⇤ 1 sup xm⇤ , y ⇤⇤ X ⇤,X ⇤⇤ y ⇤⇤ X ⇤⇤ m N |h i | 2k k 2

and therefore, for any point in the closure of R⇤⇤,inparticularfor x Jx. ⇤⇤

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem Lemma B.1.3

Another characteristic of our construction is that 1 x⇤ , x⇤⇤ Jw < , n > n > m. |h m niX ⇤,X ⇤⇤ | p p

therefore, for n > np > m,

x⇤ , x⇤⇤ Jx x⇤ , x⇤⇤ Jw + w x, x⇤ |h m iX ⇤,X ⇤⇤ |  |h m niX ⇤,X ⇤⇤ | |h n miX ,X ⇤ |

Since x is a limit point of wn n N in the weak topology, given xm⇤ { } 2 and an integer N > m there exist w with w x, x < 1 n |h n m⇤ iX ,X ⇤ | N e n > nN > m.Forthiselementwehave 2 x⇤ , x⇤⇤ Jx x⇤ , x⇤⇤ Jw + w x, x⇤ < |h m iX ⇤,X ⇤⇤ |  |h m niX ⇤,X ⇤⇤ | |h n miX ,X ⇤ | N

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem Eberlein-Smulianˇ Theorem The Krein-Smulianˇ Theorem Lemma B.1.3

and, consequently x , x Jx =0forallm.Asseen h m⇤ ⇤⇤ iX ⇤,X ⇤⇤ above 1 sup xm⇤ , x⇤⇤ Jx X ⇤,X ⇤⇤ x⇤⇤ Jx X ⇤⇤ m N |h i | 2k k 2 and therefore x⇤⇤ = Jx. This concludes the proof.

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem The Krein-Smulianˇ Theorem

Before we begin, we present an important auxiliar result.

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem The Krein-Smulianˇ Theorem Lemma B.2.1

Lemma Let X be a separable Banach space and x X . Suppose for all ⇤⇤ 2 ⇤⇤ x⇤ X ⇤ and sequence xn⇤ in X ⇤ which converges to x⇤ in the 2 { } n weak⇤ topology; this is, x, xn⇤ !1 x, x⇤ for all x X ,wehave n h i ! h i 2 that x , x !1 x , x .Thenx = Jx for some x X . h n⇤ ⇤⇤i ! h ⇤ ⇤⇤i ⇤⇤ 2 Proof: Let xj j N a dense subset of X . Suppose that x⇤⇤ / JX ; { } 2 2 this is, that d(x⇤⇤, JX )=d > 0. By the Hahn-Banach Theorem, there exist x X such that, x = 1, ⇤⇤⇤ 2 ⇤⇤⇤ k ⇤⇤⇤kX ⇤⇤⇤ JX , x =0e x , x = d.Let h ⇤⇤⇤iX ⇤⇤,X ⇤⇤⇤ h ⇤⇤ ⇤⇤⇤iX ⇤⇤,X ⇤⇤⇤

W = z⇤ : x , z⇤ < 1fori =1, , n . n { |h i iX ,X ⇤ | ··· }

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem The Krein-Smulianˇ Theorem Lemma B.2.1

By the Goldstine Theorem (JX ⇤ is dense in X ⇤⇤⇤ with the weak⇤ topology of X ), given Jx , , Jx , x X and ✏ > 0, there ⇤⇤⇤ 1 ··· n ⇤⇤ 2 ⇤⇤ exist x X , x = 1, such that ⇤ 2 ⇤ k ⇤kX ⇤

x , x ⇤ = x , x ⇤ Jx , x ⇤⇤⇤ = Jx , Jx ⇤ x ⇤⇤⇤ <✏, |h 1 i| |h 1 iX ,X ⇤ h 1 iX ⇤⇤,X ⇤⇤⇤ | |h 1 iX ⇤⇤,X ⇤⇤⇤ | ......

x , x ⇤ = x , x ⇤ Jx , x ⇤⇤⇤ = Jx , Jx ⇤ x ⇤⇤⇤ <✏, |h n i| |h n iX ,X ⇤ h 1 iX ⇤⇤,X ⇤⇤⇤ | |h n iX ⇤⇤,X ⇤⇤⇤ | x ⇤, x ⇤⇤ x ⇤⇤, x ⇤⇤⇤ = x ⇤⇤, Jx ⇤ x ⇤⇤⇤ <✏. |h iX ⇤,X ⇤⇤ h iX ⇤⇤,X ⇤⇤⇤ | |h iX ⇤⇤,X ⇤⇤⇤ |

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem The Krein-Smulianˇ Theorem Lemma B.2.1

Therefore, there exist a functional x B¯X ⇤ (0) z X : z , x d/2 W .Thesequence n⇤ 2 1 \{ ⇤ 2 ⇤ |h ⇤ ⇤⇤iX ⇤,X ⇤⇤ | }\ n x converges to zero in the weak topology of X indeed, given { n⇤} ⇤ ⇤ x X and ✏ > 0, there exist x with (x/✏) x < 1and 2 j k j k

x/✏, x⇤ x/✏ x , x⇤ + x , x⇤ < 2, for all n j. |h n iX ,X ⇤ |  |h j n iX ,X ⇤ | |h j n iX ,X ⇤ | However, x , x d/2 and this gives us a contradiction, |h n⇤ ⇤⇤iX ⇤,X ⇤⇤ | this concludes the proof of lemma.

We are now in a position to prove the Krein-Smulianˇ Theorem.

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem The Krein-Smulianˇ Theorem Krein-Smulianˇ

Using Eberlein Smulian result we the the following theorem wich is already known in the strong topology.

Theorem (Krein-Smulian)ˇ If X is a Banach space and K X is weakly compact, then the ⇢ closed convex hull coK of K is weakly compact.

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem The Krein-Smulianˇ Theorem Proof of K-S Theorem

First we reduced our prove to the case that X is separable. From Eberlein Smulian Theorem, its enough to show that every sequence of elements of coK has a weakly convergent subsequence.

Since each element of a sequence of coK is a convex finite linear combination of elements of K, the sequence will be generate by a sequence S in K.

Let Y be the closure of the subspace generate by S, its enough to show that Kˆ = K Y is weakly compact (Theorem 1) in Y . \

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem The Krein-Smulianˇ Theorem

Hence, its enough to consider X separable and K weakly compact. Define Kw as the subset K with the weak topology. Consider T : X C(K )definedbyTx (k)=x (k), k K and ⇤ ! w ⇤ ⇤ 2 the dual operator of T , T : C(K ) X . ⇤ w ⇤ ! ⇤⇤ Choose any element of C(Kw )⇤, by the Riesz representation Theorem is a regular measure µ,andlet xn⇤ n N a bounded { } 2 sequence that converges in the weak⇤ topology of X ⇤ to x⇤.

ICMC - USP SMA 5878 Functional Analysis II Eberlein Smulian Theorem The Krein-Smulianˇ Theorem

Then, by the dominated convergence Theorem,

n x⇤, T ⇤µ = x⇤(k)dµ(k) !1 x⇤(k)dµ(k)= x⇤, T ⇤µ . h n iX ⇤,X ⇤⇤ n ! h iX ⇤,X ⇤⇤ Z Z From Lemma B.2.1 we have that T µ JX . ⇤ 2 Now, take the unitary disk in C(Kw )⇤, wich is convex and compact 1 in weak⇤ topology. Then, the range of the disk by J T ⇤ is also convex, weakly compact and contains K.

This shows that co(K) is weakly compact and finishes the proof.

ICMC - USP SMA 5878 Functional Analysis II