2.3. REFLEXIVITY AND WEAK TOPOLOGY
2.3
35
Reflexivity and Weak Topology
Proposition 2.3.1. If X is a Banach space and Y is a closed subspace of
X, the (Y, Y ⇤ ) = (X, X ⇤ ) \ Y , i.e. the weak topology on Y is the weak
topology on X restricted to Y .
Theorem 2.3.2. (Theorem of Alaoglu, c.f. [Fol, Theorem 5.18] )
BX ⇤ is w⇤ compact for any Banach space X.
Sketch of a proof. Consider the map
Y
⇤
!
{ 2 K : | |  kxk},
: BX
x2X
x⇤ 7! (x⇤ (x) : x 2 X).
Then we check that
is continuous
with respect to w⇤ topology on BX ⇤
Q
and the product topology on x2X { 2 K : | |  kxk}, has a closed image,
⇤ onto its image.
and is a homeorphism from BX
Q
Since by the Theorem of Tychano↵ x2X { 2 K : | |  kxk} is compact,
1)
(BX ⇤ ) is a compact subset, which yields (via the homeomorphism
that BX ⇤ is compact in the w⇤ topology.
Theorem 2.3.3. (Theorem of Goldstein)
BX is (via the canonical embedding) w⇤ dense in BX ⇤⇤ .
The proof follows immediately from the following Lemma.
Lemma 2.3.4. Let X be a Banach space and let x⇤⇤ 2 X ⇤⇤ , with x⇤⇤  1,
and x⇤1 , x⇤2 , . . . , x⇤n 2 X ⇤ . Then
inf
kxk1
n
X
i=1
|hx⇤⇤ , x⇤i i
hx⇤i , xi|2 = 0.
P
Proof. For x 2 X put (x) = ni=1 |hx⇤⇤ , x⇤i i hx⇤i , xi|2 and = inf x2BX (x),
and choose a sequence (xj ) ⇢ BX so that (xj ) & , if j % 1. W.l.o.g we
can also assume that ⇠i = limk!1 hx⇤i , xk i exists for all i = 1, 2, . . . , n.
For any t 2 [0, 1] and any x 2 BX we note for k 2 N
((1
t)xk + tx)
n
X
=
|hx⇤⇤ , x⇤i i
i=1
=
n
X
i=1
|hx⇤⇤ , x⇤i i
(1
t)hx⇤i , xk i
thx⇤i , xi|2
hx⇤i , xk i + thx⇤i , xk
xi|2
36
CHAPTER 2. WEAK TOPOLOGIES AND REFLEXIVITY
=
n
X
i=1
|hx⇤⇤ , x⇤i i
+2t<
n
⇣X
i=1
hx⇤i , xk i|2
+ 2t<
!k!1
n
⌘
X
|hx⇤i , xk i hx⇤i , xi|2
hx⇤i , xk ihx⇤i , xk xi + t2
hx⇤⇤ , x⇤i i
n
⇣X
i=1
i=1
(hx⇤⇤ , x⇤i i ⇠i )(⇠i
|
{z
}
=:
From the minimality of
<
n
⇣X
i ⇠i
i=1
<(hx⇤ , xi) with x⇤ :=
n
⇣X
kx⇤ k  <
⌘
⇠
i i .
hx⇤ , xi = hx⇤ , e
ia
i=1
Indeed, write
hx⇤ , xi
=
reia ,
|hx⇤ , xi| = e
ia
i=1
it follows that for all x 2 BX
⌘
and thus
i
n
⌘
X
|⇠i
hx⇤i , xi) + t2
n
X
⇤
i xi ,
i=1
then
n
⇣X
xi  <
i=1
On the other hand
kx⇤ k
lim sup <(hx⇤ , xk i) = <
k!1
and thus
n
X
kx⇤ k
i ⇠i
i=1
which implies that
kx⇤ k = <
So
= lim
=
k!1
n
X
i=1
(xk )
| i |2
n
⇣X
n
⇣X
<
i=1
n
⇣X
i=1
i ⇠i
⌘
.
i=1
⌘
⇠
i i ,
i ⇠i
⌘
⌘
⇠
.
i
hx⇤i , xi|2 .
37
2.3. REFLEXIVITY AND WEAK TOPOLOGY
=
=
n
X
i i
i=1
n
X
⇤⇤ ⇤
i (hx , xi i
i=1
=<
n
⇣X
i=1
⌘
⇤⇤ ⇤
i
⇠
)
(hx
,
x
i
i
i
= <hx⇤⇤ , x⇤ i
⇤⇤
⇤
 kx k · kx k
Thus
⇠i )
<
n
⇣X
i=1
⇤
i ⇠i
⌘
(since
kx k  kx⇤ k
2 R)
kx⇤ k = 0.
= 0 which proves our claim.
Theorem 2.3.5. Let X be a Banach space. Then X is reflexive if and only
if BX is compact in the weak topology.
Proof. Let : X ,! X ⇤⇤ be the canonical embedding.
“)” If X is reflexive and thus
is onto it follows that
is an homeomorphism between (BX , (X, X ⇤ )) and (BX ⇤⇤ , (X ⇤⇤ , X ⇤ )). But by the
Theorem of Alaoglu 2.3.2 (BX ⇤⇤ , (X ⇤⇤ , X ⇤ )) is compact.
“(” Assume that BX , (X, X ⇤ ) is compact, and assume that x⇤⇤ 2 BX ⇤⇤
we need to show that there is an x 2 BX so that (x) = x⇤⇤ , or equivalently
that hx⇤ , xi = hx⇤⇤ , x⇤ i for all x⇤ 2 X ⇤ .
For any finite set A = {x⇤1 , . . . x⇤n } ⇢ X ⇤ and for any " > 0 we can,
according to Lemma 2.3.4, choose an x(A,") 2 BX so that
n
X
i=1
|hx⇤⇤
x(A,") , x⇤i i|2  ".
The set
I = {(A, ") : A ⇢ X ⇤ finite and " > 0},
is directed via (A, ")  (A0 , "0 ) : () A ⇢ A0 and "0  ". Thus, by
compactness, the net x(A,") : (A, ") 2 I must have a subnet (zj : j 2 J)
which converges weakly to some element x 2 BX .
We claim that hx⇤ , xi = hx⇤⇤ , x⇤ i, for all x⇤ 2 X ⇤ . Indeed, let j 7! ij
be the map from J to I, so that zj = xij , for all j 2 J, and so that, for
any j0 there is a i0 with ji
j0 , for i i0 . Let x⇤ 2 X ⇤ and " > 0. Put
⇤
i0 = ({x }, ") 2 I, choose j0 , so that ij
i0 , for all j
j0 , and choose
38
CHAPTER 2. WEAK TOPOLOGIES AND REFLEXIVITY
j1 2 J, j1 j0 , so that |hx zj⇤ , x⇤ i| < ", for all j j1 . It follows therefore
that (note that for ij1 = (A, "0 ) it follows that x⇤ 2 A and "0  ")
hx⇤⇤
x, x⇤ i  hx⇤⇤
xij1 , x⇤ i + hzj1
x, x⇤ i  " + " = 2".
Since " > 0 and x⇤ 2 X ⇤ were arbitrary we deduce our claim.
Theorem 2.3.6. For a Banach space X the following are equivalent.
a) X is reflexive,
b) X ⇤ is reflexive,
c) every closed subspace of X is reflexive.
Proof. “(a))(c)” Assume Y ⇢ X is a closed subspace. Proposition 2.2.5
yields that BY = BX \ Y is a (X, X ⇤ )-closed and, thus, (X, X ⇤ )-compact
subset of BX . Since, by the Theorem of Hahn-Banach (Corollary 1.4.4),
every y ⇤ 2 Y ⇤ can be extended to an element in X ⇤ , it follows that (Y, Y ⇤ )
is the restriction of (X, X ⇤ ) to the subspace Y . Thus, BY is (Y, Y ⇤ )compact, which implies, by Theorem 2.3.5 that Y is reflexive.
“(a))(b)” If X is reflexive then (X ⇤ , X ⇤⇤ ) = (X ⇤ , X). Since by the
Theorem of Alaoglu 2.3.2 BX ⇤ is (X ⇤ , X)-compact the claim follows from
Theorem 2.3.5.
“(c))(a)” clear.
“(b))(a)” If X ⇤ is reflexive, then, “(a))(b) ” X ⇤⇤ is also reflexive and thus,
the implication “(a))(c)” yields that X is reflexive.
Similar ideas as in the proof of Theorem 2.3.3 are used to show the
following result which characterizes when a Banach space X is a dual spas
of another space.
Theorem 2.3.7. Assume that X is a Banach space and Z is a closed subspace of X ⇤ so that BX is compact in the topology (X, Z).
Then Z ⇤ is isometrically isomorphic to X and the map
T : X ! Z ⇤,
x 7! fx , with fx (z) = hz, xi, for x 2 X and z 2 Z,
is an isometrical isomorphism onto Z ⇤ .
Proof. We first note that T (BX ) is (Z ⇤ , Z) dense in Z ⇤ . Indeed, if this
is not true we can apply the Geometrical Hahn Banach Theorem for locally convex spaces (Theorem 2.2.3) applied to the locally convex space
2.3. REFLEXIVITY AND WEAK TOPOLOGY
39
Z ⇤ , (Z ⇤ , Z) whose dual is by Proposition 2.1.5 Z, (Z, Z ⇤ ) , and obtain
elements z ⇤ 2 SZ ⇤ and z 2 SZ so that
sup hx, zi < hz ⇤ , zi = 1.
x2BX
But this contradicts our assumption.
Secondly, our assumption says that T (BX ) is (Z ⇤ , Z)-compact. To see
that note that if (xi )i2I is a net in X and z ⇤ 2 Z ⇤ , then
(fxi )i2I converges to z ⇤ with respect to (Z ⇤ , Z)
() limhxi , zi = hz ⇤ , zi for all z 2 Z
i2I
⇤
() z 2 T (BX ) and (X, Z ⇤ )
limhxi , zi = z ⇤ (By assumption).
i2I
Exercises
1. Show Theorem 2.3.3 using Lemma 2.3.4.
2. Prove Proposition 2.3.1 and Corollary 2.2.6.
3. Show that B`⇤1 is not sequentially compact in the w⇤ -topology.
Hint: Consider the unit vector basis of `1 seen as subsequence of B`⇤1 .
4. Prove that for a Banach space X every w⇤ -converging sequence in X ⇤
is bounded, but that if X is infinite dimensional, X ⇤ contains nets
(x⇤i : i 2 I) which converge to 0, but so that for every c > 0 and all
i 2 I there is a j0 i, with kxj k c, whenever j j0 .
5. Show that in each infinite dimensional Banach space X there is a
weakly null net in SX .
6.⇤ Prove that every weakly null sequence in `1 is norm null.
Hint: Assume that (xn ) ⇢ S`1 is weakly null. Then there is a subsequence xnk and a block sequence (zk ) so that limk!1 kxnk zk k = 0.