Next consider the weak topology. The most interesting results have to do with a reflexive
Banach space. The following lemma ties together the weak and weak ∗ topologies in the case
of a reflexive Banach space.
Lemma 23.5.7Let J : X → X^{′′}be the James map
Jx (f) ≡ f (x)
and let X be reflexive so that J is onto. Then J is a homeomorphism of
(X, weak topology)
and (X^{′′}, weak ∗ topology).This means J is one to one, onto, and both J and J^{−1}arecontinuous.
Proof:Let f ∈ X^{′} and let
Bf (x,r) ≡ {y : |f (x)− f (y)| < r}.
Thus B_{f}
(x,r)
is a subbasic set for the weak topology on X. I claim that
JBf (x,r) = Bf (Jx,r)
where B_{f}
(Jx, r)
is a subbasic set for the weak ∗ topology. If y ∈ B_{f}
(x,r)
, then
∥Jy− Jx∥
=
∥x − y∥
< r and so JB_{f}
(x,r)
⊆ B_{f}
(Jx,r)
. Now if x^{∗∗}∈ B_{f}
(Jx,r)
, then since
J is reflexive, there exists y ∈ X such that Jy = x^{∗∗} and so
∥y − x∥ = ∥Jy − Jx∥ < r
showing that JB_{f}
(x,r)
= B_{f}
(Jx,r)
. A typical subbasic set in the weak ∗ topology is of
the form B_{f}
(Jx,r)
. Thus J maps the subbasic sets of the weak topology to the
subbasic sets of the weak ∗ topology. Therefore, J is a homeomorphism as claimed.
■
The following is an easy corollary.
Corollary 23.5.8If X is a reflexive Banach space, then the closed unit ball isweakly compact.
Proof:Let B be the closed unit ball. Then B = J^{−1}
(B∗∗)
where B^{∗∗ }is the unit ball in
X^{′′} which is compact in the weak ∗ topology. Therefore B is weakly compact because J^{−1} is
continuous. ■
Corollary 23.5.9Let X be a reflexive Banach space. If K ⊆ X is compact in theweak topology and X^{′}is separable in the weak ∗ topology, then there exists a metric d,on K such that if τ_{d }is the topology on K induced by d and if τ is the topology on Kinduced by the weak topology of X, then τ = τ_{d}. Thus one can consider K with the weaktopology as a metric space.
Proof: This follows from Theorem 23.5.5 and Lemma 23.5.7. Lemma 23.5.7 implies J
(K)
is compact in X^{′′}. Then since X^{′} is separable in the weak ∗ topology, X is separable in the
weak topology and so there is a metric, d^{′′} on J
(K )
which delivers the weak ∗ topology on
J
(K)
. Let d
(x,y)
≡ d^{′′}
(Jx,Jy)
. Then
(K, τ) J→ (J (K ),τ′′) i→d(J (K ),τ ) J→−1(K,τ )
d d weak ∗ weak
and all the maps are homeomorphisms. ■
Here is a useful lemma.
Lemma 23.5.10Let Y be a closed subspace of a Banach space X and let y ∈ X∖Y.Then there exists x^{∗}∈ X^{′}such that x^{∗}
(Y)
= 0 but x^{∗}
(y)
≠0.
Proof: Define f
(x+ αy )
≡
∥y∥
α. Thus f is linear on Y ⊕ Fy. I claim that f is
also continuous on this subspace of X. If not, then there exists x_{n} + α_{n}y → 0 but
|f (xn + αny)|
≥ ε > 0 for all n. First suppose
|αn|
is bounded. Then, taking a
further subsequence, we can assume α_{n}→ α. It follows then that
{xn}
must also
converge to some x ∈ Y since Y is closed. Therefore, in this case, x + αy = 0 and
so α = 0 since otherwise, y ∈ Y . In the other case when α_{n} is unbounded, you
have
(xn∕αn + y)
→ 0 and so it would require that y ∈Y which cannot happen
because Y is closed. Hence f is continuous as claimed. It follows that for some
k,
|f (x+ αy )| ≤ k ∥x+ αy∥
Now apply the Hahn Banach theorem to extend f to x^{∗}∈ X^{′}. ■
Next is the Eberlein Smulian theorem which states that a Banach space is reflexive if and
only if the closed unit ball is weakly sequentially compact. Actually, only half the
theorem is proved here, the more useful only if part. The book by Yoshida [32]
has the complete theorem discussed. First here is an interesting lemma for its own
sake.
Lemma 23.5.11A closed subspace of a reflexive Banach space is reflexive.
Proof: Let Y be the closed subspace of the reflexive space, X. Consider the following
diagram
∗∗
Y ′′ i→1- 1 X ′′
′ i∗ onto ′
Y ←i X
Y → X
This diagram follows from Theorem 23.2.10 on Page 1625, the theorem on adjoints. Now let
y^{∗∗}∈ Y^{′′}. Then i^{∗∗}y^{∗∗} = J_{X}
(y)
because X is reflexive. I want to show that y ∈ Y . If it is not
in Y then since Y is closed, there exists x^{∗}∈ X^{′} such that x^{∗}
Theclosed unit ball in a reflexive Banachspace X, is weakly sequentially compact. By this is meant that if
{x }
n
is contained inthe closed unit ball, there exists asubsequence,
{x }
nk
and x ∈ X such that for allx^{∗}∈ X^{′},
x∗ (xnk) → x∗(x).
Proof: Let {x_{n}}⊆ B ≡B
(0,1)
. Let Y be the closure of the linear span of {x_{n}}. Thus Y
is a separable. It is reflexive because it is a closed subspace of a reflexive space so the above
lemma applies. By the Banach Alaoglu theorem, the closed unit ball B^{∗} in Y^{′} is
weak ∗ compact. Also by Theorem 23.5.5, B^{∗} is a metric space with a suitable
metric.
B∗∗ Y′′ i∗∗→1- 1 X′′
∗ ′ i∗ onto ′
weakly separable B Y ←i X
separable B Y → X
Thus B^{∗ }is complete and totally bounded with respect to this metric and it follows that
B^{∗ } with the weak ∗ topology is separable. This implies Y^{′} is also separable in the weak ∗
topology. To see this, let
{y∗n}
≡ D be a weak ∗ dense set in B^{∗} and let y^{∗}∈ Y^{′}. Let p be a
large enough positive rational number that y^{∗}∕p ∈ B^{∗}. Then if A is any finite set from Y,
there exists y_{n}^{∗}∈ D such that ρ_{A}
(y∗∕p − y∗n)
<
εp
. It follows py_{n}^{∗}∈ B_{
A}
(y∗,ε)
showing that
rational multiples of D are weak ∗ dense in Y^{′}. Since Y is reflexive, the weak and weak ∗
topologies on Y^{′} coincide and so Y^{′} is weakly separable. Since Y^{′} is weakly separable,
Corollary 23.5.6 implies B^{∗∗}, the closed unit ball in Y^{′′} is weak ∗ sequentially compact. Then
by Lemma 23.5.7B, the unit ball in Y , is weakly sequentially compact. It follows there
exists a subsequence x_{nk}, of the sequence
{xn}
and a point x ∈ Y , such that for all
f ∈ Y^{′},
f (xnk) → f (x).
Now if x^{∗}∈ X^{′}, and i is the inclusion map of Y into X,
∗ ∗ ∗ ∗ ∗ ∗
x (xnk) = ix (xnk) → ix (x) = x (x).
which shows x_{nk} converges weakly and this shows the unit ball in X is weakly sequentially
compact. ■
Corollary 23.5.13Let
{xn}
be any bounded sequence in a reflexive Banach space X.Then there exists x ∈ X and a subsequence,
{xnk}
such that for all x^{∗}∈ X^{′},
lk→im∞ x∗(xnk) = x∗(x)
Proof: If a subsequence, x_{nk} has
||xnk||
→ 0, then the conclusion follows. Simply let
x = 0. Suppose then that
||xn||
is bounded away from 0. That is,
||xn||
∈
[δ,C]
. Take a
subsequence such that
||xnk||
→ a. Then consider x_{nk}∕
||xnk||
. By the Eberlein Smulian
theorem, this subsequence has a further subsequence, x_{nk
j}∕
∥∥x ∥∥
∥ nkj∥
which converges weakly to
x ∈ B where B is the closed unit ball. It follows from routine considerations that x_{nk
j}→ ax
weakly. ■