Download MAT 578 Functional Analysis

MAT 578 Functional Analysis
John Quigg
Fall 2008
revised September 30, 2008
Weak topologies
This section develops the all-important weak and weak* topologies associated with Banach
spaces.
Reminder: all vector spaces will have scalar field F = R or C.
To begin, note that if f is a linear functional on a vector space X, then |f | is a seminorm on
X. From our earlier work with seminorms we know that if X is a topological vector space
then f is continuous if and only if |f | is.
Observation 1. A set S of linear functionals on X is separating if and only if the set
{|f | : f ∈ S} of seminorms is.
Definition 2. Let S be a separating set of linear functionals on X. The weak topology on
X generated by S is the locally convex topology generated by the family {|f | : f ∈ S} of
seminorms.
By our earlier work, the weak topology generated by a separating set S of linear functionals
coincides with the weakest topology making every f ∈ S continuous. We say U ⊂ X is
weakly open if it is open in the weak topology, the weak closure of A ⊂ X is the closure of
A in weak topology, a net {xi } in X converges weakly if it converges in the weak topology,
and similarly for other topological concepts.
Lemma 3. Let S be a separating set of linear functionals on X, and give X the associated
weak topology T . Then X ∗ = span S, the linear span of S, and T coincides with the weak
topology generated by span S.
Proof. We know that every functional in S is T -continuous, hence so is every linear combination of elements of S. Thus span S ⊂ X ∗ . On the other hand, if g ∈ X ∗ , then |g| is a
continuous seminorm on X, so by our earlier work on seminorms there exist f1 , . . . , fn ∈ S
and a > 0 such that
|g| ≤ a max |fi |.
i
Tn
Thus ker g ⊃ 1 ker fi . It is a general fact from linear algebra that this implies g ∈
span{f1 , . . . , fn }. Thus X ∗ ⊂ span S.
For the second statement, let T 0 denote the weak topology generated by span S. Since
S ⊂ span S, every functional in S is T 0 -continuous, so T is weaker than T 0 . But, as we
observed above, every functional in span S is T -continuous, so T is also stronger than T 0 ,
and we are done.
2
Suppose X is a vector space and Y is a separating vector space of linear functionals on X,
and give X the associated weak topology. As we have seen above, Y consists precisely of the
weakly continuous linear functionals on X. For each x ∈ X define a linear functional E(x)
on Y by
E(x)(f ) = f (x).
Then E(X) is a separating set of linear functionals on Y , so generates a weak topology on
Y , which is the weakest topology making each E(x) continuous. Note that E is an injective
(because Y is separating) linear map from X into the vector space of all linear functionals
(continuous or not) on Y .
Definition 4. With the above notation, we refer to the weak topology on Y generated by
E(X) as the weak topology on Y generated by X.
Corollary 5. Let X be a vector space with a separating vector space Y of linear functionals.
Give X the weak topology generated by Y , and Y the weak topology generated by X. Then
the linear map E of the above discussion is an isomorphism of X onto Y ∗ .
Definition 6. Let X be a locally convex space. The weak topology on X is the weak
topology generated by X ∗ .
When X is a locally convex space, it is useful to explicitly observe that, by Lemma 3, a
linear functional on X is continuous if and only if it is weakly continuous.
Definition 7. Let X be a Banach space. The weak* topology on X ∗ is the weak topology
generated by X (more precisely, by the canonical image of X in X ∗∗ .
In the above definition we restricted X to be a Banach space for the following reason: if X is
an incomplete normed space then every functional in X ∗ extends uniquely to the completion
X̄, so X ∗ = (X̄)∗ . However, by Corollary 5 the weak topologies on X ∗ generated by X and
X ∗ are different. Since we routinely replace a given normed space by its completion, without
the restriction of the above definition the weak* topology would be ambiguous.
The dual X ∗ of a Banach space X now carries the norm, weak, and weak* topologies.
In general these are all distinct. More precisely, if X is finite-dimensional then all three
topologies coincide1, while if X is infinite-dimensional then the norm topology is strictly
stronger than the weak and the weak*-topologies. The weak topology on X ∗ is strictly
stronger than the weak* topology unless X is reflexive (recall that this means the canonical
embedding X ,→ X ∗∗ is surjective.
The following is the single most important fact about weak* topologies:
Theorem 8 (Alaoglu). Let X be a Banach space. Then the closed unit ball of X ∗ is weak*
compact.
Proof. Let A = {f ∈ X ∗ : kf k ≤ 1}, with the weak* topology. Recall that for a > 0 we
defined B̄a = {t ∈ F : |t| ≤ a}. We also let B̄0 = {0} ⊂ F. Put
Y
P =
B̄kxk ,
x∈X
1Generalizing
the corresponding fact for norms, a finite-dimensional vector space has exactly one topology
making it a topological vector space.
3
which by Tychonoff’s Theorem is a compact topological space with the product topology.
We have A ⊂ P , and the weak* topology on A coincides with the relative product topology.
It suffices to show that A is closed in P . Let fi → f in P , with fi ∈ A for all i. Since A
consists precisely of the linear elements of P , it suffices to show that f is linear. For x, y ∈ X,
since P has the topology of pointwise convergence on X we have
f (x + y) = lim fi (x + y)
i
= lim fi (x) + fi (y)
i
= f (x) + f (y).
A similar computation shows that f (cx) = cf (x) for c ∈ F, and we are done.
Theorem 9. Let X be a locally convex space and A ⊂ X be convex . Then A coincides with
the weak closure of A.
w
Proof. Let T denote the given topology on X, and let A denote the weak closure of A.
w
w
Since the weak topology on X is weaker than T , A is T -closed, so A ⊂ A . On the other
hand, if x ∈ X \ A, then by the Hahn-Banach Separation Theorem there exists f ∈ X ∗ such
w
w
that Ref (x) < inf Ref (A). Thus x ∈
/ A . Therefore A ⊃ A , and we are done.