Basic Analysis – Gently Done Topological Vector Spaces

9: The Dual Space of a Normed Space 109
Proposition 9.22 The unit ball {ℓ : �ℓ� ≤ 1} in X ∗ is closed in the w ∗ -topology.
Proof Let ℓ ν → ℓ in the w ∗ -topology. Then ℓ ν (x) → ℓ(x), for each x ∈ X. But if
�ℓ ν � ≤ 1 for all ν, it follows that |ℓ ν (x)| ≤ �x� and so |ℓ(x)| ≤ �x� for all x ∈ X.
That is, �ℓ� ≤ 1 and we conclude that the unit ball is closed.
We will use Tychonov’s theorem to show that the unit ball in the dual space of
a normed space is actually compact in the w ∗ -topology. To do this, it is necessary
to consider the unit ball of the dual space in terms of a suitable cartesian product.
By way of a preamble, let us consider the dual space X ∗ of the normed space X
in such terms. Each element ℓ in X ∗ is a (linear) function on X. The collection
of values ℓ(x), as x runs over X, can be thought of as an element of a cartesian
product with components given by the ℓ(x). Specifically, for each x ∈ X, let Y x
be a copy of K, equipped with its usual topology. Let Y = �
x∈X Y x
= �
x∈X K,
equipped with the product topology. To each element ℓ ∈ X ∗ , we associate the
element γℓ ∈ Y given by γℓ (x) = ℓ(x), i.e., the x-coordinate of γℓ is ℓ(x) ∈ K = Yx .
If ℓ1 ,ℓ2 ∈ X∗ , and if γℓ1 = γℓ2 , then γℓ1 and γ have the same coordinates so
ℓ2
that ℓ1 (x) = γℓ1 (x) = γℓ2 (x) = ℓ2 (x) for all x ∈ X. In other words, ℓ1 = ℓ2 , and
so the correspondence ℓ ↦→ γℓ of X∗ → Y is one–one. Thus X∗ can be thought of
as a subset of Y = �
x∈X K.
Suppose now that {ℓα } is a net in X∗ such that ℓα → ℓ in X∗ with respect to
the w∗-topology. This is equivalent to the statement that ℓα (x) → ℓ(x) for each
x ∈ X. But then ℓα (x) = px (γℓα ) → ℓ(x) = px (γℓ ) for all x ∈ X, which, in turn,
is equivalent to the statement that γℓα → γℓ with respect to the product topology
on �
x∈X K.
We see, then, that the correspondence ℓ � γℓ respects the convergence of nets
when X∗ is equipped with the w∗-topology and Y with the product topology. It
will not come as a surprise that this also respects compactness.
Consider now X∗ 1 , the unit ball in the dual of the normed space X∗ . For any
x ∈ X and ℓ ∈ X∗ 1 , we have that |ℓ(x)| ≤ �x�. Let Bx denote the ball in K given
by
B x = {t ∈ K : |t| ≤ �x�}.
Thentheaboveremarkisjust theobservationthatℓ(x) ∈ Bx foreveryℓ ∈ X∗ 1 . We
equip B x with its usual metric topology, so that it is compact. Let Y = �
x∈X B x
equipped with the product topology. Then, by Tychonov’s theorem, Theorem 3.8,
Y is compact.
Let ℓ ∈ X ∗ 1 . Then, as above, ℓ determines an element γ ℓ
γ ℓ (x) = p x (γ ℓ ) = ℓ(x) ∈ B x .
The mapping ℓ ↦→ γ ℓ is one–one. Let � Y denote the image of X ∗ 1
�Y = {γ ∈ Y : γ = γ ℓ , some ℓ ∈ X ∗ 1 }.
of Y by setting
under this map,