Basic Analysis – Gently Done Topological Vector Spaces

110 Basic Analysis
According to the discussion above, we see that a net ℓα converges to ℓ in X∗ 1 with
respect to the w∗-topology if and only if γℓα converges to γℓ in Y. In other words,
the correspondence ℓ � γℓ is a homeomorphism between X∗ 1 and � Y when these
are equipped with the induced topologies.
Proposition 9.23 � Y is closed in Y.
Proof Let (γλ ) be a net in � Y such that γλ → γ in Y. Then px (γλ ) → px (γ) in
Bx , for each x ∈ X. Each γλ is of the form γℓλ for some ℓλ ∈ X∗ 1 . Hence
px (γℓλ ) = ℓλ (x) → γ(x)
for each x ∈ X ∗ 1 . It follows that for any a ∈ K and elements x 1 ,x 2
γ(ax 1 +x 2 ) = limℓ λ (ax 1 +x 2 )
= limaℓ λ (x 1 )+ℓ λ (x 2 )
= aγ(x 1 )+γ(x 2 ).
That is, the map x ↦→ γ(x) is linear on X. Furthermore, γ(x) = p x (γ) ∈ B x ,
i.e., |γ(x)| ≤ �x�, for x ∈ X. We conclude that the mapping x ↦→ γ(x) defines an
element of X ∗ 1 . In other words, if we set ℓ(x) = γ(x), x ∈ X, then ℓ ∈ X∗ 1 and
γ ℓ = γ. That is, γ ∈ � Y and so � Y is closed, as required.
The compactness result we seek is now readily established.
Theorem 9.24 (Banach-Alaoglu) Let X be a normed space and let X ∗ 1 denote
the unit ball in the dual X ∗ ,
X ∗ 1 = {ℓ ∈ X∗ : �ℓ� ≤ 1}.
Then X ∗ 1 is a compact subset of X∗ with respect to the w ∗ -topology.
∈ X
Proof Using the notation established above, we know that Y is compact and
since � Y is closed in Y, we conclude that � Y is also compact. Now, X∗ 1 and � Y are
homeomorphic when given the induced topologies so it follows that X∗ 1 is compact
in the induced w∗-topology. However, a subset of a topological space is compact if
and only if it is compact with respect to its induced topology and so we conclude
that X∗ 1 is w∗-compact. Department of Mathematics King’s College, London