Basic Analysis – Gently Done Topological Vector Spaces

9: The Dual Space of a Normed Space 107
form on X is norm continuous. Nonetheless, the weak and norm topologies on X
are not (always) equal, as we shall now show.
Proposition 9.16 Every weak neighbourhood of 0 in an infinite dimensional
normed space contains a one dimensional subspace.
Proof Let U be any weak neighbourhood of 0 in the infinite dimensional normed
space X. Then there is r > 0 and elements ℓ 1 ,...,ℓ n in X ∗ , the dual of X, such
that
V(0,|ℓ 1 |,...,|ℓ n |;r) ⊆ U.
Since X is infinite dimensional, so is X ∗ . (If X ∗ were finite dimensional, then
X ∗∗ would also be finite dimensional. However, this is impossible because we
have seen that X is linearly isometrically isomorphic to a subspace of X ∗∗ .) By
Proposition 5.16, there is z ∈ X such that z �= 0 and ℓ i (z) = 0 for all 1 ≤ i ≤ n.
Thus tz ∈ U for all t ∈ K.
Theorem 9.17 The weak topology on an infinite dimensional normed space X is
strictly weaker than the norm topology.
Proof Every element of X ∗ is continuous when X is equipped with the norm
topology. The weak topology is the weakest topology on X with this property, so
it is immediately clear that the weak topology is weaker than the norm topology.
Now suppose that X is infinite-dimensional. We shall exhibit a set which is
open with respect to the norm topologybut not with respect to the weak topology.
Consider the ‘open’ unit ball
G = {x ∈ X : �x� < 1}.
Then clearly G is open with respect to the norm topology on X. We claim that
G is not weakly open. If, on the contrary, G were weakly open, then, since 0 ∈ G,
there would be some z ∈ X with z �= 0 such that tz ∈ G for all t ∈ K, by
Proposition 9.16. Clearly this is not possible for values of t with |t| ≥ 1/�z�. We
conclude that G is not weakly open and the result follows.
Remark 9.18 We have shown that no weak neighbourhood of 0 can be norm
bounded. Since boundedness is preserved under translations, we conclude that no
non-empty weakly open set is norm bounded.
We now turn to a discussion of a topology on X ∗ , the dual of the normed space
X. The idea is to consider X as a family of maps : X ∗ → C given by x : ℓ ↦→ ℓ(x),
for x ∈ X and ℓ ∈ X ∗ —this is the map ϕ x defined earlier.