460 CHAPTER 17. BANACH SPACES

17.5 Weak And Weak ∗ Topologies17.5.1 Basic Definitions

Let X be a Banach space and let X ′ be its dual space.1 For A′ a finite subset of X ′, denoteby ρA′ the function defined on X

ρA′ (x)≡ maxx∗∈A′|x∗ (x)| (17.5.13)

and also let BA′ (x,r) be defined by

BA′ (x,r)≡ {y ∈ X : ρA′ (y− x)< r} (17.5.14)

Then certain things are obvious. First of all, if a ∈ F and x,y ∈ X ,

ρA′ (x+ y) ≤ ρA′ (x)+ρA′ (y) ,

ρA′ (ax) = |a|ρA′ (x) .

Similarly, letting A be a finite subset of X , denote by ρA the function defined on X ′

ρA (x∗)≡max

x∈A|x∗ (x)| (17.5.15)

and let BA (x∗,r) be defined by

BA (x∗,r)≡{

y∗ ∈ X ′ : ρA (y∗− x∗)< r

}. (17.5.16)

It is also clear that

ρA (x∗+ y∗) ≤ ρ (x∗)+ρA (y

∗) ,

ρA (ax∗) = |a|ρA (x∗) .

Lemma 17.5.1 The sets, BA′ (x,r) where A′ is a finite subset of X ′ and x ∈ X form a basisfor a topology on X known as the weak topology. The sets BA (x∗,r) where A is a finitesubset of X and x∗ ∈ X ′ form a basis for a topology on X ′ known as the weak ∗ topology.

Proof: The two assertions are very similar. I will verify the one for the weak topology.The union of these sets, BA′ (x,r) for x∈ X and r > 0 is all of X . Now suppose z is containedin the intersection of two of these sets. Say

z ∈ BA′ (x,r)∩BA′1(x1,r1)

Then let C′ = A′∪A′1 and let

0 < δ ≤min(

r−ρA′ (z− x) ,r1−ρA′1(z− x1)

).

1Actually, all this works in much more general settings than this.