17.5. WEAK AND WEAK ∗ TOPOLOGIES 461

Consider y ∈ BC′ (z,δ ) . Then

r−ρA′ (z− x)≥ δ > ρC′ (y− z)≥ ρA′ (y− z)

and sor > ρA′ (y− z)+ρA′ (z− x)≥ ρA′ (y− x)

which shows y ∈ BA′ (x,r) . Similar reasoning shows y ∈ BA′1(x1,r1) and so

BC′ (z,δ )⊆ BA′ (x,r)∩BA′1(x1,r1) .

Therefore, the weak topology consists of the union of all sets of the form BA (x,r).

17.5.2 Banach Alaoglu TheoremWhy does anyone care about these topologies? The short answer is that in the weak ∗topology, closed unit ball in X ′ is compact. This is not true in the normal topology. Thiswonderful result is the Banach Alaoglu theorem. First recall the notion of the producttopology, and the Tychonoff theorem, Theorem 14.3.6 on Page 391 which are stated herefor convenience.

Definition 17.5.2 Let I be a set and suppose for each i ∈ I, (Xi,τ i) is a nonempty topolog-ical space. The Cartesian product of the Xi, denoted by ∏i∈I Xi, consists of the set of allchoice functions defined on I which select a single element of each Xi. Thus f ∈ ∏i∈I Ximeans for every i ∈ I, f (i) ∈ Xi. The axiom of choice says ∏i∈I Xi is nonempty. Let

Pj (A) = ∏i∈I

Bi

where Bi = Xi if i ̸= j and B j = A. A subbasis for a topology on the product space consistsof all sets Pj (A) where A ∈ τ j. (These sets have an open set from the topology of X j in thejth slot and the whole space in the other slots.) Thus a basis consists of finite intersectionsof these sets. Note that the intersection of two of these basic sets is another basic set andtheir union yields ∏i∈I Xi. Therefore, they satisfy the condition needed for a collection ofsets to serve as a basis for a topology. This topology is called the product topology and isdenoted by ∏τ i.

Theorem 17.5.3 If (Xi,τ i) is compact, then so is (∏i∈I Xi,∏τ i).

The Banach Alaoglu theorem is as follows.

Theorem 17.5.4 Let B′ be the closed unit ball in X ′. Then B′ is compact in the weak ∗topology.

Proof: By the Tychonoff theorem, Theorem 17.5.3

P≡∏x∈X

B(0, ||x||)