7. Consequences of the Hahn-Banach theorems - Aarhus Universitet

Weak vector space topologies
Proof.
Let (x ′ ,y ′ ) be an element of R. It suffices to show that there are
X ′ -open sets S,T such that (x ′ ,y ′ ) ∈ S × T ⊆ R. To this end
note that there are pairs (li,Vi),i = 1,2,... ,n, such that li ∈ X ′ ,
Vi ⊆ Φ is open and
x ′ + y ′ ∈ l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn) ⊆ U.
In particular, li(x ′ ) + li(y ′ ) ∈ Vi,i = 1,2,... ,n. Since the vector
space operations are continuous on Φ, there are open sets Si,Ti in
Φ such that li(x ′ ) ∈ Si,li(y ′ ) ∈ Ti and Si + Ti ⊆ Vi,i = 1,2,... ,n.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems