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

Weak vector space topologies
Proof.
X is locally convex: Consider an X ′ -open set U such that 0 ∈ U.
There are then pairs (li,Vi),i = 1,2,... ,n, such that
0 ∈ l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn) ⊆ U,
where li ∈ X ′ and the Vi’s are open in Φ.
Note that 0 = li(0) ∈ Vi for all i. Since Φ is a locally convex
topological space, there are open and convex neighborhoods Wi of
0 in Φ such that Wi ⊆ Vi.
Since the li’s are linear, l −1
1 (W1) ∩ l −2
2 (W2) ∩ · · · ∩ l −1
n (Wn) is
convex.
Since l −1
1 (W1) ∩ l −2
2 (W2) ∩ · · · ∩ l −1
n (Wn) is X ′ -open and
0 ∈ l −1
1 (W1) ∩ l −2
2 (W2) ∩ · · · ∩ l −1
n (Wn) ⊆ U,
we have shown that X is locally convex in the X ′ -topology.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems