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

Weak vector space topologies
Proof.
Conversely, let Λ ∈ X ∗ . Then Λ −1 ({z ∈ Φ : |z| < 1}) is X ′ open
and contains 0.
There are therefore 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 ∈ Vi for all i. It follows then that Λ(x) = 0 when
x ∈ ker l1 ∩ ker l2 ∩ · · · ∩ ker ln. Indeed, if li(x) = 0 for all i, we see
that
tx ∈ l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn) ⊆ U
for all t ∈ Φ, which implies that |t||Λ(x)| = |Λ(tx)| < 1 for all
t ∈ Φ. This is only possible when Λ(x) = 0.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems