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

Weak vector space topologies
Proof.
Conversely, assume that M separates the points of X. Let
x0,y0 ∈ X, x0 = y0.
There is then a functional l ∈ M such that l(x0) = l(y0).
Set ǫ = 1/2 |l(x0) − l(y0)| > 0, and note that
Since
x0 ∈ {y ∈ X : |l(y) − l(x0)| < ǫ} ,
y0 ∈ {y ∈ X : |l(y) − l(y0)| < ǫ} .
{y ∈ X : |l(y) − l(y0)| < ǫ} ∩ {y ∈ X : |l(y) − l(x0)| < ǫ} = ∅,
the proof is complete.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems