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

Weak vector space topologies
Proof.
(c) ⇒ (a): Define π : X → Φ n such that
π(x) = (l1(x),l2(x),... ,ln(x)).
It follows from (c) that we can define l0 : π(X) → Φ such that
l0 (π(x)) = l(x).
Let P : Φ n → π(X) be a linear projection such that P (Φ n ) = π(X).
Then F = l0 ◦ P : Φ n → Φ is a linear map such that
F ◦ π(x) = l0 ◦ P ◦ π(x) = l0 (π(x)) = l(x).
Recall from linear algebra that the linear map F : Φ n → Φ is of the
form
F(x1,x2,... ,xn) = α1x1 + α2x2 + · · · + αnxn.
It follows that
l(x) = F ◦ π(x) = F (l1(x),l2(x),... ,ln(x))
= α1l1(x) + α2l2(x) + · · · + αnln(x), x ∈ X.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems