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

Weak vector space topologies
Theorem
Theorem 3.10. Let X be a vector space and X ′ a separating vector
space of linear functional on X. Then the X ′ -topology makes X a
locally convex topological vector space whose dual is X ′ .
The proof is based on the following lemma of independent interest:
Lemma
Let l1,l2,... ,ln and l be linear functional on the vector space X.
The following are equivalent:
(a) There are scalars α1,α2,... ,αn such that
l = α1l1 + α2l2 + · · · + αnln.
(b) There is a γ > 0 such that |l(x)| ≤ γ max1≤i≤n |li(x)| for all
x ∈ X.
(c) For all x ∈ X, l1(x) = l2(x) = · · · = ln(x) = 0 ⇒ l(x) = 0.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems