Initial Topology on Vector Space Generated by Linear Functionals is Locally Convex

Theorem
Let $\GF \in \set {\R, \C}$.

Let $X$ be a vector space over $\GF$.

Let $F$ be a set of linear functionals on $X$ that form a vector space over $\GF$.

That is, for each $\lambda, \mu \in \GF$ and $f, g \in F$, we have:


 * $\lambda f + \mu g \in F$

Let $\tau$ be the initial topology on $X$ generated by $F$.

For each $f \in F$, define $p_f : X \to \R_{\ge 0}$ by:


 * $\map {p_f} x = \cmod {\map f x}$

and let:


 * $\PP = \set {p_f : f \in F}$

Then the standard topology on the locally convex space $\struct {X, \PP}$ is precisely $\tau$.

Proof
Let $\tau'$ be the standard topology on the locally convex space.

From definition, a sub-basis for $\tau'$ is given by:


 * $\SS' = \set {\map {B_{p_f} } {\epsilon, x} : f \in F, \, \epsilon > 0, \, x \in X}$

where:


 * $\map {B_{p_f} } {\epsilon, x} = \set {y \in X : \map {p_f} {y - x} < \epsilon}$

We show that every $f \in F$ is $\tau'$-continuous.