Hahn-Banach Theorem for Continuous Linear Functional on Locally Convex Space

Theorem

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

Let $\struct {X, \PP}$ be a locally convex space over $\GF$ with its standard topology.

Let $X_0$ be a linear subspace of $X$.

Let $f_0 : X_0 \to \GF$ be a continuous linear functional.

Then there exists a continuous linear functional $f : X \to \GF$ such that $f$ extends $f_0$.

Proof

By Normed Vector Space is Locally Convex Space and Norm on Vector Space is Seminorm, we can view the normed vector space $\struct {\GF, \cmod {\, \cdot \,} }$ as the locally convex space $\struct {\GF, \set {\cmod {\, \cdot \,} } }$.

From Characterization of Continuous Linear Transformations between Locally Convex Spaces, there exists $C > 0$, $n \in \N$ and $p_1, \ldots, p_n \in \PP$ such that:

$\ds \cmod {\map {f_0} x} \le C \max_{1 \le i \le n} \map {p_i} x$

for each $x \in X_0$.

Define $p : X \to \R_{\ge 0}$ by:

$\ds \map p x = C \max_{1 \le i \le n} \map {p_i} x$

for each $x \in X$.

From Pointwise Maximum of Finite Family of Seminorms is Seminorm and Non-Negative Scalar Multiple of Seminorm on Vector Space is Seminorm, we have that $p$ is a seminorm.

So we have:

$\cmod {\map {f_0} x} \le \map p x$

for each $x \in X_0$.

From:

the Hahn-Banach Theorem: Real Vector Space: Corollary 1 if $\GF = \R$

the Hahn-Banach Theorem: Complex Vector Space: Corollary if $\GF = \C$

there exists a linear functional $f$ extending $f_0$ such that:

$\ds \cmod {\map f x} \le \map p x = C \max_{1 \le i \le n} \map {p_i} x$

for each $x \in X_0$.

From Characterization of Continuous Linear Transformations between Locally Convex Spaces, $f$ is continuous.

So $f$ is a continuous linear functional satisfying the demand.

$\blacksquare$