Existence of Non-Zero Continuous Linear Functional vanishing on Proper Closed Subspace of Hausdorff Locally Convex Space

Theorem

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

Let $\struct {X, \PP}$ be a Hausdorff locally convex space over $\GF$.

Let $x_0 \in X \setminus Y$.

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

$\map f y = 0$ for each $y \in Y$

and:

$\map f {x_0} \ne 0$

Let:

$X_0 = \map \span {Y \cup \set x}$

$X_0$ is a linear subspace of $X$.

Note that we can then write any $u \in X_0$ in the form:

$u = y + \alpha x$

for $y \in Y$ and $\alpha \in \mathbb F$.

We want to define a map in terms of this representation, so we show that this representation is unique.

Let:

$u = y_1 + \alpha_1 x = y_2 + \alpha_2 x$

Then:

$\paren {\alpha_2 - \alpha_1} x = y_1 - y_2$

If $\alpha_1 = \alpha_2$, then we have $y_1 = y_2$ as required.

Aiming for a contradiction, suppose suppose that $\alpha_1 \ne \alpha_2$, then we would have:

$\ds x = \frac 1 {\alpha_2 - \alpha_1} \paren {y_1 - y_2}$

and so $x \in Y$, from the definition of a linear subspace, contradiction.

So, we must have $\alpha_1 = \alpha_2$ and $y_1 = y_2$, and so the representation is unique.

Now, define $f_0 : X_0 \to \GF$ by:

$\map {f_0} {y + \lambda x_0} = \lambda$

for each $y \in Y$, $\lambda \in \GF$.

We have $\map {f_0} x = 0$ if and only if $x \in Y$, so that:

$\ker f_0 = Y$

Then we have:

$\map f {x_0} = \map {f_0} {x_0} = 1 \ne 0$

and:

$\map f y = \map {f_0} y = 0$

for each $y \in Y$.

So $f$ is a continuous linear functional satisfying our requirements.

$\blacksquare$