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 $Y$ be a proper closed linear subspace of $X$.

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:


 * $\map f {x_0} \ne 0$

Proof
Let:


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

From Linear Span is Linear Subspace, we have:


 * $X_0$ 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.

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 $\map {f_0} x = 0$ $x \in Y$, so that:


 * $\ker f_0 = Y$

This is closed, so by Characterization of Continuous Linear Functionals on Topological Vector Space, we have:


 * $f_0$ is continuous.

From Hahn-Banach Theorem for Continuous Linear Functional on Locally Convex Space, there exists a continuous linear functional $f$ extending $f_0$.

Then we have:


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

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