A closed space is the kernel of some map I'm working on the following problems in Royden:

Let $X_0$ be a codimension 1 subspace of a normed linear space $X$. Show that $X_0$ is closed with respect to the strong topology if and only if the $X_0$ = $\ker \phi $ for some $\phi \in X^*$

The reverse direction is clear -- continuous functions pull back closed sets. The forward direction I'm a bit unsure of -- I set up a function: $$\phi: X_0 \mapsto 0 \\ a + X_0 \mapsto a$$
I show then that its well defined, linear, and continuous -- and so sits in $X^*$. Is there an easier way see this without explicitly exhibiting a map? I haven't really used the fact that the maps was co-dimension 1. My construction works for any codimension of $X_0$, which is leading me to doubt my method.

Suppose $X_0$ is closed.  Since $X_0$ is a proper subspace, there is some nonzero linear functional $\phi\in X^*$ such that $X_0\subset\ker\phi$ (by the Hahn-Banach theorem).  Suppose the inclusion is strict.  Then there is some $x\in \ker\phi\setminus X_0$.  But $X=X_0\oplus\text{span}\{x\}$, so $X=\ker\phi$, a contradiction, and therefore $X_0=\ker\phi$.
