# Implication from Hahn-Banach theorem

I have to prove that if $X$ is a normed vector space over $K$, and $M$ its closed subspace, then there exists a linear and continuous functional $\phi$ from $X$ to $K$ such that $ker\phi = M$. I've got a hint that says 'use Hahn-Banach theorem'.

• That's hardly enough. The kernel of a linear funcitonal has codimension $1$. What you can find is a functional whose kernel contains $M$ (as long as the closure of $M$ is a proper subspace). – tomasz May 14 '17 at 18:24
• For the version with containment, Hint: Use Hahn-Banach theorem. – Berci May 14 '17 at 23:03