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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'.

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    $\begingroup$ 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). $\endgroup$ – tomasz May 14 '17 at 18:24
  • $\begingroup$ For the version with containment, Hint: Use Hahn-Banach theorem. $\endgroup$ – Berci May 14 '17 at 23:03

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