# Best approximation. Hahn Banach theorem.

Let $A$ be a normed space over R.

Let $B$ be a proper closed subspace of $A$.

If $a_0 \in A$ and $b_0 \in B$, $||a_0-b|| \geq ||a_0 - b_0||$ for all $b \in B$

if and only if there is a $f \in V^*$ with $||f|| = 1$ such that $f(b)=0$ for all $b \in B$ and $f(a_0) = ||a_0-b_0||$

I thought $f(a_0)=$inf $f${$||a_0-b||$: for all $b \in B$}

Then I realized that f is not a linear functional.

Is there other approach?

Thank you.

Edited : Now I'm thinking about proving just the existence of such linear functional without showing what such linear functional looks like by using Hahn Banach theorem somehow.

Hint: if $f(a_0) = \alpha = \|a_0 - b_0\|$, you want $f(t a_0 + b) = t \alpha$ for $t \in \mathbb R$, $b \in B$. Show this has norm $1$ on the span of $a_0$ and $B$. Then use Hahn-Banach ...