# Application of Hahn Banach theorem to best approximation

This is a two part question and I'm really stuck. I know I have to apply to the Hahn-Banach Theorem/one of its corollaries but I'm really uncertain as to how $$\\$$

1) Let V be a normed vector space over R, and let W be a proper closed subspace of V . A basic tool in linear approximation theory provides a way of proving that one has a best approximation. It goes as follows: If $$v_0 ∈ V$$ and if $$w_0 ∈W$$,then $$w_0$$ is a best approximation in W to $$v_0$$ if and only if there is a $$φ∈V∗$$ with$$∥φ∥=1$$ such that $$φ(w)=0$$ for all $$w∈W$$and $$φ(v_0) = ∥v_0 − w_0∥$$. Prove this assertion.

2)With notation as above, prove that for a given $$v_0 ∈ V$$ its (possibly empty) set of best approximations in $$W$$ is a norm-closed convex set.