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.


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