Let $W$ be a closed linear subspace of a normed vector space $V$. Let $i_V: V \to V^{**}$. and $i_W: W \to W^{**}$ be the canonical embeddings of V and W into their second duals. Prove that there exists an isometric embedding $\Phi: W^{**} \to V^{**}$. Show that $\Phi(W^{**}) = (W^{\perp})^{\perp}$. Can you help me to prove this?
$(W^{\perp})^{\perp}=\{\Gamma \in V^* | F(f) = 0 \quad \text{for all} \quad f \in V^* s.t. f(W)=0\}$ I think I have to use Hahn Banach theorem, but I don't know how.