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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.

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    There is a natural inclusion map $i:W\to V$ that is obviously distance-preserving. It induces a map $i^{\ast\ast}:W^{\ast\ast}\to V^{\ast\ast}$ that might be worth looking at.2012-04-04

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