Anihilators question

Question

Let $W \subset V$ is a subspace. Prove that anihilator $W^{\perp } = \{f \in V^{*}:\forall _{w\in W} f(w)=0\}$ space $W$ in $V$ is a subspace. Prove, that $W^{*} \cong V^{*}/W^{\perp} $, and if $\dim V < \infty$, then $(W^{\perp})^{\perp} = W$ (considering $(V^{*})^{*}$ with $V$).

Answer 1

To see that $W^* \simeq V^*/W^\perp$, read it off the sequence $$0 \rightarrow W^\perp \rightarrow V^* \rightarrow^{res} W^* \rightarrow 0$$

If we dualize the above sequence, we obtain the commutative diagram

In the case that $\dim V < \infty$, the vertical arrows are isomorphisms, whence the result.