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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 $V$).

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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 diagramenter image description here

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

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