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Let $V$ be a finite dimentional vector space over the field $F.$ If $L$ is a linear functional on the dual space $V^*$ of $V$, then there is a unique vector $\alpha$ in $V$ such that

$L(f)=f(\alpha)$ for every $f$ in $V^*.$

Can anyone tell me how for any linear functional on the dual space $V^*$ of $V$ there is a unique vector $\alpha$ in $V$ which satisfy the above condition?

Thanks!

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$V^{**}$ has the same dimension as $V^*$, and thus the same dimension as $V$. The linear map from $V$ to $V^{**}$ taking $\alpha$ to the linear functional "evaluation at $\alpha$" is injective, and therefore surjective.

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