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Is it true that a vector space is just the set of maps from the underlying field to the space itself. I.e. if $V$ is a vector space of the field $k$ then $$ V\cong \operatorname{Hom}_k(k,V) $$ if so then this would make an intuitive understanding of the dual space $V^*$ somewhat trivial since $$ V^{**}\cong V\cong \operatorname{Hom}_k(k,V)\implies V^*=\operatorname{Hom}_k(V,k) $$ If true, an explanation of why $V\cong \operatorname{Hom}_k(k,V)$ with a simple example or two would provide a lot of clarity for me since I could easily grasp the dual vector space idea from that point.

Edit: I actually had to read two proposed answers a couple times for the idea to sink in but I could only pick one answer.

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  • $\begingroup$ That isomorphism is indeed valid. I am not at all clear about what your "intuitive understanding is", per the implication you've written. I suppose you're saying that $[\operatorname{Hom}_k(k,V)]^* \cong \operatorname{Hom}_k(V,k)$ is "obviously true" for some reason. Is that so? $\endgroup$ – Omnomnomnom Aug 4 '16 at 18:20
  • $\begingroup$ Yes, $[\operatorname{Hom}_k(k,V)]^* \cong \operatorname{Hom}_k(V,k)$ is how I arrived at the implication. $\endgroup$ – Bob Aug 4 '16 at 18:28
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    $\begingroup$ Is there an intuitive justification you have for the isomorphism $[\operatorname{Hom}_k(k,V)]^* \cong \operatorname{Hom}_k(V,k)$? Note that this isomorphism (generally) fails to hold if $V$ is infinite dimensional. $\endgroup$ – Omnomnomnom Aug 4 '16 at 18:28
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    $\begingroup$ What @Omnomnomnom says generally fails for infinite-dimensional spaces is $[\operatorname{Hom}_k(V,W)]^* \cong \operatorname{Hom}_k(W,V)$. $\endgroup$ – Henning Makholm Aug 4 '16 at 18:42
  • $\begingroup$ More generally if $R$ is any unital ring and $M$ is an $R$-module, there is a natural isomorphism between $\hom_R(R,-)$ and the identity functor. (Also, @Omnomnomnom's last comment is incorrect.) $\endgroup$ – arctic tern Aug 4 '16 at 19:37
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This works even for infinite-dimensional vector spaces (or for that matter for general modules over unital rings): The map $$ f \in \operatorname{Hom}_k(k,V) \mapsto f(1) \in V $$ is always vector space isomorphism. You don't need duals for that.

This is clearly injective and a homomorphism; to see that it is surjective, note that $v\in V$ corresponds to the map $t\in k\mapsto t\cdot v\in V$.

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Indeed, we have $\DeclareMathOperator{\Hom}{Hom} V \cong \Hom_k(k,V)$. To see that this is the case, note that we have the (natural) isomorphism $\Phi: V \to \Hom_k(k,V)$ given by $$ [\Phi(v)](t) = t \cdot v $$ for $v \in V$ and $t \in k$.

In terms of matrices, this amounts to saying that a column-vector $v = (v_1,\dots,v_n) \in k^n$ can be naturally identified with the $n \times 1$ matrix $$ \pmatrix{v_1\\ \vdots \\ v_n} $$ which denotes a linear map from $k^1$ to $k^n$.

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  • $\begingroup$ I'd appreciate feedback if I've said something inaccurate. $\endgroup$ – Omnomnomnom Aug 4 '16 at 22:26
  • $\begingroup$ Makes sense to me as the OP. I marked it up in fact. Just wish I could mark both your answer and the other answer as correct. $\endgroup$ – Bob Aug 5 '16 at 1:30

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