$\operatorname{Hom}_k(k,V)$ is a vector space? 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.

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 $t\in k\mapsto t\cdot v\in V$.
