complex valued real-linear functionals

Question

Let $V$ be a vector space over $\mathbb{R}$ and $V^{*}$ denote its dual space. Why is $V^* \otimes \mathbb{C}$ the space of complex-valued real-linear functionals on $V$?

Answer 1

The space $V^\ast \otimes \mathbb{C}$ is not equal to the space of complex valued real linear functionals, however in finite dimensions they are canonically isomorphic. One way to think of the space of complex valued real linear functionals on $V$ is to regard it as $Hom(V,\mathbb{C})$. Here I mean to regard $\mathbb{C}$ as a 2-dimensional real vector space and think of homomorphisms of real vector spaces.

In the setting where the vector spaces are finite dimensional we have a canonical isomorphism $Hom(V,W)\cong V^\ast\otimes W$ and the answer to your question is just a specific instance of this isomorphism. In case you are unfamiliar, the isomorphism from $V^\ast\otimes W$ is given by $(\phi\otimes w)(v)=\phi(v)w$.

In the infinite dimensional setting the answer appears to be more subtle, as is explained here.