Vector spaces with complex field as scalar.

Question

Sorry for stating the question informally. If we have a vector space whose scalars are the field $\mathbb{R}$, if we change the field to be $\mathbb{C}$ and "adapt" the addition and scalar multiplication operations, does it necessarily yield a new vector space?

Are there theorems developed in this manner or similar tests?

Answer 1

You cannot do it in general because a finite-dimensional vector space over $\mathbb C$ has even dimension over $\mathbb R$. Of course, not every vector space over $\mathbb R$ has even dimension.

On the other hand, you can always complexify a real vector space $V$:

Take $V \times V$ and let $a+bi$ act on $(v,w)$ as if $(v,w)=v+wi$, following the natural rule.

This construction works even if the dimension of $V$ is infinite.

Answer 2

There is a general construction, informally stated: convert the scalars from $\mathbb R$ to $\mathbb C$.
In fancy language: if $V$ is a real vector space, then $$ V_1 = V \otimes_{\mathbb R} \mathbb C $$ is the corresponding complex vector space.