Let $(K,V)$ be a vector space, is there a special name for a vector space $(F,V)$ where $F$ is a sub-field of $K$? Given a vector space $(K,V)$ where $K$ is the scalar field and $V$ is the abelian group, then another vector space $(K,W)$ is a subspace of $(K,V)$ if $W\subseteq V$. My question is, is there a special name for a vector space $(F,V)$ where $F$ a sub-field of $K$?
 A: It is common to prefix the name of the vector space with the field it's over, instead of using pairs. That could give you want you want here. So for example:


*

*Let $V$ be a $K$-vector space. Consider $V$ as an $F$-vector space, for any $F \subset K$. Then...

*Every $K$-vector space is an $F$-vector space, therefore...
This seems like it could be sufficient for your purposes.
Anyway, I don't know of any special term other than this and I'm not sure one is needed.
A: Operations $(F, V)\mapsto (K, V)$ and $(K,V)\mapsto (F,V)$ where $F$ is a subfield of $K$ are called respectively the extension and restriction of scalars. In case of $\Bbb R$ and $\Bbb C$ they are called respectively complexification and realification. It's OK to say "$(\Bbb R,V)$ is the realification of $(\Bbb C,V)$".
Restriction and extension of scalars can be naturally extended to homomorphisms and actually are functors $\mathsf{Vec}_K\to\mathsf {Vec}_F$ and $\mathsf{Vec}_F\to\mathsf {Vec}_K$. It also can be defined for more general modules. Restriction is just a "forgetting" the multiplication by some elements of the base ring. Extension is "forcing" the multiplication by elements of the extended ring, i. e. $(K, V)=K \otimes_F (F,V)$.
Note also, that notation $(K, V)$ is very rarely used, usually people just say "$K$-vector space", or "over $K$".
See more details in Kostrikin, Manin "Linear Algebra and Geometry".
