Why do we consider only real or complex Banach spaces?

Question

In wikipedia, normed vector space is defined as a vector space over a subfield of $\mathbb{C}$ equipped with a norm.

However, Banach space is defined as a complete normed space over $\mathbb{R}$ or $\mathbb{C}$.

Is there a reason that we consider only real or complex Banach space?

Since every normed space over a subfield $F$ of $\mathbb{C}$ can be completed to a complete normed space over $F$, I don't get why we are not considering these complete normed spaces over $F$.

Answer 1

It is not too hard to show that if $V$ is a complete $\Bbb{Q}$ vector space, one can extend the scalar multiplication uniquely continuously to $\Bbb{R}\times V\to V$, so that $V$ is also a $\Bbb{R}$ vector space. Hence, we assume this to begin with.

Also, if the vector space is complete, it is natural to assume that the underlying field is complete too.

EDIT: Here, I consider $\Bbb{Q}$ to be equipped with the usual absolute value $|x|=\max \{x,-x\}$ and assume that the norm $\Vert \cdot \Vert$ on $V$ satisfies $\Vert \alpha x \Vert = |\alpha| \cdot \Vert x\Vert$ for $x\in V$ and $\alpha \in \Bbb{Q}$.