In "Analyse fonctionnelle" of Brezis, in chapter III why do we need Banach spaces ? (especially for Kakutani's theorem)

Indeed the equivalence still holds if $E$ is an incomplete normed space (over $\mathbb{R}$ or $\mathbb{C}$): both sides are false. This is pretty easy to see directly and really misses the point of the theorem. So the authors probably just decided not to bother to include this relatively uninteresting case.

It's pretty common in functional analysis to write theorems that only cover Banach spaces, even when normed spaces could also be included. This can be for any of several reasons:

• In most applications, you are working with Banach spaces

• The theorem may become trivial for incomplete spaces

• For an incomplete space $X$, the theorem gives you the "right" conclusion if you apply it to the completion of $X$.

Consider $\mathbb{R}$ as an vector space over $\mathbb{Q}$ the rationals space endowed with the norm of absolute value, then $\mathbb{R}$ over $\mathbb{Q}$ is an infinite dimensional normed vector space where the ball is compact.