Remark

— Let F be a Banach space over the field $\mathbf{C}$ of complex numbers; then, for every function $f \in \mathscr{F}_\mathbf{F}^p$ and every complex number $\alpha$, $\alpha f$ belongs to $\mathscr{F}_\mathbf{F}^p$ and $N_p(\alpha f) = |\alpha| N_p(f)$; in other words, $\mathscr{F}_\mathbf{F}^p$ is also a vector space over $\mathbf{C}$, and $N_p(f)$ is a semi-norm on this complex vector space (TVS, II, §1).