Remark 2

In the definitions of the comparison relations $f_1 \preccurlyeq f_2$ and $f_1 \ll f_2$ the norms on the spaces $V_1, V_2$ where $f_1$ and $f_2$ respectively take their values, are involved only apparently; in reality only the topologies on $V_1$ and $V_2$ are involved, for the relations $f_1 \preccurlyeq f_2$ and $f_1 \ll f_2$ are replaced by equivalent relations when one replaces the norm on $V_1$ or $V_2$ by an equivalent norm (\emph{Gen. Top.}, IX, p. 170, def. 7).