Remark 1

With the above notations, let $F_n$ be the vector space generated by $H_n$, with a norm equal to the gauge of $H_n$. We shall see (III, p. 8, cor.) that $F_n$ is a Banach space. The injection from $F_n$ into $E_{n+1}$ is compact, hence a fortiori also the injection $w_n$ from $F_n$ into $F_{n+1}$. Further, $E$ is the inductive limit of the inductive system $(F_n, w_n)$ of Banach spaces. For, a convex balanced neighbourhood $V$ of 0 in $E$ is such that $V \cap E_n$ absorbs $H_{n-1}$ for all $n \geq 1$, and conversely, if a convex balanced set $W$ in $E$ is such that $W \cap E_n$ absorbs $H_{n-1}$, then $W \cap E_{n-1}$ contains a set homothetic to $W_{n-1}$ for all $n \geq 1$, and hence $W$ is a neighbourhood of 0 in $E$.