Proposition 5

The locally convex space $\mathscr{F}_\mathbf{F}^p$ is complete.

Since the Hausdorff space associated with $\mathscr{F}_\mathbf{F}^p$ is a normed space, it suffices to prove that every Cauchy sequence $(f_n)$ in $\mathscr{F}_\mathbf{F}^p$ has a limit for the topology of convergence in mean of order $p$ (GT, IX, §2, No. 6, Prop. 9). By hypothesis, for every $\varepsilon > 0$ there exists an integer $m_0$ such that the relations $m \geq m_0,\ n \geq m_0$ imply $N_p(f_n - f_m) \leq \varepsilon$. One may therefore define, by induction on $k$, a strictly increasing sequence $(n_k)$ of integers $\geq 0$ such that $N_p(f_{n_{k+1}} - f_{n_k}) \leq 2^{-k}$. If we show that the series with general term $g_k = f_{n_{k+1}} - f_{n_k}$ ($k \geq 1$) is convergent in mean of order $p$, then it will have a sum $g \in \mathscr{F}_\mathbf{F}^p$, and $f = g + f_{n_1}$ will be the limit of the sequence $(f_{n_k})$ in $\mathscr{F}_\mathbf{F}^p$; $f$ will then be a cluster point of the sequence $(f_n)$; since this sequence is a Cauchy sequence, it will have $f$ as limit, and Prop. 5 will have been proved (GT, II, §3, No. 2, Cor. 2 of Prop. 5).

Prop. 5 is thus a consequence of the following proposition: