Proposition 3

a) Let $(f_n)_{n\in\mathbf{N}}$ be a sequence in $\mathscr{F}_{\pi}^p(G)$ such that the series

converges. Then the series with general term $f_n(g)$ is absolutely convergent for $g$ outside a set T which is negligible modulo H. Let $f$ be the function on G with values in E which is equal to the sum of this series on G - T and equal to 0 on T. Then one has $f\in \mathscr{F}_{\pi}^p(G)$ and the series with general term $f_n$ converges to $f$ in the space $\mathscr{F}_{\pi}^p(G)$;

b) Let $(f_n)$ be a sequence converging in $\mathscr{L}_{\pi}^p(G)$ to a function $f$. There exists a subsequence $(f_{n_k})$ extracted from $(f_n)$ such that $f_{n_k}(g)$ converges to $f(g)$ for every $g$ outside a set negligible modulo H;

c) The semi-normed spaces $\mathscr{F}_{\pi}^p(G)$ and $\mathscr{L}_{\pi}^p(G)$ are complete.

In assertion a), to say that the series with general term $f_n$ converges to $f$ in the space $\mathscr{F}_{\pi}^p(G)$ means that the sequence of partial sums $f_0+\cdots +f_n$ converges to $f$ in $\mathscr{F}_{\pi}^p(G)$. One then also says that $f$ is a sum of this series.

Let us prove a). By prop. 6 of INT, IV, p. 128, § 3, n$^o3$, there exists a $\nu$-negligible set $S\subset G/H$ such that the series with general term $\|f_n(gH)\|$ converges absolutely for $gH\notin S$. Moreover, the function $h$ which is equal to the sum of this series for $gH\notin S$ and which is zero for $gH\in S$ satisfies $N_p(h)<+\infty$.

The set $T =\overset{-1}{\varpi}(S)$ is negligible modulo H. For every $g \notin T$, the series with general term $f_n(g)$ is absolutely convergent in E. Define $f(g) =\sum f_n(g)$ for $g \notin T$ and $f(g) = 0$ otherwise. We have $f\in$ $\mathscr{F}_{\pi}(G)$. Note that $\|f(gH)\|\leqslant h(gH)$ for every $g\in G$, whence $N_p(f)\leqslant$ $N_p(h)<+\infty$. Thus $f\in \mathscr{F}_{\pi}^p(G)$. Similarly, we obtain

for every $k\in \mathbf{N}$, hence the series $\sum f_n$ converges to $f$ in $\mathscr{F}_{\pi}^p(G)$. Assertion a) is proved.

Let us prove b). The sequence $(\|f_n-f\|)_{n\in\mathbf{N}}$ converges to 0 in the space $\mathscr{L}^p(G/H)$. By INT, IV, p. 131, § 4, n$^o3$, th. 3, there exists a subsequence $(\|f_{n_k}-f\|)_{k\in\mathbf{N}}$ which converges $\nu$-almost everywhere to 0. The sequence $(f_{n_k}(g))_{k\in\mathbf{N}}$ then converges to $f(g)$ for every $g$ outside a set negligible modulo H.

Finally, let us prove c). Let $(f_n)_{n\in\mathbf{N}}$ be a Cauchy sequence in $\mathscr{F}_{\pi}^p(G)$. There exists a strictly increasing sequence of integers $(n_k)_{k\in\mathbf{N}}$ such that $N_p(f_{n_{k+1}}-f_{n_k})\leqslant 2^{-k}$ for every $k\in \mathbf{N}$. For every $k\in \mathbf{N}$, put $h_k$ = $f_{n_{k+1}}-f_{n_k}\in \mathscr{F}_{\pi}^p(G)$. By a), the series with general term $h_k$ converges in $\mathscr{F}_{\pi}^p(G)$; let $h$ denote its sum. For every $\ell \in \mathbf{N}$, we obtain

hence $f_{n_0}+h$ is a cluster value of the sequence $(f_n)$. It therefore converges (TG, II, p. 14, cor. 2 of prop. 5). Thus the space $\mathscr{F}_{\pi}^p(G)$ is complete; since the space $\mathscr{L}_{\pi}^p(G)$ is closed in $\mathscr{F}_{\pi}^p$(G), it is likewise complete.