Lemma 4

The law of operation $H\times \mathscr{S}^r(X; E)\rightarrow \mathscr{S}^r(X; E)$ is continuous.

In view of the definition of the topology of $\mathscr{S}^r(X; E)$ and General Topology, Chap. X, §3, no. 4, Th. 3, it suffices to prove that for any integers $k\leq r$, the map $f: H\times X\times \mathscr{S}^k(X; E)\rightarrow J^k(X; E)$ such that $f(h, x, s) =j_x^k(^hs)$ is continuous. For $h\in H$, denote by $\tau_h$ (resp. $\theta_h$) the automorphism $x \rightarrow hx$ of X (resp. of E). Define maps

by $f_1(h, x) =j^k_x(\tau_h), f_2(h, v) =j_v^k(\theta_h), g(h, x, s) =j_{hx}^k(s)$. We have

and consequently it suffices, by Differentiable and Analytic Manifolds, Results, 12.3.6, to prove that $f_1, f_2$ and $g$ are continuous.

Now $g$ is the composite map

with $\varepsilon (x, u) =u(x)$; the map $\varepsilon$ being continuous (General Topology, Chap. X, §3, no. 4, Cor. 1 of Th. $3$)$,g$ is continuous.

Let $(h_0, x_0)\in H\times X$; we shall prove that $f_1$ is continuous at $(h_0, x_0)$. There exist charts $(U, \psi ,F)$ and $(V, \chi ,F')$ of X and an open subset $\Omega$ of H such that $x_0\in U, h_0\in \Omega$ and $m(\Omega \times U)\subset V$. By using the expression for $J^k(X,X)$ in these charts, we are reduced to proving, for $1\leq l\leq k$, the continuity at $(h_0, x_0)$ of the map $(h, x) \rightarrow \Delta^l_x(\tau_h)$ from $\Omega \times U$ to $P_l(F; F')$, with $\Delta^l_x(\tau_h)(v) =$ $\frac{1}{l!}D^l\tau_h(x).v$ for $v\in F ($Differentiable and Analytic Manifolds, Results, 12.2). But $D^l\tau_h(x)$ is simply the $l$th partial derivative of $m(h, x)$ with respect to $x$, which is continuous by hypothesis; consequently, $f_1$ is continuous. The proof that $f_2$ is continuous is similar, hence the lemma.