Corollary 3

Let F be a Banach space, $p$ an integer $\geq 0,^k\Omega^p(X; F)$ the space of differential forms of degree $p$ on X, with values in F, and of class $C^k(2\leq k+ 1\leq r)$. For $\omega \in^k\Omega^p(X; F)$, put $\omega^{\sharp}=\int_H\tau_h^*\omega dh$. Then the map $\omega \rightarrow \omega^{\sharp}$ is a projection on $^k\Omega^p(X; F)$ whose image is the subspace of H-invariant forms. We have $d(\omega^{\sharp}) = (d\omega )^{\sharp}$ for all $\omega \in^k\Omega^p(X; F)$.

The first assertion follows from the proposition applied to the vector H-bundle Alt$^p(T(X); F)$ (Chap. III, §1, no. 8, Examples). To prove the second assertion, it suffices, in view of Integration, Chap. III, §3, no. 2, Prop. 2, to prove that the map $d:^k\Omega^p(X; F)\rightarrow^{k-1}\Omega^{p+1}(X; F)$ is continuous when the first (resp. second) space is given the topology of compact $C^k$-convergence (resp. $C^{k-1}$-convergence). But this follows immediately from the definition of these topologies by means of semi-norms (Spectral Theories, in preparation) and the fact that $d$ is a differential operator of order $\leq 1 ($Differentiable and Analytic Manifolds, Results, 14.4.2).