Example

With the hypotheses and notation of Proposition 16, let F be a complete normable space. Then $\mathscr{L}((TX)^p; F)$ is a vector G-bundle of class C^{r-1}; so is Alt^p(TX; F) if K is of characteristic zero or X is finite-dimensional (Differentiable and Analytic Manifolds, R, 7.7, 7.8). If X is finite-dimensional, $\bigotimes^p (TX) \otimes \bigotimes^q (TX)^*$ is a vector G-bundle of class C^{r-1}.