Remark

We apply Corollary 1 with $G = \mathbf{GL}(F)$ ($F$ a complete normable space), $\pi_1 = \pi_2 = \mathrm{Id}_G$ and $\pi_3$ the trivial representation of $G$ on $K$. We obtain an analytic representation $\pi$ of $\mathbf{GL}(F)$ on $\mathscr{L}(F, F; K)$. We assume that $F$ is finite-dimensional and that $K$ is of characteristic 0. Applying Corollary 2 to Proposition 39 to $\pi$, we recover part of Corollary 1 to Proposition 37.