Example 2

With the notations and hypothesis of prop. 7 of III, p. 6, the space E, being the inductive limit of Banach spaces, is barrelled (III, p. 25); moreover, every bounded subset of E is relatively compact (III, p. 6, prop. 7). In other words, E is a Montel space. In particular, Gevrey spaces (III, p. 10) are Montel spaces. \* This is true for the space $\mathscr{H}(K)$ consisting of germs of functions analytic in a neighbourhood of a compact subset K of $\mathbf{C}^n$ (III, p. 10).*