A barrelled space (in particular, a Fréchet space) in which each closed bounded set is compact. The space $H(G)$ of all holomorphic functions in a domain $G$, with the topology of uniform convergence on compact sets, is a Fréchet space and, in view of a theorem of Montel (cf. Montel theorem, 2), every bounded sequence of holomorphic functions is relatively compact in $H(G)$, so $H(G)$ is a Montel space. The space $C^\infty(\Omega)=\mathcal E(\Omega)$ of all infinitely-differentiable functions in a domain $\Omega\subset\mathbf R^n$, the space $D(\Omega)$ of all functions of compact support and the space $S(\mathbf R^n)$ of differentiable functions that are rapidly decreasing at infinity, are also Montel spaces in their natural topologies.
A Montel space is reflexive (cf. Reflexive space). The strong dual of a Montel space is a Montel space; in particular, the spaces of generalized functions $\mathcal E'(\Omega)$, $D'(\Omega)$ and $S'(\Omega)$ are Montel spaces. A normed space is a Montel space if and only if it is finite-dimensional.
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Montel space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Montel_space&oldid=32479