# Montel space

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.

#### References

 [1] N. Bourbaki, "Elements of mathematics. Topological vector spaces" , Addison-Wesley (1977) (Translated from French) [2] A.P. Robertson, W.S. Robertson, "Topological vector spaces" , Cambridge Univ. Press (1964) [3] R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)