Montel space
A barrelled space (in particular, a Fréchet space) in which each closed bounded set is compact. The space of all holomorphic functions in a domain
, 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
, so
is a Montel space. The space
of all infinitely-differentiable functions in a domain
, the space
of all functions of compact support and the space
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 ,
and
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) |
Comments
References
[a1] | H. Jarchow, "Locally convex spaces" , Teubner (1981) (Translated from German) |
[a2] | G. Köthe, "Topological vector spaces" , 1 , Springer (1969) |
[a3] | H.H. Schaefer, "Topological vector spaces" , Springer (1971) |
Montel space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Montel_space&oldid=13981