Montel room

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The mathematical term Montel space describes a special class of locally convex spaces . They carry its name from the Montel's theorem from the theory of functions . Many locally convex spaces from the theory of distributions are Montel spaces.


A locally convex space is called a Montel space if it is quasitonneliert and the closure of each bounded set is compact .


  • A normalized space is a Montel space if and only if it is finite-dimensional.
  • If is a domain and is the space of the holomorphic functions on G with the semi-norms , where the compact subsets of G runs through, then by Montel's theorem each in bounded set has a compact closure. Since the Fréchet room is also quasi-tiled , it turns out to be a Montel room.
  • Be open and the space of any number of differentiable functions with the semi-norms , that's a Montel space. It was for the multiindex used notation.
  • Be open and the subspace of any number of differentiable functions with a compact carrier in . For compact, let the space of the functions with carrier in K with the subspace topology induced by . Then there is a finest locally convex topology that makes all embeddings continuous. with this topology is the space of the test functions and is an example of a non- metrizable Montel space.
  • Be the space of all functions for which all suprema are finite. The multi-index notation was used again. The room with the semi-norms is called the room of rapidly falling functions and is a Montel room.
  • Complete, quasi-tiled Schwartz rooms are Montel rooms.
  • Every locally convex space with the finest locally convex topology , that is to say with the topology generated by all absolutely convex , absorbing sets as a zero neighborhood basis , is a Montel space.

Properties of Montel rooms

  • Montel rooms are reflexive and therefore barreled .
  • Montel spaces are quasi-complete ; H. every bounded Cauchy network converges. There are incomplete Montel rooms.
  • Direct products (with the product topology ) and direct sums (with the final topology ) of Montel spaces are again Montel spaces.
  • In general, neither closed subspaces nor quotients of Montel spaces are again Montel spaces.
  • If E is a Montel space, so is the strong dual space E '. So in particular, those in the distribution theory spaces occur , and Montel spaces.