Montel space

The mathematical term Montel- space refers to a special class of locally convex spaces. Your name they bear, by the theorem of Montel from the theory of functions. Many locally convex spaces in the theory of distributions are Montelräume.

Definition

A locally convex space is called a Montel space if it is quasi-barreled and the completion of every bounded set is compact.

Examples

  • A normed space if and only Montel space if it is finite- dimensional.
  • Once a site and is the space of holomorphic functions on G with the semi-norms, with the compact subsets of G passes, so by the theorem of Montel in any bounded set has a compact statements. There is a Fréchet space and quasi-barreled, proves to be a Montel space.
  • Be open and the space of infinitely differentiable functions with the semi-norms, so is a Montel space. It was used for the multi- index notation.
  • Be open and the subspace of infinitely differentiable functions with compact support in. For compact is the space of functions with support in K with the induced subspace topology. Then there is a finest locally convex topology that makes all embeddings steadily. with this topology is the space of test functions and is an example of a non - metrizable Montel space.
  • Be the space of all functions for which all suprema are finite. It was taken again by the multi- index notation use. The space with the seminorms is called space of functions quickly and falling is a Montel space.
  • Full quasi- barreled Schwartz spaces are Montel spaces.
  • Every locally convex space with the finest locally convex topology, that is, with the absolutely convex of all, absorbing quantities as base of neighborhoods generated topology is a Montel space.

Properties of Montelräumen

  • Montel spaces are reflexive and therefore barreled.
  • Montel spaces are quasi-complete, that is, every bounded Cauchy net converges. There are incomplete Montel spaces.
  • Direct products (with the product topology ) and direct sums ( with the final topology) of Montel spaces are back Montel spaces.
  • In general, neither closed subspaces nor quotient of Montel spaces back Montel spaces.
  • If E is a Montel space, as well as the strong dual space E '. In particular, therefore, the spaces that occur in the distribution theory, and Montel spaces.

Swell

  • K. Floret, J. Wloka: Introduction to the theory of locally convex spaces, Lecture Notes in Mathematics 56, 1968
  • H. Jarchow: Locally Convex Spaces, Teubner, Stuttgart 1981 ISBN 3-519-02224-9
  • R. Meise, D. Vogt: Introduction to Functional Analysis, Vieweg, 1992 ISBN 3-528-07262-8
  • Locally convex space
  • Functional Analysis
Complex analysis Closed set Domain (mathematical analysis) Smooth function Multi-index notation Bump function Absorbing set Base (topology) Quotient space (linear algebra)
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