Montel room

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.

definition

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

Examples

• 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. ${\displaystyle G\subset {\mathbb {C} }}$ ${\displaystyle H(G)}$ ${\displaystyle \textstyle p_{K}(f):=\sup _{z\in K}|f(z)|}$ ${\displaystyle K\subset G}$ ${\displaystyle H(G)}$ ${\displaystyle H(G)}$ ${\displaystyle H(G)}$
• 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. ${\displaystyle \Omega \subset {\mathbb {R} }^{n}}$ ${\displaystyle {\mathcal {E}}(\Omega )}$ ${\displaystyle f\colon \Omega \rightarrow {\mathbb {R} }}$ ${\displaystyle \textstyle p_{K,m}(f):=\sup _{|\alpha |\leq m}\sup _{x\in K}|D^{\alpha }f(x)|}$ ${\displaystyle {\mathcal {E}}(\Omega )}$ ${\displaystyle \alpha =(\alpha _{1},\ldots ,\alpha _{n})}$
• 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. ${\displaystyle \Omega \subset {\mathbb {R} }^{n}}$ ${\displaystyle {\mathcal {D}}(\Omega )\subset {\mathcal {E}}(\Omega )}$ ${\displaystyle \Omega }$ ${\displaystyle K\subset \Omega }$ ${\displaystyle {\mathcal {D}}_{K}(\Omega )}$ ${\displaystyle {\mathcal {E}}(\Omega )}$ ${\displaystyle {\mathcal {D}}(\Omega )}$ ${\displaystyle {\mathcal {D}}_{K}(\Omega )\subset {\mathcal {D}}(\Omega )}$ ${\displaystyle {\mathcal {D}}(\Omega )}$
• 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. ${\displaystyle {\mathcal {S}}({\mathbb {R} }^{n})}$ ${\displaystyle f\colon {\mathbb {R} }^{n}\rightarrow {\mathbb {R} }}$ ${\displaystyle \textstyle p_{k,m}(f):=\sup _{|\alpha |\leq k}\sup _{x\in {\mathbb {R} }^{n}}|(1+|x|^{2})^{m}D^{\alpha }f(x)|}$ ${\displaystyle {\mathcal {S}}({\mathbb {R} }^{n})}$ ${\displaystyle \{p_{k,m};\,k,m\in {\mathbb {N} }_{0}\}}$
• 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. ${\displaystyle {\mathcal {E}}'(\Omega )}$ ${\displaystyle {\mathcal {D}}'(\Omega )}$ ${\displaystyle {\mathcal {S}}'({\mathbb {R} }^{n})}$

swell

• Klaus Floret, Joseph Wloka : Introduction to the theory of locally convex spaces (= Lecture Notes in Mathematics. Vol. 56, ISSN  0075-8434 ). Springer, Berlin et al. 1968, doi : 10.1007 / BFb0098549 .
• HH Schaefer: Topological Vector Spaces, Springer, 1971 ISBN 0-387-98726-6
• 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