# 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})}$