(DF) room

(DF) spaces are a class of special locally convex spaces considered in the mathematical branch of functional analysis , which play an important role in the duality theory of Fréchet spaces . Dual spaces of Fréchet spaces are (DF) spaces, and dual spaces of (DF) spaces are again Fréchet spaces. This explains the designation (DF) introduced by Grothendieck in 1954 .

definition

The definition of the (DF) -spaces is motivated by the following two properties of dual spaces that can be metrizable locally convex spaces . If the dual space of a metrizable locally convex space then applies: ${\displaystyle E=F\,'}$ ${\displaystyle F}$

1. There is a sequence of bounded sets in , so that for each bounded set there is one and one with . ${\displaystyle (B_{n})_{n}}$ ${\displaystyle E}$ ${\displaystyle B\subset E}$ ${\displaystyle n\in {\mathbb {N} }}$ ${\displaystyle \lambda >0}$ ${\displaystyle B\subset \lambda B_{n}}$
2. If a sequence of absolutely convex neighbors is in , and there is a with for every bounded set , then a neighbors of zero is in . ${\displaystyle (V_{n})_{n}}$ ${\displaystyle E}$ ${\displaystyle B\subset E}$ ${\displaystyle \lambda >0}$ ${\displaystyle B\subset \lambda \bigcap _{n}V_{n}}$ ${\displaystyle \bigcap _{n}V_{n}}$ ${\displaystyle E}$

Hence one defines

A (DF) space is a locally convex space E that has the properties (1) and (2) mentioned above.

Examples

• The definition is laid out in such a way that dual spaces are metrizable locally convex vector spaces (DF) -spaces, even complete (DF) -spaces.
• Every quasitonnelierte space that fulfills the first property of the above definition is a (DF) space. In particular, all normalized spaces are (DF) spaces. There are therefore (DF) spaces that are not complete and there are complete (DF) spaces that are not a dual space.
• The space of all real sequences with the topology of point-wise convergence is not a (DF) -space. ${\displaystyle {\mathbb {R} }^{\infty }}$

Inheritance properties

• Completions of (DF) spaces are again (DF) spaces.
• If there is a closed subspace in the (DF) space E, the factor space is again a (DF) space. The (DF) property is generally not inherited by (closed) subspaces. ${\displaystyle F\subset E}$ ${\displaystyle E/F}$
• If there is a sequence of (DF) spaces, the direct sum with the final topology is again a (DF) space. The (DF) property inherited generally not on product spaces . ${\displaystyle (E_{n})_{n}}$ ${\displaystyle \bigoplus _{n}E_{n}}$
• The projective tensor product of two (DF) -spaces is again a (DF) -space.

Other properties

• The strong dual space of a (DF) space is a Fréchet dream. From this it follows easily that the bidual space of a Fréchet dream is again a Fréchet dream. Another important conclusion is that a Fréchet dream is reflexive if and only if its strong dual space is reflexive.
• Every separable (DF) space is quasi-delineated.
• The topology of a (DF) -space E can be localized in the following sense: An absolutely convex set is a null neighborhood if and only if for every absolutely convex, bounded set the average is a neighborhood of 0 in the subspace topology to . ${\displaystyle U\subset E}$ ${\displaystyle B\subset E}$ ${\displaystyle B\cap U}$ ${\displaystyle B}$
• A (DF) -space carries the finest locally convex topology, which makes the inclusions of the bounded sets from part (1) of the above definition continuous. ${\displaystyle B_{n}\subset E}$ ${\displaystyle B_{n}}$

gDF rooms

If a locally convex space only has the property (1) of the above definition and if it has the finest locally convex topology, which makes the inclusions of the bounded sets from part (1) of the above definition continuous, then this space is called gDF space (generalized DF). Every (DF) room is a gDF room. ${\displaystyle B_{n}\subset E}$

Like (DF) -spaces, gDF-spaces are closed with respect to the operations completion, formation of quotient spaces, countable summation and projective tensor products.

If and are two locally convex spaces, then let the space of continuous, linear mappings be provided with the topology of uniform convergence on limited sets, i.e. H. it is , if for any bounded set and each zero neighborhood one out there, so that for all and . With this concept formation, gDF spaces can be characterized as follows: ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle L_{b}(E,F)}$ ${\displaystyle L(E,F)}$ ${\displaystyle E\rightarrow F}$ ${\displaystyle T_{i}\to T}$ ${\displaystyle B\subset E}$ ${\displaystyle U\subset F}$ ${\displaystyle i_{0}}$ ${\displaystyle T_{i}(x)-T(x)\in U}$ ${\displaystyle x\in B}$ ${\displaystyle i\geq i_{0}}$

For a locally convex space are equivalent: ${\displaystyle E}$

• ${\displaystyle E}$ is a gDF room.
• For every Fréchet dream there is a Fréchet dream. ${\displaystyle F}$ ${\displaystyle L_{b}(E,F)}$
• For every Banach room there is a Fréchet dream. ${\displaystyle F}$ ${\displaystyle L_{b}(E,F)}$

literature

• R. Meise, D. Vogt: Introduction to Functional Analysis, Vieweg, 1992 ISBN 3-528-07262-8
• HH Schaefer: Topological Vector Spaces, Springer, 1971 ISBN 0-387-98726-6
• A. Grothendieck: Sur les espaces (F) et (DF). Summa Brasil. Math. 3, 57-123 (1954)
• H. Jarchow: Locally Convex Spaces, Teubner, Stuttgart 1981 ISBN 3-519-02224-9