Radon-Nikodym property
The Radon-Nikodym property , named after Johann Radon and Otton Marcin Nikodým , is a property of Banach spaces or vectorial measures considered in the mathematical sub-area of functional analysis . A Banach space has the Radon-Nikodym property, often abbreviated to RNP (after the English term " Radon-Nikodym property "), if a statement analogous to the classical Radon-Nikodym theorem applies to vector dimensions with values .
Definitions
Let it be a Banach space, a measurable space and a vectorial measure. It is said to have the Radon-Nikodym property if the following applies:
- is of limited variation .
- If there is a finite, positive measure with , then there is a Bochner-integrable function with for all .
The notation means, as usual, that is absolutely continuous with respect to , that is, that from already follows for all . In the above definition, the two measures fulfill a vector-valued variant of the classical Radon-Nikodym theorem.
Finally, a Banach space is defined to have the Radon-Nikodym property if every vectorial measure of limited variation with values in has the Radon-Nikodym property.
Examples
- The Banach space has the Radon-Nikodym property. That is exactly what the Radon-Nikodym theorem says.
- Every reflective space has the Radon-Nikodym property. Thus the sequence spaces and the L p spaces for and all Hilbert spaces have the Radon-Nikodym property.
- Dunford-Pettis theorem : Every separable dual space has the Radon-Nikodym property. Examples of this are or the space of nuclear operators on the Hilbert space . More generally, every dual space that is a subspace of a weakly compact generated Banach space has the Radon-Nikodym property.
- Is any index set, then has the Radon-Nikodym property.
- If the Banach space has an equivalent very smooth norm , its dual space has the Radon-Nikodym property. In particular, locally weakly uniformly convex dual spaces have the Radon-Nikodym property.
- The space of zero sequences , the space of bounded sequences and function rooms , , have not the Radon Nikodym property.
properties
- Closed subspaces of rooms with radon-nikodymium properties again have the radon-nikodymium property.
- The Radon-Nikodym property is not inherited by quotient spaces . The space is the quotient of , because every separable Banach space is the quotient of , and this one has the Radon-Nikodym property, but that one does not.
- The set of Davis Huff Maynard Phelps is a geometric characterization of the Radon-Nikodym property. A Banach space has the Radon-Nikodym property if and only if there is one for every bounded set and for every one that is not in the closed convex hull of . Here referred to the -ball order .
- The set of Lewis Stegall characterized spaces with the Radon Nikodym property by operators: A Banach space has exactly the Radon Nikodym property if for each measure space with a positive, finite extent of each continuous , linear operator on factored. The latter means that for every continuous, linear operator there are continuous, linear operators and with .
The Kerin-Milman property
Motivated by the Kerin-Milman theorem , one says that a Banach space has the Krein-Milman property if every closed, bounded, convex set is equal to the closure of the convex hull of its extremal points . Note that no compactness requirement is made here. This is often abbreviated as KMP after the English name " Krein-Milman property ".
According to a statement by Lindenstrauss , every room with the Radon-Nikodym property also has the Kerin-Milman property. The inversion of this statement is an open mathematical problem, but it is known for dual spaces; more precisely, the following statements about a Banach space are equivalent:
- (the dual space of ) has the Radon-Nikodym property.
- has the Kerin-Milman property.
- If is a separable subspace of , then is separable.
Individual evidence
- ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , page 106
- ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 6, §3, page 213
- ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Corollary 5.45
- ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Corollary 5.42
- ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 6, §4: The Dunford-Pettis Theorem
- ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 6, §4, Theorem 2
- ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 6, §4, Corollary 4
- ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , (5.13) + (5.15)
- ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , sentence 5.49
- ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 6, §3: The Davis-Huff-Maynard-Phelps Theorem
- ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Theorem 5.36
- ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 6, §5, Theorem 1
- ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , page 118
- ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 6, §6, Corollary 1