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 . ${\displaystyle X}$ ${\displaystyle X}$

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: ${\displaystyle X}$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle \mu \colon {\mathcal {A}}\rightarrow X}$ ${\displaystyle \mu }$

1. ${\displaystyle \mu }$is of limited variation .
2. If there is a finite, positive measure with , then there is a Bochner-integrable function with for all . ${\displaystyle \lambda \colon {\mathcal {A}}\rightarrow \mathbb {R} }$ ${\displaystyle \mu \ll \lambda }$ ${\displaystyle \lambda }$ ${\displaystyle f\colon \Omega \rightarrow X}$ ${\displaystyle \mu (A)=\int _{A}f\mathrm {d} \lambda }$ ${\displaystyle A\in {\mathcal {A}}}$

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. ${\displaystyle \mu \ll \lambda }$ ${\displaystyle \mu }$ ${\displaystyle \lambda }$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle \lambda (A)=0}$ ${\displaystyle \mu (A)=0}$

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. ${\displaystyle X}$ ${\displaystyle X}$

Examples

• The Banach space has the Radon-Nikodym property. That is exactly what the Radon-Nikodym theorem says. ${\displaystyle \mathbb {R} }$
• 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. ${\displaystyle \ell ^{p}}$ ${\displaystyle 1<p<\infty }$
• 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. ${\displaystyle \ell ^{1}}$ ${\displaystyle {\mathcal {N}}(\ell ^{2})}$ ${\displaystyle \ell ^{2}}$
• Is any index set, then has the Radon-Nikodym property. ${\displaystyle I}$ ${\displaystyle \ell ^{1}(I)}$
• 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. ${\displaystyle c_{0}}$ ${\displaystyle \ell ^{\infty }}$ ${\displaystyle C([0,1])}$ ${\displaystyle L^{1}([0,1])}$ ${\displaystyle L^{\infty }([0,1])}$

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. ${\displaystyle c_{0}}$ ${\displaystyle \ell ^{1}}$ ${\displaystyle \ell ^{1}}$
• 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 . ${\displaystyle X}$ ${\displaystyle B\subset X}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle x\in B}$ ${\displaystyle B\setminus K_{\varepsilon }(x)}$ ${\displaystyle K_{\varepsilon }(x)}$ ${\displaystyle \varepsilon }$ ${\displaystyle x}$
• 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 . ${\displaystyle X}$ ${\displaystyle (\Omega ,{\mathcal {A}},\lambda )}$ ${\displaystyle \lambda }$ ${\displaystyle L^{1}(\Omega ,{\mathcal {A}},\lambda )\rightarrow X}$ ${\displaystyle \ell ^{1}}$ ${\displaystyle T\colon L^{1}(\Omega ,{\mathcal {A}},\lambda )\rightarrow X}$ ${\displaystyle R\colon L^{1}(\Omega ,{\mathcal {A}},\lambda )\rightarrow \ell ^{1}}$ ${\displaystyle S\colon \ell ^{1}\rightarrow X}$ ${\displaystyle T=S\circ R}$

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: ${\displaystyle X}$

• ${\displaystyle X'}$(the dual space of ) has the Radon-Nikodym property. ${\displaystyle X}$
• ${\displaystyle X'}$ has the Kerin-Milman property.
• If is a separable subspace of , then is separable. ${\displaystyle Y\subset X}$ ${\displaystyle X}$ ${\displaystyle Y'}$

Individual evidence

1. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , page 106
2. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 6, §3, page 213
3. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Corollary 5.45
4. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Corollary 5.42
5. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 6, §4: The Dunford-Pettis Theorem
6. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 6, §4, Theorem 2
7. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 6, §4, Corollary 4
8. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , (5.13) + (5.15)
9. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , sentence 5.49
10. ^ 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
11. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Theorem 5.36
12. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 6, §5, Theorem 1
13. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , page 118
14. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 6, §6, Corollary 1