Radon Riesz property
The Radon-Riesz property , named after Johann Radon and Frigyes Riesz , is a property of normalized spaces considered in the mathematical sub-area of functional analysis. It describes a relationship between weak - convergent and norm -konvergenten consequences . Other names are Kadets-Klee property , after MI Kadets and Victor Klee or simply property (H) , which originally comes from an alphabetical list of properties and z. B. is used in the Mahlon Day textbook given below .
A normalized space has the Radon-Riesz property if it fulfills the following condition: If there is a sequence in this space which converges weakly to a and for which applies, then it already follows . In this case, the room is also called a Radon-Riesz room .
- Every room with the Schur property has the Radon-Riesz property, since in the case of the former, the norm convergence alone follows from the existence of the weak convergence of the sequence.
- If and is a measure space with positive measure , then the L p -space has the Radon-Riesz property. This statement, proven by J. Radon and F. Riesz, is also known as the Radon-Riesz theorem , from which the later naming of this property resulted.
- Every uniformly convex space , even every locally uniformly convex space, has the Radon-Riesz property. Since the L p spaces are uniformly convex, this generalizes the previous example. In particular, every Hilbert space has the Radon-Riesz property.
- Strongly convex spaces have the Radon-Riesz property.
- There are Banach spaces with the Radon-Riesz property that are not strictly convex . To do renormalize the sequence space for one through . The Banach space is then an example of the desired kind.
- The sequence space of the zero sequences with the supremum norm does not have the Radon-Riesz property. If the sequence denotes a 1 in the nth position and a 0 everywhere else, then obviously weak and applies , but because of there is no norm convergence.
An equivalent formulation is obtained by restricting the vectors in the definition of the Radon-Riesz property to those of length 1. Denotes the unit sphere of a standardized space , then:
- A normalized X has the Radon-Riesz property if and only if for every sequence in that converges weakly to a , it already follows.
- MM Day: Normed linear spaces , Springer-Verlag (1973), ISBN 3-540-06148-7
- Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , definition 2.5.26
- Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , Theorem 5.3.7
- Vasile I. Istratescu: Strict Convexity and Complex Strict Convexity: Theory and Applications , Lecture Notes in Pure and Applied Mathematics (1984), ISBN 0-824-71796-1 , example 2.4.46