# 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 .

## definition

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 . ${\ displaystyle (X, \ | \ cdot \ |)}$${\ displaystyle (x_ {n}) _ {n \ in \ mathbb {N}}}$${\ displaystyle x \ in X}$${\ displaystyle \ | x_ {n} \ | \ rightarrow \ | x \ |}$${\ displaystyle \ | x_ {n} -x \ | \ rightarrow 0}$

## Examples

• 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.${\ displaystyle (\ Omega, {\ mathfrak {S}}, \ mu)}$${\ displaystyle 1 ${\ displaystyle L ^ {p} (\ Omega, {\ mathfrak {S}}, \ mu)}$
• Every uniformly convex space , even every locally uniformly convex space, has the Radon-Riesz property. Since the L p spaces are uniformly convex, this generalizes the previous example. In particular, every Hilbert space has the Radon-Riesz property.
• Strongly convex spaces have the Radon-Riesz property.
• There are Banach spaces with the Radon-Riesz property that are not strictly convex . To do renormalize the sequence space for one through . The Banach space is then an example of the desired kind.${\ displaystyle \ ell ^ {p}}$${\ displaystyle 1 ${\ displaystyle \ | (\ alpha _ {n}) _ {n} \ |: = \ max \ {| \ alpha _ {1} |, \ | (0, \ alpha _ {2}, \ alpha _ { 3}, \ ldots) \ | _ {p} \}}$${\ displaystyle (\ ell ^ {p}, \ | \ cdot \ |)}$
• The sequence space of the zero sequences with the supremum norm does not have the Radon-Riesz property. If the sequence denotes a 1 in the nth position and a 0 everywhere else, then obviously weak and applies , but because of there is no norm convergence.${\ displaystyle c_ {0}}$ ${\ displaystyle \ | \ cdot \ | _ {\ infty}}$${\ displaystyle e_ {n}}$${\ displaystyle e_ {1} -e_ {n + 1} \ rightarrow e_ {1}}$${\ displaystyle 1 = \ | e_ {1} -e_ {n + 1} \ | _ {\ infty} \ rightarrow \ | e_ {1} \ | _ {\ infty} = 1}$${\ displaystyle \ | (e_ {1} -e_ {n + 1}) - e_ {1} \ | _ {\ infty} = 1}$

## characterization

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: ${\ displaystyle S_ {X}}$${\ displaystyle \ {x \ in X; \, \ | x \ | = 1 \}}$${\ displaystyle X}$

• A normalized X has the Radon-Riesz property if and only if for every sequence in that converges weakly to a , it already follows.${\ displaystyle (x_ {n}) _ {n \ in \ mathbb {N}}}$${\ displaystyle S_ {X}}$${\ displaystyle x \ in S_ {X}}$${\ displaystyle \ | x_ {n} -x \ | \ rightarrow 0}$

Is the relative weak topology on bounded sets metrizable , for example, when the dual space separable , so that means that the weak topology and the standard topology coincide on the unit sphere.

## Individual evidence

1. ^ MM Day: Normed linear spaces , Springer-Verlag (1973), ISBN 3-540-06148-7
2. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , definition 2.5.26
3. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , Theorem 5.3.7
4. ^ 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