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. ${\displaystyle (\Omega ,{\mathfrak {S}},\mu )}$ ${\displaystyle 1<p<\infty }$ ${\displaystyle L^{p}(\Omega ,{\mathfrak {S}},\mu )}$
• 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. ${\displaystyle \ell ^{p}}$ ${\displaystyle 1<p<\infty }$ ${\displaystyle \|(\alpha _{n})_{n}\|:=\max\{|\alpha _{1}|,\|(0,\alpha _{2},\alpha _{3},\ldots )\|_{p}\}}$ ${\displaystyle (\ell ^{p},\|\cdot \|)}$
• 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