Radon–Riesz property

Last updated May 04, 2019

The Radon–Riesz property is a mathematical property for normed spaces that helps ensure convergence in norm. Given two assumptions (essentially weak convergence and continuity of norm), we would like to ensure convergence in the norm topology.

In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to". If such a limit exists, the sequence is called convergent. A sequence which does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of analysis ultimately rests.

Definition

Suppose that (X, ||·||) is a normed space. We say that X has the Radon–Riesz property (or that X is a Radon–Riesz space) if whenever $(x_{n})$ is a sequence in the space and $x$ is a member of X such that $(x_{n})$ converges weakly to $x$ and $\lim _{n\to \infty }\Vert x_{n}\Vert =\Vert x\Vert$ , then $(x_{n})$ converges to $x$ in norm; that is, $\lim _{n\to \infty }\Vert x_{n}-x\Vert =0$ .

In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis.

Other names

Although it would appear that Johann Radon was one of the first to make significant use of this property in 1913, M. I. Kadets and V. L. Klee also used versions of the Radon–Riesz property to make advancements in Banach space theory in the late 1920s. It is common for the Radon–Riesz property to also be referred to as the Kadets–Klee property or property (H). According to Robert Megginson, the letter H does not stand for anything. It was simply referred to as property (H) in a list of properties for normed spaces that starts with (A) and ends with (H). This list was given by K. Fan and I. Glicksberg (Observe that the definition of (H) given by Fan and Glicksberg includes additionally the rotundity of the norm, so it does not coincides with the Radon-Riesz property itself). The "Riesz" part of the name refers to Frigyes Riesz. He also made some use of this property in the 1920s.

Johann Karl August Radon was an Austrian mathematician. His doctoral dissertation was on the calculus of variations.

Mikhail Iosiphovich Kadets was a Soviet-born Jewish mathematician working in analysis and the theory of Banach spaces.

In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

It is important to know that the name "Kadets-Klee property" is used sometimes to speak about the coincidence of the weak topologies and norm topologies in the unit sphere of the normed space.

Examples

1. Every real Hilbert space is a Radon–Riesz space. Indeed, suppose that H is a real Hilbert space and that $(x_{n})$ is a sequence in H converging weakly to a member $x$ of H. Using the two assumptions on the sequence and the fact that

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.

\langle x_{n}-x,x_{n}-x\rangle =\langle x_{n},x_{n}\rangle -\langle x_{n},x\rangle -\langle x,x_{n}\rangle +\langle x,x\rangle ,

and letting n tend to infinity, we see that

\lim _{n\to \infty }{\langle x_{n}-x,x_{n}-x\rangle }=0.

Thus H is a Radon–Riesz space.

2. Every uniformly convex Banach space is a Radon-Riesz space. See Section 3.7 of Haim Brezis' Functional analysis.

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References

Megginson, Robert E. (1998), An Introduction to Banach Space Theory, New York Berlin Heidelberg: Springer-Verlag, ISBN 0-387-98431-3

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