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.
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 is a sequence in the space and is a member of X such that converges weakly to and , then converges to in norm; that is, .
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. The "Riesz" part of the name refers to Frigyes Riesz. He also made some use of this property in the 1920s.
Every real Hilbert space is a Radon–Riesz space. Indeed, suppose that H is a real Hilbert space and that is a sequence in H converging weakly to a member of H. Using the two assumptions on the sequence and the fact that
and letting n tend to infinity, we see that
Thus H is a Radon–Riesz space.
- Johann Radon
- Frigyes Riesz
- Hilbert space or Banach space theory
- Weak topology
- Normed space
- Functional analysis
- Schur's property