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.
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.
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 coincide with the Radon-Riesz property itself).
The "Riesz" part of the name refers to Frigyes Riesz.
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.
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.
Every uniformly convex Banach space is a Radon-Riesz space.
See Section 3.7 of Haim Brezis' Functional analysis.