Radon–Riesz property

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 → ∞ ∥ x n ∥ = ∥ x ∥ , then ( x n ) converges to x in norm; that is, lim n → ∞ ∥ x n − x ∥ = 0 .

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.

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.

Example

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

and letting n tend to infinity, we see that

Thus H is a Radon–Riesz space.