Radon–Riesz property

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.

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, $\lim _{{n\to \infty }}\Vert x_{n}-x\Vert =0$.

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. The "Riesz" part of the name refers to Frigyes Riesz. He also made some use of this property in the 1920s.

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

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

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.

See also

• Johann Radon
• Frigyes Riesz
• Hilbert space or Banach space theory
• Weak topology
• Normed space
• Functional analysis
• Schur's property

References

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