# Radon-Riesz theorem

The **Radon-Riesz** is a mathematical theorem in measure theory , the statements is true about when the weak convergence and convergence in pth mean by functional consequences are equivalent. In this context, the convergence in the p-th mean is also referred to as norm convergence or strong convergence in , as is usual in functional analysis . The set is named after Johann Radon and Frigyes Riesz .

## statement

Let it be and off and denote the norm. Then converges in the p-th mean if and only if weakly converges and is.

## Radon Riesz property

The Radon-Riesz theorem gives its name to the Radon-Riesz property . This is a property of normalized spaces in functional analysis . A normalized space has the Radon-Riesz property if and only if in this space the norm convergence of a sequence is equivalent to the fact that the sequence converges weakly and the sequence of norms converges to the norm of the limit value.

## literature

- Jürgen Elstrodt: Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .