Radon-Riesz theorem

The Radon-Riesz is a mathematical theorem in measure theory , the statements is true about when the weak convergence ${\displaystyle {\mathcal {L}}^{p}}$ 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 . ${\displaystyle {\mathcal {L}}^{p}}$

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. ${\displaystyle p\in (1,\infty )}$ ${\displaystyle f\colon X\to \mathbb {K} ,(f_{n}\colon X\to \mathbb {K} )_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {L}}^{p}}$ ${\displaystyle \|\cdot \|_{p}}$ ${\displaystyle {\mathcal {L}}^{p}}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \lim _{n\to \infty }\|f_{n}\|_{p}=\|f\|_{p}}$

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 .