Radon-Riesz theorem

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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 .


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.