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 . ${\ displaystyle {\ mathcal {L}} ^ {p}}$
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}}}$ ${\ displaystyle \ lim _ {n \ to \ infty} \ | f_ {n} \ | _ {p} = \ | f \ | _ {p}}$