Radon Riesz property

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The Radon-Riesz property is a mathematical property for normed spaces that helps ensure convergence in norm. Essentially, given two assumptions (essentially weak convergence and continuity of norm), we would like to ensure convergence in the norm topology.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 (xn) is a sequence in the space and x is a member of X such that (xn) converges converges weakly to x and lim_{ntoinfty} Vert x_n Vert = Vert xVert , then (xn) converges to x in norm; that is, lim_{ntoinfty} Vert x_n - xVert = 0 . Although it would appear that Johann Radon was one of the first to make significant use of the 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).