Radon–Riesz property

The Radon–Riesz property is a mathematical property for normed spaces that helps ensure convergence in norm. 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 ${\displaystyle (x_{n})}$ is a sequence in the space and ${\displaystyle x}$ is a member of X such that ${\displaystyle (x_{n})}$ converges weakly to ${\displaystyle x}$ and ${\displaystyle \lim _{n\to \infty }\Vert x_{n}\Vert =\Vert x\Vert }$, then ${\displaystyle (x_{n})}$ converges to ${\displaystyle x}$ in norm; that is, ${\displaystyle \lim _{n\to \infty }\Vert x_{n}-x\Vert =0}$.

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