Schur's property

In mathematics, Schur's property, named after Issai Schur, is the property of normed spaces that is satisfied precisely if weak convergence of sequences entails convergence in norm.

When we are working in a normed space X and we have a sequence ${\displaystyle (x_{n})}$ that converges weakly to ${\displaystyle x}$ (see weak convergence), then a natural question arises. Does the sequence converge in perhaps a more desirable manner? That is, does the sequence converge to ${\displaystyle x}$ in norm? A canonical example of this property, and commonly used to illustrate the Schur property, is the ${\displaystyle \ell _{1}}$ sequence space.

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