# 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.

## Motivation

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

## Definition

Suppose that we have a normed space (X, ||·||), $x$ an arbitrary member of X, and $(x_{n})$ an arbitrary sequence in the space. We say that X has Schur's property if $(x_{n})$ converging weakly to $x$ implies that $\lim _{n\to \infty }\Vert x_{n}-x\Vert =0$ . In other words, the weak and strong topologies share the same convergent sequences. Note however that weak and strong topologies are always distinct in infinite-dimensional space.

## Name

This property was named after the early 20th century mathematician Issai Schur who showed that 1 had the above property in his 1921 paper.

## See also

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