# 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 that converges weakly to , then a natural question arises. Does the sequence converge in perhaps a more desirable manner? That is, does the sequence converge to in norm? A canonical example of this property, and commonly used to illustrate the Schur property, is the sequence space.

## Definition

Suppose that we have a normed space (*X*, ||·||), an arbitrary member of *X*, and an arbitrary sequence in the space. We say that *X* has **Schur's property** if converging weakly to implies that . 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 *ℓ ^{1}* had the above property in his 1921 paper.[1]

## See also

- Radon-Riesz property for a similar property of normed spaces
- Schur's theorem

## Notes

- J. Schur, "Über lineare Transformationen in der Theorie der unendlichen Reihen",
*Journal für die reine und angewandte Mathematik*,**151**(1921) pp. 79-111

## References

- Megginson, Robert E. (1998),
*An Introduction to Banach Space Theory*, New York Berlin Heidelberg: Springer-Verlag, ISBN 0-387-98431-3