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 that converges weakly to , then a natural question arises. Does the sequence converge in perhaps a more desirable manner? If so, 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.
Suppose that we have a normed space , an arbitrary member of , and an arbitrary sequence in the space. We say that 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.
The space âÂÂ<sup>1</sup> of sequences whose series is absolutely convergent has the Schur property.
This property was named after the early 20th century mathematician Issai Schur who showed that âÂÂ<sup>1</sup> had the above property in his 1921 paper.