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.
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.
This property was named after the early 20th century mathematician Issai Schur who showed that ℓ1 had the above property in his 1921 paper.
- J. Schur, "Über lineare Transformationen in der Theorie der unendlichen Reihen", Journal für die reine und angewandte Mathematik, 151 (1921) pp. 79-111