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 ( x n ) that converges weakly to 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 x in norm? A canonical example of this property, and commonly used to illustrate the Schur property, is the ℓ 1 sequence space.
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 → ∞ ∥ x n − x ∥ = 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.
This property was named after the early 20th century mathematician Issai Schur who showed that ℓ1 had the above property in his 1921 paper.
