Schur's property

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In mathematics, Schur's property, named after Issai Schur, is the property of normed spaces that is satisfied precisely if weak convergence[disambiguation needed] of sequences entails convergence in norm.

Motivation

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

Definition

Suppose that we have a normed space (X, ||·||), ${\displaystyle x}$ an arbitrary member of X, and ${\displaystyle (x_{n})}$ an arbitrary sequence in the space. We say that X has Schur's property if ${\displaystyle (x_{n})}$ converging weakly to ${\displaystyle x}$ implies that ${\displaystyle \lim _{n\to \infty }\Vert x_{n}-x\Vert =0}$. 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.[1]

Notes

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