In mathematics, Schur's property, named after Issai Schur, is the property of

normed space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "lengt ...

s 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

(x_)

that converges weakly to

x

, 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

\ell_1

sequence space In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural ...

Definition

Suppose that we have a normed space (''X'', , , ·, , ),

x

an arbitrary member of ''X'', and

(x_)

an arbitrary sequence in the space. We say that ''X'' has Schur's property if

(x_)

converging weakly to

x

implies that

\lim_ \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.

Examples

The space ''ℓ¹'' of sequences whose series is absolutely convergent has the Schur property.

Name

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

Notes

References