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 of sequences entails convergence in norm.

Motivation[edit]

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 Shur property, is the l1 sequence space.

Definition[edit]

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\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[edit]

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]

See also[edit]

Notes[edit]

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

References[edit]

  • Megginson, Robert E. (1998), An Introduction to Banach Space Theory, New York Berlin Heidelberg: Springer-Verlag, ISBN 0-387-98431-3