Schur's property

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

When we are working in a normed space X and we have a sequence $(x_{n})$ 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.

Definition

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.

Examples

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

Schur's Property in Group Theory

Finite Groups

Consider the symmetric group S3. This group has irreducible representations of dimensions 1 and 2 over C. If ρ is an irreducible representation of S3 of dimension 1 (trivial representation), then Schur's Lemma tells us that any S3-homomorphism from this representation to any other representation (including itself) is either an isomorphism or zero. In particular, if ρ is a 1-dimensional representation and σ is a 2-dimensional representation, any homomorphism from ρ to σ must be zero because these two representations are not isomorphic.

Infinite Groups

For the group Z (the group of integers under addition), every irreducible representation is 1-dimensional. If V and W are 1-dimensional representations of Z, then Schur’s Lemma implies that any homomorphism between them is an isomorphism (unless the homomorphism is zero, which is not possible in this case).

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.^[1]

Notes

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

References

Megginson, Robert E. (1998), An Introduction to Banach Space Theory, New York Berlin Heidelberg: Springer-Verlag, ISBN 0-387-98431-3
Simon, B. (2015), Representations of Finite and Compact Groups. Springer.

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

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