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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

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When we are working in a normed space X and we have a sequence that converges weakly to , then a natural question arises. Does the sequence converge in perhaps a more desirable manner? That is, does the sequence converge to in norm? A canonical example of this property, and commonly used to illustrate the Schur property, is the sequence space.

Definition

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Suppose that we have a normed space (X, ||·||), an arbitrary member of X, and an arbitrary sequence in the space. We say that X has Schur's property if converging weakly to implies that . 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

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The space 1 of sequences whose series is absolutely convergent has the Schur property.

Schur's Property in Group Theory

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Finite Groups

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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

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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

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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

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Notes

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  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

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  • 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.