Schur's property
Encyclopedia
In mathematics
, Schur's property, named after Issai Schur
, is the property of normed space
s that is satisfied precisely if weak convergence
of sequences entails convergence in norm.
that converges weakly to 
(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
in norm?
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.
who showed that ℓ1 had the above property in his 1921 paper.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Schur's property, named after Issai Schur
Issai Schur
Issai Schur was a mathematician who worked in Germany for most of his life. He studied at Berlin...
, is the property of normed space
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
s that is satisfied precisely if weak convergence
Weak convergence
In mathematics, weak convergence may refer to:* The weak convergence of random variables of a probability distribution.* The weak convergence of a sequence of probability measures....
of sequences entails convergence in norm.
Motivation
When we are working in a normed space X and we have a sequence that converges weakly to (see weak convergenceWeak convergence
In mathematics, weak convergence may refer to:* The weak convergence of random variables of a probability distribution.* The weak convergence of a sequence of probability measures....
), then a natural question arises. Does the sequence converge in perhaps a more desirable manner? That is, does the sequence converge to
in norm?Definition
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.Name
This property was named after the early 20th century mathematician Issai SchurIssai Schur
Issai Schur was a mathematician who worked in Germany for most of his life. He studied at Berlin...
who showed that ℓ1 had the above property in his 1921 paper.