In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Schur's property, named after
Issai Schur
Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the Humboldt University of Berlin, University of Berlin. He obtained his doctorate in 1901, became lecturer i ...
, is the property of
normed space
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898.
The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war p ...
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
that converges weakly to
, then a natural question arises. Does the sequence converge in perhaps a more desirable manner? If so, 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
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 num ...
.
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.
Examples
The space ''ℓ
1'' of sequences whose series is absolutely convergent has the Schur property.
Name
This property was named after the early 20th century mathematician
Issai Schur
Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the Humboldt University of Berlin, University of Berlin. He obtained his doctorate in 1901, became lecturer i ...
who showed that ''ℓ
1'' 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]
See also
*
Radon-Riesz property for a similar property of normed spaces
*
Schur's theorem
Notes
References
*
{{DEFAULTSORT:Schur's Property
Functional analysis
Issai Schur