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.
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 Schur property, is the
ℓ
1
sequence space.
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
→
∞
∥
x
n
−
x
∥
=
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.
This property was named after the early 20th century mathematician Issai Schur who showed that ℓ1 had the above property in his 1921 paper.
