Dunford-Schwartz theorem
Encyclopedia
In mathematics
, particularly functional analysis
, the Dunford–Schwartz theorem, named after Nelson Dunford
and Jacob T. Schwartz states that the averages of powers of certain norm-bounded operator
s on L1
converge
in a suitable sense.
Theorem. Let
be a linear operator from 
to 
with 
and 
. Then
exists almost everywhere
for all
.
The statement is no longer true when the boundedness condition is relaxed to even
.
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...
, particularly functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, the Dunford–Schwartz theorem, named after Nelson Dunford
Nelson Dunford
Nelson Dunford was an American mathematician, known for his work in functional analysis, namely integration of vector valued functions, ergodic theory, and linear operators. The Dunford decomposition, Dunford–Pettis property, and Dunford-Schwartz theorem bear his name.He studied mathematics at the...
and Jacob T. Schwartz states that the averages of powers of certain norm-bounded operator
Bounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
s on L1
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
converge
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
in a suitable sense.
Theorem. Let
be a linear operator from to with and . Then
exists almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
for all
.The statement is no longer true when the boundedness condition is relaxed to even
.