 Consider a family of operators
formula_3, formula_4,
with each formula_3
a bounded linear operator from formula_6 to formula_7.Denote

The family of operators
formula_9, formula_10
is "almost orthogonal" if

The Cotlar–Stein lemma states that if formula_3
are almost orthogonal,
then the series
formula_13
converges in the strong operator topology,
and that

If "R"1, ..., "R""n" is a finite collection of bounded operators, then

So under the hypotheses of the lemma,

It follows that 

and that

Hence the partial sums 

form a Cauchy sequence.The sum is therefore absolutely convergent with limit satisfying the stated inequality.