In analysis, a branch of mathematics, Hilbert's inequality states that
 formula_1
for any sequence "u"1,"u"2... of complex numbers. It was first demonstrated by David Hilbert with the constant instead of ; the sharp constant was found by Issai Schur. It implies that the discrete Hilbert transform is a bounded operator in "ℓ"2.
Formulation.
Let ("u""m") be a sequence of complex numbers. If the sequence is infinite, assume that it is square-summable:
formula_2
Hilbert's inequality (see ) asserts that
 formula_1
Extensions.
In 1973, Montgomery &amp; Vaughan reported several generalizations of Hilbert's inequality, considering the bilinear forms
 formula_4
and
 formula_5
where "x"1,"x"2...,"x""m" are distinct real numbers modulo 1 (i.e. they belong to distinct classes in the quotient group R/Z) and "λ"1...,"λ""m" are distinct real numbers. Montgomery &amp; Vaughan's generalizations of Hilbert's inequality are then given by
 formula_6
and
 formula_7
where
formula_8
formula_9
is the distance from "s" to the nearest integer, and min+ denotes the smallest positive value. Moreover, if
formula_10
then the following inequalities hold:
 formula_11
and
 formula_12