Hilbert's Inequality
Given a Positive Sequence 
,
where the
s are Real and ``square summable.''Another Inequality known as Hilbert's applies to Nonnegative sequences and 
,
unless all
or all 
are 0. If 
and 
are Nonnegative integrable functions, then the integralform is
The constant
is the best possible, in the sense that counterexamples can be constructed for anysmaller value.
References
Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 308-309, 1988.
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