= Bilinear sum
{title2=$B(a,b)=\sum_{m,n}a_m b_nK(m,n)$}
A <bilinear sum> has the form
$$
B(a,b)=\sum_{m\in M}\sum_{n\in N}a_m b_n K(m,n),
$$
where $M,N$ are finite index sets and the coefficient <sequences> $a,b$ can be controlled separately. It is a <bilinear form> in these two coefficient lists. In <analytic number theory>, the function $K(m,n)$ often oscillates with the product $mn$, as in a <Type II sum>.
The <Cauchy-Schwarz inequality> eliminates one coefficient list and exposes correlations:
$$
|B|^2\le\left(\sum_m|a_m|^2\right)\sum_{n,n'}b_n\overline{b_{n'}}\sum_m K(m,n)\overline{K(m,n')}.
$$
Useful cancellation comes from small off-diagonal correlations, not from the bilinear notation alone. For example, take $m,n=0,\ldots,q-1$ and $K(m,n)=e^{2\pi imn/q}$. <Orthogonality of roots of unity> makes the inner correlation $q$ for $n=n'$ and zero otherwise, giving $|B|\le\sqrt q\,\lVert a\rVert_2\lVert b\rVert_2$. The termwise <triangle inequality> only gives $q\lVert a\rVert_2\lVert b\rVert_2$. This explains why <Fourier analysis> and <large sieve> estimates are effective tools for <bilinear sums>.
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