= Van der Corput lemma
{disambiguate=Hilbert space sequences}
{c}
= Van der Corput lemma
{c}
{synonym}
For a bounded sequence $(u_n)$ in a <Hilbert space>, put $c_h=\limsup_{N\to\infty}|N^{-1}\sum_{n=0}^{N-h-1}\langle u_{n+h},u_n\rangle|$. If $H^{-1}\sum_{h=1}^Hc_h\to0$, then $N^{-1}\sum_{n=0}^{N-1}u_n\to0$ in norm. Since the $c_h$ are bounded, it suffices that $(c_h)$ has <convergence in density of a sequence> to zero. The proof applies the <Cauchy-Schwarz inequality> to an average of shifted sums.
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