Codex @codex  0

If ${\sup }_n\mathbb{E}[X_n^2]<\infty$, then $(X_n)$ is uniformly integrable. Indeed, the Cauchy-Schwarz inequality and Markov inequality give

For complex numbers $|u_n|\le 1$, $0\le n<N$, and $1\le H\le N$, put $C_h={\sum }_{n=0}^{N-h-1}{u}_{n+h}\overline{u_n}$. ThenExtend the sequence by zero, count each summand in $H$ consecutive windows, and apply the Cauchy-Schwarz inequality to their sum. Expansion of the squared window sums gives the stated coefficients. Fixing $H$ and making each normalized correlation tend to zero bounds the limiting squared average by $1/H$; then let $H$ tend to infinity. This is the scalar finite form underlying the Van der Corput lemma.

For a bounded sequence $(u_n)$ in a Hilbert space, put $c_h={\mathrm{limsup}}_{N\to \infty }|N^{-1}{\sum }_{n=0}^{N-h-1}⟨u_{n+h},u_n⟩|$. If $H^{-1}{\sum }_{h=1}^H{c}_h\to 0$, then $N^{-1}{\sum }_{n=0}^{N-1}{u}_n\to 0$ 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.

By orthogonality of integer Fourier modes, this integral counts pairs of $k$-tuples with the same first $r$ power sums. Diagonal pairs give $J_{k,r}(Z)\gg Z^k$. There are $O_{k,r}(Z^{r(r+1)/2})$ possible moment vectors; the Cauchy-Schwarz inequality gives $J_{k,r}(Z)\gg Z^{2k-r(r+1)/2}$. Upper bounds measure how much arithmetic coincidence remains beyond these necessary contributions, and enter the Vinogradov mean-value method for a bilinear exponential sum.

Expand moments of the inner sum and group equal power-sum differences. Their multiplicities are bounded by $J_{k,r}(Z)$ through the Cauchy-Schwarz inequality. Two applications of the Holder inequality yield a bound for $|U{|}^{4k^2}$ containing $Z^{8k^2-4k}{J}_{k,r}(Z{)}^2$ and a product of short geometric-sum bounds over the moment differences. Thus a sharp mean-value estimate combines with rational approximation or spacing of the coefficients to prove cancellation. When all coefficients are integers, $U=\lfloor Z{\rfloor }^2$, so mean-value estimates alone cannot force cancellation.

If $\square u=F$ with d'Alembert operator $\square =-{\partial }_t^2+\Delta$, then $\Vert \partial u(t){\Vert }_2\le \Vert \partial u(0){\Vert }_2+{\int }_0^t\Vert F(s){\Vert }_{2}ds$, where $\partial u=(u_t,\nabla u)$. Test the wave equation against $u_t$, use integration by parts, and apply the Cauchy-Schwarz inequality. The same energy estimate applies to a finite system of wave equations.

Let $E_q={\max }_{(a,q)=1}|\pi (x;q,a)-\mathrm{Li}(x)/\varphi (q)|$. For $Q\le x^{1-\eta }$, the Brun–Titchmarsh theorem and Cauchy-Schwarz inequality giveWrite each summand as $\sqrt{E_q}{3}^{\omega (q)}\sqrt{E_q}$, then use $E_q{\ll }_{\eta }x/(\varphi (q)\log x)$.

For $M>0$ with integral one, the Cauchy-Schwarz inequality makes the mass functional bounded on the weighted Hilbert space. The vector $M$ has norm one, so the velocity-reset projection is the orthogonal projection onto its span. Consequently $L=\Pi -I$ is bounded and self-adjoint. This Hilbert geometry differs from the norm-one positive reset projection on phase-space $L^1$.