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A martingale is square integrable when its value at every time belongs to L2 space. This implies ordinary integrability by Cauchy-Schwarz inequality. The condition is separate from a uniform bound on these second moments over an infinite horizon. An Itô integral is square integrable on a finite interval when its predictable integrand has finite expected integrated square, by Itô isometry.

For an adapted stationary process with the displayed bounds, the Itô formula and stationarity give ${\int }_0^t(\mathbb{E}[Z_s{g}_s]+\lambda )ds=0$. The Cauchy-Schwarz inequality makes each integrand nonnegative, so equality holds almost everywhere. Then $\mathbb{E}(g_s+\lambda Z_s{)}^2\le 0$, proving $g=-\lambda Z$ for almost every time and outcome. Thus the process solves the Ornstein-Uhlenbeck stochastic differential equation and its initial value, rather than stationarity alone, determines its coupling with W.

For a fitted local first-order response $\hat{m}(x)={\hat{\beta }}_0+{\hat{b}}^{T}x$, the unit direction maximizing its directional derivative in coded Euclidean distance is $d=\hat{b}/\Vert \hat{b}\Vert$, by the Cauchy-Schwarz inequality. Run sequential feasible settings $x_0+hd$, increase $h$ while measured response improves, and refit near promising settings. Physical increments are $h_{j}h{d}_j$, so coded and unscaled physical gradients need not give the same experimental path. A near-zero gradient calls for further modeling rather than division by zero.

For a nonzero gradient, this unit direction minimizes the directional derivative among Euclidean unit directions. The Cauchy-Schwarz inequality gives $\nabla f(x{)}^{T}p\ge -\Vert \nabla f(x){\Vert }_2$, with equality for the displayed direction. The steepest descent method usually uses the equivalent unnormalized direction $-\nabla f(x)$ and chooses a separate step length. A different choice of norm can change the steepest direction.

On the graph of a function $z=f(x,y)$, assume $f$ is differentiable, its planar gradient $p$ is nonzero, and speed $U>0$ is measured by the three-dimensional Euclidean norm. The upward unit normal vector is $n=(-p,1)/\sqrt{1+|p{|}^2}$. Projecting the downward vertical vector $-e_z$ onto the tangent plane gives $-(p,|p{|}^2)/(1+|p{|}^2)$. By the Cauchy-Schwarz inequality, normalizing this projected vector produces the tangent direction with greatest height decrease at the prescribed speed, giving the displayed velocity. Its vertical component is $-U|p|/\sqrt{1+|p{|}^2}$, and its horizontal direction is parallel to $-\nabla f$. Consequently its planar paths agree with negative gradient flow up to a change of time, although the speeds differ. Where $p=0$, every tangent direction has zero first-order height change and this normalized formula is undefined.

For density of states proportional to ${ϵ}^a$, $a>-1$, and positive temperature, set $q=\mu /(kT)$, $n(x)=(1+e^{x-q}{)}^{-1}$, $b=a+1$ and $M_j={\int }_0^{\infty }{x}^{a+j}n(1-n)dx$. Differentiation and integration by parts giveThe inequality is strict Cauchy-Schwarz inequality, since $x$ is not constant under this positive weight. Thus fixed entropy per particle forces fixed $\mu /T$. This is the injectivity step behind the adiabatic scaling of an ideal Fermi gas.

A function into a Banach space is strongly measurable if, outside a null set, it is a pointwise norm limit of measurable simple functions. A measurable function with values in a separable Banach space is strongly measurable: approximate values by a countable collection of balls of shrinking radii and then truncate the resulting countably valued approximants to simple functions. A strongly measurable function is Bochner integral integrable exactly when its norm is integrable. Finite second moments under a probability measure therefore imply existence of its Bochner integral expected value by the Cauchy-Schwarz inequality.

For a weak solution of $-\mathrm{div}(A\nabla u)=0$ with real symmetric coefficients satisfying the two-sided uniform ellipticity bounds $\lambda I\le A\le \Lambda I$, the displayed Caccioppoli inequality holds with $C=4{\Lambda }^2/{\lambda }^2$ for a smooth cutoff function $\phi$ of compact support. Put $w=(u-k{)}^{+}$ and test the weak formulation with ${\phi }^{2}w$. The Sobolev chain rule gives $\nabla w=1_{\{u>k\}}\nabla u$, and Cauchy-Schwarz inequality yields $\lambda \int {\phi }^2|\nabla w{|}^2\le 2\Lambda (\int {\phi }^2|\nabla w{|}^2{)}^{1/2}(\int w^2|\nabla \phi {|}^2{)}^{1/2}$. Divide and square. The constant is independent of the level $k$.

Let $S$ be a real symmetric positive definite matrix, $C>0$ and $q\ne 0$. The Cauchy-Schwarz inequality in the $S$-weighted inner product gives $q^{T}x=(S^{-1}q{)}^{T}Sx\le \sqrt{q^T{S}^{-1}q}\sqrt{x^{T}Sx}$. Equality is attained by a positive multiple of $S^{-1}q$. Thus the displayed maximum holds. For a translated ellipsoid, add $q^{T}a$ for its centre $a$. This computes a coordinate extremum without constructing a rotation matrix.

For a symmetric positive-definite matrix $A$, solving $A{x}_{\ast }=b$ is equivalent to minimizing the quadratic form with linear termWriting the error vector $e=x-x_{\ast }$ and expanding gives the exact energy norm identityThe cross term vanishes because $A{x}_{\ast }=b$. With $\eta =Se$ and $B=S^{-1}A{S}^{-1}$, the same quantity is ${\eta }^{T}B\eta /2$. Symmetric positive-definite preconditioning changes coordinates without changing the energy assigned to a corresponding error vector.

In an orthonormal eigenbasis of $A$, the level set $e^{T}Ae=t>0$ has semiaxis lengths $\sqrt{t/{\lambda }_i}$. Its longest-to-shortest axis ratio is $\sqrt{{\kappa }_2(A)}$. A large spectral condition number of a positive-definite matrix therefore appears geometrically as a thin ellipsoid. Good scaling makes the transformed ellipsoid rounder, allowing a simple search direction to improve all coordinates together.

The negative gradient of $F$ is the linear-system residual $r$. Relative to the norm $\Vert v{\Vert }_M=\sqrt{v^{T}Mv}$, a steepest direction is $z=M^{-1}r$: the Cauchy-Schwarz inequality applied to $r^{T}v=(S^{-1}r{)}^T(Sv)$ shows that the greatest decrease per unit $M$-length occurs along this direction. Equivalently, ordinary gradient descent in $y=Sx$ becomes preconditioned steepest descent in $x$.^[ref]

Suppose a probability density function and its third derivative are bounded. A square-integrable unit-integral kernel for density estimation with first two moments zero and ${\mu }_3=\int |u{|}^3|K(u)|du<\infty$ has bias of a kernel density estimator at most $\Vert f^{(3)}{\Vert }_{\infty }{\mu }_3{h}^3/6$, by Taylor theorem with Lagrange remainder and moment cancellation. Independence gives variance at most $\Vert f{\Vert }_{\infty }\Vert K{\Vert }_2^2/(nh)$. The Cauchy-Schwarz inequality bounds the expected centred absolute deviation by the square root of this variance. Adding the bias proves the inequality uniformly in $x$. The choice $h=n^{-1/7}$ gives expected absolute error $O(n^{-3/7})$.

For a complex function on $X_1\times X_2\times X_3$, its eighth power is the average of the conjugated product over the eight vertices of a coordinate cube. Three successive applications of the Cauchy-Schwarz inequality show that it bounds correlation with a product of three bounded functions, each omitting one coordinate. This is the analytic estimate behind the tetrahedron counting lemma.

For forward motion with nonnegative speed $v(t)$ over duration $T>0$ and distance $L={\int }_0^{T}v(t)dt>0$, define the time-averaged speed $V_t=L/T$ and distance-averaged speed $V_x=L^{-1}\int vdx$. The change of variables $dx=vdt$ gives $V_x={\mathbb{E}}_t[v^2]/{\mathbb{E}}_t[v]$, where ${\mathbb{E}}_t$ averages uniformly over time. ThereforeEquality holds exactly for constant speed almost everywhere. This also follows directly from the Cauchy-Schwarz inequality applied to $v$ and one on the time interval. The reason for the difference is that distance averaging puts more weight on periods when the vehicle covers more distance per unit time. These formulas assume a finite second time moment; if that moment is infinite, the distance average is infinite and the inequality holds in the extended sense.

For the unnormalized Hellinger distance, $\mathrm{TV}(P,Q)\le h(P,Q)\le \sqrt{\mathrm{KL}(P,Q)}$. Factor $|p-q|=|\sqrt{p}-\sqrt{q}|(\sqrt{p}+\sqrt{q})$ and use the Cauchy-Schwarz inequality to bound the total variation distance. For the other bound, $\log t\le t-1$ gives $p\log (p/q)\ge 2(p-\sqrt{pq})$, whose integral is the squared unnormalized Hellinger distance. Infinite Kullback-Leibler divergence is allowed.

Let a tripartite graph have nonempty parts $X,Y,Z$, pair edge density of a bipartite graph values $\alpha ,\beta ,\gamma$ on $XY,YZ,XZ$, and $\Vert G(X,Y)-\alpha {\Vert }_{\square }\le c$. If every $z\in Z$ has exactly $\beta |Y|$ neighbors in $Y$, its normalized triangle count $\tau$ obeysThe constant-degree assumption makes the contribution of the constant $\alpha$ exactly $\alpha \beta \gamma$. For each fixed $z$, apply the bilinear correlation bound for the box norm to the indicator functions of its two vertex neighbourhoods. Their squared $L^2$ norms are $\beta$ and the relative $X$-degree of $z$. Average over $z$ and use the Cauchy-Schwarz inequality to bound the mean square root of that degree by $\sqrt{\gamma }$.

For a nonnegative random variable with finite mean $\mu$ and $0\le \theta \le 1$, the part of the expectation on $X\le \theta \mu$ is at most $\theta \mu$. Subtracting gives $\mathbb{E}[X{1}_{\{X>\theta \mu \}}]\ge (1-\theta )\mu$. If the second moment is positive and finite, applying the Cauchy-Schwarz inequality to $X$ and the indicator variable gives the Paley-Zygmund inequality $\mathbb{P}(X>\theta \mu )\ge (1-\theta {)}^2{\mu }^2/\mathbb{E}[X^2]$.

For a nonnegative submartingale with maximum $M_t$, the first-moment maximal estimate is $\lambda \mathbb{P}(M_t\ge \lambda )\le \mathbb{E}[X_t{1}_{\{M_t\ge \lambda \}}]$. Integrate $2\lambda$ times the tail probability only up to $R$, obtaining $\mathbb{E}(M_t\wedge R{)}^2\le 2\mathbb{E}[X_t(M_t\wedge R)]$. The Cauchy-Schwarz inequality gives $\Vert M_t\wedge R{\Vert }_2\le 2\Vert X_t{\Vert }_2$, and monotone convergence gives the result as $R\to \infty$. Truncation avoids assuming beforehand that the maximum has a finite second moment.

// Target: brownian-motion.bigb

A Type II sum has the form ${\sum }_{d\sim D}{\sum }_{m\sim M}{a}_d{b}_{m}f(dm)$ with two nontrivially weighted variables, both long enough for a Cauchy-Schwarz inequality and a large sieve estimate. The Vaughan identity supplies this bilinear structure for sums weighted by the Von Mangoldt function.

For self-adjoint operators $A,B$ and a normalized quantum state in their relevant product operator domains, put $u=A\psi$, $v=B\psi$, $z=⟨u,v⟩$. With the inner product linear in its second entry, $⟨C⟩=2\mathrm{Im}z$. The Cauchy-Schwarz inequality givesEquality holds exactly when $u,v$ are linearly dependent and $\mathrm{Re}z=0$. This includes either zero vector; if both vectors are nonzero it is equivalent to $B\psi =i\lambda A\psi$ for real nonzero $\lambda$. Applying this inequality to the centred observables $A-⟨A⟩I$ and $B-⟨B⟩I$ gives the Robertson uncertainty principle.

For nonzero $f$ with $f,xf,f^{\prime }\in L^2(\mathbb{R})$, integration by parts gives $\Vert f{\Vert }_2^2=-2\mathrm{Re}⟨f^{\prime },xf⟩$. The Cauchy-Schwarz inequality and Plancherel theorem yield the moment product bound $1/4$ with the angular-frequency Fourier convention. Equality occurs exactly for $f(x)=C{e}^{-a{x}^2/2}$ with $a>0$ real and $C\ne 0$. Translation, modulation or a quadratic phase increase one of these uncentred moments; the equality family is narrower than that for centred variances.