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For a nonnegative bounded weak subsolution and $\beta >1$, test with ${\eta }^2(u+\epsilon {)}^{\beta -1}$. Uniform ellipticity and Cauchy-Schwarz inequality give $\int |Du{|}^2{u}^{\beta -2}{\eta }^2\le 4{\Lambda }_{\ast }^2/[{\lambda }^2(\beta -1{)}^2]\int u^{\beta }|D\eta {|}^2$, with ${\Lambda }_{\ast }$ an operator norm bound for the coefficient matrix. The Sobolev chain rule and $\epsilon \downarrow 0$ justify fractional powers and zero values. An entrywise bound $\Lambda$ yields ${\Lambda }_{\ast }\le n\Lambda$.

For $a,b\ge 0$ and $q>2$, Cauchy-Schwarz inequality applied to the integral of $(q/2)s^{q/2-1}$ gives $(a^{q/2}-b^{q/2}{)}^2\le q^2(a^{q-1}-b^{q-1})(a-b)/(4(q-1))$. Integrating against the symmetric two-point kernel measure and taking the energy limit proves the displayed inequality.

// Target: analysis.bigb

For reduced $a/q$ with $|\alpha -a/q|\le q^{-2}$ and $1\le q\le N$, the displayed bound follows by splitting Vaughan identity with parameters $U=V=N^{2/5}$. Type I sums use short geometric-sum bounds; type II sums use the Cauchy-Schwarz inequality and rational spacing to control bilinear cancellation. A badly approximable number has a Dirichlet approximant with $q\asymp N^{1/2}$, yielding $O(N^{4/5}{\log }^{4}N)$ and hence $O(N^{9/10})$.

// Target: analytic-number-theory.bigb

For two normalized pure states, the difference of their projectors has two eigenvalues $\pm \sqrt{1-|⟨u|v⟩{|}^2}$, giving that trace distance. Write a mixed density operator as $\rho ={\sum }_j{q}_j|u_j⟩⟨u_j|$. Convexity of the trace norm and weighted Cauchy-Schwarz inequality give $D(\rho ,|v⟩⟨v|)\le {\sum }_j{q}_j\sqrt{1-|⟨u_j|v⟩{|}^2}\le \sqrt{1-⟨v|\rho |v⟩}$. Thus a small probability of failing the pure-state projector test forces trace-distance proximity, including for correlated multipartite outputs.

Let $B$ be a real square matrix and suppose $x^{T}Bx\ge c|x{|}^2$ for every real vector $x$, with $c>0$. The Cauchy-Schwarz inequality gives $x^{T}Bx\le |x||Bx|$, hence $|Bx|\ge c|x|$ for all $x$. In particular the kernel is zero, so $B$ is an invertible matrix. Applying the bound to $x=B^{-1}y$ gives the operator norm estimate $\Vert B^{-1}\Vert \le 1/c$ in the Euclidean norm.

Symmetry of $B$ is unnecessary: $x^{T}Bx=x^{T}Hx$ for its symmetric part of a matrix $H=(B+B^T)/2$. Thus the hypothesis is exactly a positive lower bound on that symmetric matrix, even when the antisymmetric part is nonzero.

Let ${\delta }_t=\sqrt{p_t}-\sqrt{p}=(t/2)g\sqrt{p}+r_t$, with $\Vert r_t{\Vert }_2=o(|t|)$. Since $p_t-p=2\sqrt{p}{\delta }_t+{\delta }_t^2$, the Cauchy-Schwarz inequality gives $\Vert p_t-p-tpg{\Vert }_1\le 2\Vert r_t{\Vert }_2+\Vert {\delta }_t{\Vert }_2^2=o(|t|)$. Thus differentiability in quadratic mean suffices to differentiate bounded density integrals, even when the density itself has no useful pointwise derivative.

Suppose $S=S_I+S_R$ has finite positive variance $v$, and a matching quota share reinsurance contract exists with fraction ${\alpha }^{\ast }=\sqrt{\mathrm{Var}(S_I)/v}\in [0,1]$. The Cauchy-Schwarz inequality bounds $\mathrm{Cov}(S,S_I)\le {\alpha }^{\ast }v$. Expanding $\mathrm{Var}(S-S_I)$ then proves that the matching quota share minimizes the other party's variance, and hence the sum of party variances. Claimwise retentions $0\le h(x)\le x$ in a compound Poisson distribution automatically satisfy the required variance range, because $\mathrm{Var}(S_I)=\lambda \mathbb{E}[h(X{)}^2]\le \lambda \mathbb{E}[X^2]$.

Let a solenoidal flow be confined to a sphere $V$, with the insulating boundary condition for the radial magnetic scalar. Define $N={\int }_V{P}^2$, $D={\int }_{{\mathbb{R}}^3}|\nabla P{|}^2$, $M={\int }_{{\mathbb{R}}^3}|\mathbf{B}{|}^2$ and $q={\max }_V|\mathbf{u}\cdot \mathbf{x}|$. Integration by parts inside and outside the sphere givesThe Cauchy-Schwarz inequality therefore makes $q^2\ge {\eta }^{2}D/M$ necessary whenever $N^{\prime }\ge 0$ and $D>0$. A uniform strict violation with a positive gap forces decay, using a Sobolev inequality to bound $N$ by a constant times $D$. A bounded statistically steady field instead requires $q_{\ast }^2\ge {\eta }^2⟨D⟩/⟨M⟩$, with $q_{\ast }={\sup }_{t}q(t)$. This is a necessary condition for dynamo action, not a sufficient criterion or a pointwise assertion at every instant of an arbitrary time-dependent solution.

// Target: analysis.bigb

For a radial Dirichlet solution on a fixed three-dimensional shell $a<r<b$, $(r^2{U}^{\prime }{)}^{\prime }=r^{2}F$. Thus $r^2{U}^{\prime }=C_0+{\int }_a^r{s}^{2}F(s)ds$, and $U(a)=U(b)=0$ bounds $|C_0|$ by the supremum of that integral. Cauchy-Schwarz inequality in the radial volume measure bounds the integral by $C\Vert f{\Vert }_2$. Since $r\ge a>0$, the displayed gradient bound follows. The estimate exploits radiality and does not extend to arbitrary forcing in three dimensions.

The radical of a Hermitian form is the vector subspace orthogonal to the entire space. The form descends to the quotient vector space by its radical: adding a radical vector to either argument does not change the value. For a positive semidefinite Hermitian form, the Cauchy-Schwarz inequality identifies this radical with its zero-norm vectors, so the quotient form is positive definite. This conclusion does not hold for a general indefinite Hermitian form: a zero-norm vector need not belong to its radical.

If $\rho$ is a density operator with decreasing eigenvalues ${\lambda }_j$ and $T$ has matrix rank at most $k$, let $\Pi$ project onto the image of $T$. The Cauchy-Schwarz inequality for the Hilbert-Schmidt inner product givesThe Hermitian effect variational principle bounds $\mathrm{Tr}(\rho \Pi )$ by ${\sum }_{j<k}{\lambda }_j$, proving the claimed estimate without assuming $\rho$ is invertible.

Suppose $|a_u|\le \tau (u)$, $|b_d|\le \log (2N)$, $u,d>X\ge 1$, $ud\le N$, and $|\theta -a/q|\le q^{-2}$ with $(a,q)=1$. A dyadic decomposition reduces $S=\sum a_u{b}_{d}e(\theta ud)$ to blocks $u\asymp U,d\asymp D$. The Cauchy-Schwarz inequality, divisor-square summatory bound and the reciprocal fractional-part sum near a rational giveThe condition $ud\le N$ makes the allowed $u$ values for each pair $d_1,d_2$ an interval, so the exponential geometric sum bound applies despite the hyperbolic cutoff. Since $UD\le N$ and $U,D\gg X$, take square roots and sum the $O({\log }^2(2N))$ blocks to get the displayed estimate.

For nonzero vectors in a complex inner product space, the displayed quotient defines the ordinary real angle in the underlying real inner product space, whose inner product is $\mathrm{Re}⟨u,v⟩$. The complex Cauchy-Schwarz inequality bounds it between $-1$ and $1$. This convention depends on the vectors' phase factors. The different phase-invariant convention $|⟨u,v⟩|/(\Vert u\Vert \Vert v\Vert )$ measures an angle between complex one-dimensional subspaces; the two conventions should not be interchanged.

When $\beta \rho (A)<1$, summing the adjacency-matrix bound for a discrete SIR epidemic gives the displayed Neumann series bound on the expected total infected count $Z$. For an undirected graph, the adjacency matrix of a graph is a symmetric matrix, so its Euclidean operator norm equals its spectral radius. The Cauchy-Schwarz inequality then yields $\mathbb{E}Z\le \sqrt{nm}/(1-\beta \rho )$ for $m$ initially infected vertices.

For rotational motion about a fixed axis, let $r$ be cylindrical radius, $v$ the signed tangential velocity, and $dm$ the mass measure. Define the kinetic energy, axial angular momentum, and moment of inertia byThe Cauchy-Schwarz inequality gives $L^2\le (\int v^{2}dm)(\int r^{2}dm)=2KI$. More precisely, direct expansion gives the nonnegative energy gapThus the minimum at fixed $L$ and fixed mass distribution occurs exactly when $v=(L/I)r$ almost everywhere with respect to $dm$. This means constant angular velocity $\Omega =L/I$, or solid-body rotation. If the profile is continuous, the equality holds at every point in the mass support.

For a uniform unit-density cylinder of radius $a$, height $h$, and volume $V=\pi a^{2}h$, the moment of inertia is $I=V{a}^2/2$. Therefore $K\ge L^2/(V{a}^2)$, with equality for $\omega (r)=2L/(V{a}^2)$. Multiplying by a uniform nonunit density changes the mass measure and $I$ consistently.

The Robertson-Schrödinger uncertainty relation keeps the real part discarded in the proof of the Robertson uncertainty principle. Write $A^{\prime }=A-⟨A⟩I$ and $B^{\prime }=B-⟨B⟩I$. Their quantum covariance is $C_{AB}=\mathrm{Re}⟨A^{\prime }\psi |B^{\prime }\psi ⟩$. On the same finite-moment and product operator domains used above, the Cauchy-Schwarz inequality givesFor position and momentum, form the real, symmetrized covariance matrixA larger coordinatewise product can therefore encode a tilted covariance ellipse, rather than a larger determinant. This explains why the equality case of the Heisenberg uncertainty relation is more restrictive than equality in the Robertson-Schrödinger uncertainty relation.

The Robertson uncertainty principle bounds the product of the standard deviations of two observables in the same quantum state. For a normalized pure state $|\psi ⟩$ and self-adjoint operators $A,B$, write their expectation values and variances asand similarly for $B$, where $I$ is the identity operator. The quantum uncertainties $\Delta A,\Delta B$ then obeyHere $[A,B]$ is the commutator, and every expectation value is taken in the same $|\psi ⟩$. These standard deviations describe the outcome spreads when $A$ and $B$ are measured on separate ensembles prepared in that same quantum state. They are not instrument errors or the change caused by a preceding measurement in quantum mechanics. The definition and bound are developed in Barton Zwiebach, Uncertainty Principle and Compatible Observables, 2013, §§1–2.

For unbounded self-adjoint operators, the operator domains matter. Assume $\psi \in D(A)\cap D(B)$, so that both second moments are finite; $⟨A^2⟩$ here means the second moment of the measured probability distribution, equal to $\Vert A\psi {\Vert }^2$, even if $A^2\psi$ is undefined. To use the displayed operator commutator, also assume $\psi \in D(AB)\cap D(BA)$. Without these product-domain assumptions, the always meaningful version on $D(A)\cap D(B)$ uses $|\mathrm{Im}⟨A\psi |B\psi ⟩|$ as its right-hand side. These qualifications prevent a formal commutator identity from being applied to an inadmissible quantum state.

The brief proof is Cauchy-Schwarz inequality applied to $u=(A-⟨A⟩I)\psi$ and $v=(B-⟨B⟩I)\psi$:The final equality uses self-adjointness and the product-domain assumptions. Thus a geometric statement about inner products becomes a constraint on experimental spreads. For nonzero $\Delta A,\Delta B$, equality requires $u,v$ to be proportional with a purely imaginary proportionality factor; the equality case of the Heisenberg uncertainty relation turns this condition into a Gaussian wave packet.

The bound is useful because computing a single commutator and its expectation value can constrain two entire outcome distributions. Its right-hand side depends on the quantum state: the spin uncertainty state dependence example gives both a positive saturated bound and a zero bound for the same noncommuting observables. Robertson's paper, titled "The Uncertainty Principle", appeared on 1 July 1929 in Physical Review, volume 34, starting at page 163. ^[ref].

There is an absolute constant $c>0$ such that, for every $\delta >0$ and all sufficiently large $N$, a subset $A\subseteq [N]$ whose density of a finite subset is $\delta$ and which has no nonconstant three-term arithmetic progression has an arithmetic progression $P\subseteq [N]$ satisfying

The Fourier transform of the balanced function $f=1_A-\delta 1_{[N]}$ has a coefficient of magnitude at least $c_0{\delta }^{2}N$. Indeed, the trilinear countsatisfies $\Lambda (1_A,1_A,1_A)=|A|$, whereas $\Lambda (1_{[N]},1_{[N]},1_{[N]})\gg N^2$. Expanding $1_A=\delta 1_{[N]}+f$ and using the Parseval identity and the Cauchy-Schwarz inequality to bound every error term forces $\Vert \hat{f}{\Vert }_{\infty }\gg {\delta }^{2}N$ once $N$ is sufficiently large in terms of $\delta$.

Choose $\theta$ at which this large coefficient occurs. The Dirichlet approximation theorem gives $1\le d\le \sqrt{N}$ with $\Vert d\theta {\Vert }_{\mathbb{R}/\mathbb{Z}}\le N^{-1/2}$. Partition $[N]$ into progressions of common difference $d$ and lengths between $\eta \sqrt{N}$ and $2\eta \sqrt{N}$, where $\eta >0$ is a sufficiently small multiple of ${\delta }^2$. The linear phase $e(\theta x)$ varies by $O(\eta )$ on each cell. The large Fourier coefficient then givesSince ${\sum }_{x\in [N]}f(x)=0$, the total positive discrepancy is half the total absolute discrepancy, so at least one cell has average of $f$ at least $c{\delta }^2$. This is the required density increment.

For each row, the weighted Cauchy-Schwarz inequality gives $|(Bx{)}_i{|}^2\le ({\sum }_j|b_{ij}|)({\sum }_j|b_{ij}||x_j{|}^2)$. Summing over rows bounds the result by the maximum absolute row sum times the maximum absolute column sum times $\Vert x{\Vert }_2^2$. Taking square roots and the supremum proves the displayed induced matrix norm inequality. It gives a Euclidean estimate directly from entries without computing singular values.

When the continuous Fourier transform $F$ of an integrable function vanishes outside $[-\pi ,\pi ]$, its periodic Fourier coefficients are the integer samples of the continuous representative of $f$, with the sign of the index reversed. Projecting $F$ in $L^2[-\pi ,\pi ]$ and then applying the Fourier inversion theorem gives the displayed reconstruction using the normalized sinc function, with each quotient defined as one when $t=n$. The Cauchy-Schwarz inequality controls the reconstruction error uniformly in $t$. Bessel's inequality bounds the vector of shifted sinc functions, giving absolute uniform tails when the sample sequence belongs to ${\ell }^2$.