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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.

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.

When the continuous Fourier transform $F$ of an integrable function vanishes outside $[-\pi ,\pi ]$, its periodic Fourier coefficients are the integer samples 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 sinc function reconstruction. The Cauchy-Schwarz inequality controls the reconstruction error uniformly in $t$. Bessel's inequality bounds the shifted sinc coefficient vector, giving absolute uniform tails when the sample sequence belongs to ${\ell }^2$.

For a matrix with nonnegative entries, row sums at most $R$ and column sums at most $C$ imply an ${\ell }^2$ operator norm at most $\sqrt{RC}$. Indeed the Cauchy-Schwarz inequality gives $|{\sum }_j{A}_{ij}{x}_j{|}^2\le ({\sum }_j{A}_{ij}){\sum }_j{A}_{ij}|x_j{|}^2$. Summing over $i$ bounds the result by $RC{\sum }_j|x_j{|}^2$. Integral kernels and positive weight functions give analogous forms. The symmetric case has equal row and column bounds.

Put $L=\sqrt{D}$, $h(t)={\prod }_{p\mid t}(g(p{)}^{-1}-1)$ and $y_t={\sum }_{\begin{array}{c}m\le L\\ t\mid m\end{array}}g(m){\rho }_m$, where $g$ is a multiplicative arithmetic function supported on squarefree integers. For squarefree integers $m,n$,Thus the quadratic form is $\Sigma ={\sum }_{t\le L}h(t)y_t^2$. Möbius inversion gives $g(d){\rho }_d={\sum }_{d\mid t\le L}\mu (t/d)y_t$. On the permitted squarefree integers, the constraint ${\rho }_1=1$ becomes ${\sum }_t\mu (t)y_t=1$. The Cauchy-Schwarz inequality yields $\Sigma \ge 1/J$, where $J={\sum }_{t}1/h(t)$ over those integers; equality holds at $y_t=\mu (t)/(Jh(t))$.

The spatial Galerkin method for the heat equation gives a mass matrix $M$ and a diffusion stiffness matrix $S$. With zero forcing, $\frac{d}{dt}(c^{T}Mc)=-2c^{T}Sc\le 0$. The quadratic form $c^{T}Mc$ is the squared L2 norm of the represented finite-element function, proving an energy contraction. For nonzero forcing, the corresponding identity has the additional term $2c^{T}F$, which can be estimated using the Cauchy-Schwarz inequality.

For positive traffic rates in single-server colonies, a total service budget $F>{\sum }_j{\alpha }_j$ minimizes the stationary mean population when the spare rates ${\varphi }_j-{\alpha }_j$ are proportional to $\sqrt{{\alpha }_j}$. The Cauchy-Schwarz inequality proves the optimum directly.

The continuous dual space of $H^s({\mathbb{R}}^d)$ is $H^{-s}({\mathbb{R}}^d)$ under the extension of the L2 inner product. The Fourier transform definition $\Vert f{\Vert }_{H^s}^2=\int (1+|\xi {|}^2{)}^s|\hat{f}(\xi ){|}^{2}d\xi$ proves boundedness of the pairing by the Cauchy-Schwarz inequality, and the Riesz representation theorem proves that every continuous functional has this form. Consequently a bounded linear operator $K:H^r\to H^{r+a}$ has an adjoint operator, relative to this pairing, from $H^{-r-a}$ to $H^{-r}$. For an operator that smooths by two orders for every $r$, this yields $K^{\ast }:H^2\to H^4$ by taking $r=-4$.

Let $H=M-\delta J$ be the centered biadjacency matrix of an equal-part biregular graph, and put $s=\Vert H{\Vert }_{\mathrm{op}}$. For subsets $A,B$ of the two parts, their edge discrepancy is $1_A^{\mathsf{T}}H{1}_B$. The Cauchy-Schwarz inequality proves the displayed bound. If the part size is $n$, centering the two indicator vectors improves it to $sn\sqrt{\alpha (1-\alpha )\beta (1-\beta )}$. The centered four-cycle identity for a biregular graph shows that $t_4(G)\le {\delta }^4(1+c^4)$ implies $s\le c\delta n$ for $c\ge 0$.

For equally sized Hermitian matrices and a real test function with Lipschitz bound one, the difference of its averages under their empirical spectral measures is at most $N^{-1/2}$ times the Frobenius norm of their difference. Pair the increasing eigenvalues, apply the triangle inequality and Cauchy-Schwarz inequality, then use the Hoffman–Wielandt inequality.

Positivity makes $A-z$ bounded below with dense closed range for every $z<0$, excluding negative spectral values. Conversely if the spectrum is nonnegative, $R=(I+A{)}^{-1}$ has spectrum in $[0,1]$ by the resolvent spectral mapping identity. The numerical-range endpoint approximate-eigenvalue theorem gives $0\le R\le I$, and positive-form Cauchy-Schwarz inequality gives $\Vert Rf{\Vert }^2\le ⟨Rf,f⟩$. For $u=Rf$, $⟨Au,u⟩=⟨Rf,f⟩-\Vert Rf{\Vert }^2\ge 0$.

For mean-zero $f$, apply the mean ergodic theorem on an ergodic product system to $f\otimes \overline{f}$, paired with $g\otimes \overline{g}$. Its correlation is $|⟨U^{n}f,g⟩{|}^2$. The Cauchy-Schwarz inequality then gives averaged absolute correlation decay, the definition of weak mixing.

Every finite point set in the unit square admits a cyclic visiting order with ${\sum }_i\Vert x_{i+1}-x_i{\Vert }^2\le 4$. Split the square into two right triangles, apply the quadratic path bound in a right triangle, and remove auxiliary corners using the law of cosines. The Cauchy-Schwarz inequality then gives $\mathrm{TS}(X)\le 2\sqrt{|X|}$. The tour minimizing ordinary length need not minimize the sum of squared lengths.

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 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 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 $\mathbb{P}(X>\theta \mu )\ge (1-\theta {)}^2{\mu }^2/\mathbb{E}[X^2]$.

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.