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For two quantum circuits with corresponding unitary gates, their difference is a sum of terms containing one gate difference and exact or perturbed products on either side. Unitary invariance and the triangle inequality for the operator norm therefore bound the circuit difference by the sum of individual gate errors. Tensoring a gate with an identity leaves that norm unchanged. On a normalized input, the same bound controls the output vector distance with gate phases retained. This is a worst-case coherent bound; cancellation can make a particular circuit less sensitive, but cannot be assumed without further information.

A quasi-norm on a vector space is nonnegative, vanishes only at zero, is absolutely homogeneous, and satisfies a triangle inequality with one constant factor $C\ge 1$. For $0<p<1$, $(\int |f{|}^p{)}^{1/p}$ on an Lp space is a quasi-norm: $\Vert f+g{\Vert }_p^p\le \Vert f{\Vert }_p^p+\Vert g{\Vert }_p^p$ gives a quasi-triangle inequality. Measure-preserving composition preserves this quantity even though the Minkowski inequality is unavailable below exponent one.

On closed Riemann surfaces of genus at least two, if a Teichmüller map $f_0:X\to Y$ has unit-area source differential $q$ and dilatation $K_0$, any quasiconformal map $f:X\to Y$ in the same homotopy class satisfies $K_0\le {\int }_X|q||1+{\mu }_{f}q/|q|{|}^2/(1-|{\mu }_f{|}^2)$. The plus sign corresponds to a Teichmüller map with ${\mu }_{f_0}=k_0|q|/q$ and $K_0=(1+k_0)/(1-k_0)$. This fundamental inequality yields Teichmüller's uniqueness theorem by the pointwise triangle inequality and its equality case. See Gardiner and Hu, §5, equation (11).

For $|r|<1$, let $S_N$ contain the first $N$ terms, with $S_0=0$. The remainder after those terms isThis is the exact absolute error. A convenient bound replaces the denominator by $1-|r|$, giving $|R_N|\le |a||r{|}^N/(1-|r|)$ through the triangle inequality. The exact denominator is preferable when the sign or complex phase of $r$ is known.

For $a>0$ and $0<r<1$, the omitted fraction of the entire sum is $R_N/(a/(1-r))=r^N$. Capturing at least $99$ percent therefore means $r^N\le 0.01$. Taking logarithms and applying the ceiling function gives the smallest integer $N$:This gives $7,21,90,$ and $459$ terms for $r=0.5,0.8,0.95,$ and $0.99$, respectively. The difference is a concrete reason to care about the common ratio, even when all four series converge.

An absolute target has a different scale. For $a=1$, $r=0.8$, and required absolute error at most $0.01$, the total sum is $5$ andThe first acceptable term count is $N=28$: the remainder is approximately $0.0096714$, while at $N=27$ it is approximately $0.0120893$. Counting terms and checking the tail avoids an ambiguous instruction to stop when a term merely looks small.

In a metric space, applying the triangle inequality in both directions gives $|d(x,z)-d(y,z)|\le d(x,y)$. In a normed vector space, its corresponding form is $|\Vert x\Vert -\Vert y\Vert |\le \Vert x-y\Vert$. In particular, distance to a fixed point is a Lipschitz continuous real function.

Rotational Raman spectroscopy observes a frequency change of scattered light associated with molecular rotation. The coupling uses the induced electric dipole moment, ${\mathbf{d}}_{\mathrm{ind}}=\mathbf{\alpha }\mathbf{E}$, where $\mathbf{\alpha }$ is the electric polarizability tensor. For a linear molecule, the useful anisotropy is ${\alpha }_{\parallel }-{\alpha }_{\perp }$: the response differs for an electric field parallel or perpendicular to the molecular axis. A permanent electric dipole moment is not required.^[ref]

For the spin-free linear rigid rotor in one internal quantum state, the anisotropic tensor depends quadratically on the molecular direction. Its angular part has even parity and degree two in spherical harmonics. The triangle inequality for coupled angular momenta permits $|\Delta J|\le 2$, and even parity removes odd $\Delta J$, leaving $\Delta J=0,\pm 2$. The $\Delta J=0$ contribution is unshifted light; shifted rotational lines have $\Delta J=\pm 2$. Consequently the rigid-model Stokes shifts, in ordinary frequency units, areSuccessive allowed shifts are separated by $4B$, rather than the $2B$ absorption spacing of pure rotational electric-dipole selection rules. In spectroscopic wavenumber units replace $B$ by $\tilde{B}=B/c$ and $\Delta \nu$ by $\Delta \tilde{\nu }=\Delta \nu /c$, using the speed of light in $\mathrm{cm}{\mathrm{s}}^{-1}$ for $\mathrm{c}{\mathrm{m}}^{-1}$.

A homonuclear diatomic molecule such as molecular hydrogen has no permanent electric dipole moment in its symmetric internal state, so it lacks the ordinary electric-dipole microwave ladder. Its anisotropic electric polarizability can nevertheless give rotational Raman spectroscopy lines. Nuclear-spin symmetry can change the populations of even and odd $J$ levels, so the spin-free population example above must not be reused unchanged for molecular hydrogen. The method of coupling, as well as the energy eigenvalues, determines what the experiment can see.

The displayed Thomas algorithm requires every modified pivot element to be nonzero in exact arithmetic. Nonzero original diagonal entries alone are insufficient: later pivot elements can vanish after elimination. Two useful sufficient contexts are a real symmetric positive-definite matrix, and strict diagonal dominance by rows:For the symmetric positive-definite matrix case, elimination leaves positive-definite remaining blocks by the Schur complement, so each $p_i>0$. Equivalently, the initial principal minors ${\Delta }_i=\det A_{1:i,1:i}$ are positive, and $p_i={\Delta }_i/{\Delta }_{i-1}$ with ${\Delta }_0=1$.

For strict diagonal dominance, a short induction shows $|p_i|>|c_i|$. The first row gives $|p_1|=|b_1|>|c_1|$. Assuming the preceding bound, the triangle inequality givesIn particular $p_n\ne 0$. These are structural guarantees against a zero pivot element, not universal bounds on forward error in floating-point arithmetic. A poorly conditioned linear system can still amplify roundoff error; numerical stability and the condition number must be distinguished.

For a scalar BV space function, clipping to $[-M,M]$ partitions its coarea formula for BV functions into middle and tail levels. The clipped function has the original superlevel sets at heights in $(-M,M)$. The positive tail levels of $u-T_{M}u$ correspond to original heights above $M$, and its negative tail levels to heights below $-M$. Thus their total variation seminorms add exactly. This is stronger than the triangle inequality and is specific to scalar monotone clipping. It supports residual-preserving clipping of an ROF minimizer.

Let $K$ be a nonempty compact convex set in a Banach space and $T:K\to K$ a continuous map. For every $\epsilon >0$, there is a finite-dimensional linear subspace $L_{\epsilon }$ and a continuous map $T_{\epsilon }:K\to K\cap L_{\epsilon }$ that approximates $T$ uniformly to within $\epsilon$ in norm.

Choose $y_1,\dots ,y_N\in T(K)$ whose open radius-$\epsilon$ balls cover the compact image $T(K)$. Set $w_i(x)=\max \{0,\epsilon -\Vert Tx-y_i\Vert \}$ andThe denominator is positive. This convex combination belongs to $K\cap \mathrm{span}\{y_1,\dots ,y_N\}$ and depends continuously on $x$. Every contributing $y_i$ has distance less than $\epsilon$ from $Tx$, so the triangle inequality gives $\Vert T_{\epsilon }(x)-Tx\Vert <\epsilon$.

A seminorm on a real or complex vector space $E$ is a finite-valued function $p:E\to [0,\infty )$ satisfying absolute homogeneity and the triangle inequality:In particular $p(0)=0$ and $p(-x)=p(x)$. A seminorm is a norm exactly when $p(x)=0$ implies $x=0$: this is the extra definiteness of a norm axiom. The zero function is a valid seminorm. The missing definiteness is useful when a measurement deliberately ignores some directions, such as an individual coordinate or the constant term of a function whose derivative is measured.

For example, on $C^1([0,1])$ the function $p(f)={\sup }_{t\in [0,1]}|f^{\prime }(t)|$ is a seminorm that vanishes on every constant function. Adding the value measurement gives the norm $q(f)=|f(0)|+p(f)$: if $q(f)=0$, the fundamental theorem of calculus gives $f(t)=f(0)+{\int }_0^t{f}^{\prime }(s)ds=0$. A family of such partial measurements defines a seminorm topology.

A family of seminorms separates points when every nonzero vector $z$ has $p(z)>0$ for some member $p$. Equivalently, its kernels of seminorms have intersection $\{0\}$, or any two distinct vectors $x,y$ have $p(x-y)>0$ for some $p$. Individual seminorms can have large kernels of seminorms; the whole family must have no common invisible direction.

The associated seminorm topology is Hausdorff if and only if the family separates points. Indeed, if $p(x-y)=a>0$, the neighbourhoods $\{u:p(u-x)<a/3\}$ and $\{v:p(v-y)<a/3\}$ are disjoint by the triangle inequality. Conversely, if every $p$ vanishes on a nonzero $z$, then zero and $z$ belong to exactly the same open sets and cannot be separated. This criterion keeps separation distinct from completeness: detecting all directions does not guarantee that all Cauchy sequences have limits.

Suppose two pieces occupy a common connected component of a graph in an unweighted undirected graph. For integers $a>b\ge 0$, in each round an evader moves at most $b$ edges, then a pursuer moves up to $a$ edges along a shortest path to the new evader position, stopping upon capture. Let $d_t$ be their finite graph distance at a surviving round boundary.

The triangle inequality bounds the separation after the evader moves by $d_t+b$. If that separation is at most $a$, the pursuer captures it. Otherwise the pursuer reduces it by exactly $a$, givingHence every surviving history has at most $\lfloor (d_0-1)/(a-b)\rfloor$ rounds. With a finite branching factor, the surviving search tree is finite and depth-first tree search becomes a complete search algorithm. This does not make its first goal optimal for arbitrary rewards. Initial reachability is essential: an infinite graph distance supplies no decreasing finite bound.

Two controlled walkers on the same connected undirected graph can each traverse one edge or wait in each unit-cost step. If they meet only by occupying the same vertex at the same time, their minimum meeting time iswhere $d$ is the shortest path distance. A common meeting point after $t$ steps implies $d(p,q)\le 2t$ by the triangle inequality. Conversely, choose a middle vertex of a shortest connecting graph path; both walkers reach it within $\lceil d/2\rceil$ steps, and the earlier arrival waits. The wait is essential when the path length is an odd integer. This distance on the joint Cartesian product state space is a consistent heuristic: each joint move changes the walkers’ separation by at most two. Replacing $d$ by an obstacle-free Manhattan distance preserves admissible heuristic and consistent heuristic guarantees.

For equally sized matrices, corresponding ordered singular values differ by at most the spectral norm of the perturbation. Indeed, let $B_{j-1}$ be a best approximation of matrix rank at most $j-1$. The Eckart-Young-Mirsky theorem and the triangle inequality give ${\sigma }_j(A)\le \Vert A-B_{j-1}{\Vert }_2\le \Vert A-B{\Vert }_2+{\sigma }_j(B)$. Interchanging $A,B$ proves the bound.

For a compact Hausdorff space $K$, $C(K)$ consists of continuous scalar-valued functions with the supremum norm. It is a Banach space and, with complex scalars, pointwise multiplication and conjugation make it a commutative unital C-star algebra.

Why does completeness hold? A Cauchy sequence in the supremum norm is Cauchy at every point, hence has a pointwise limit. The uniform Cauchy estimate passes to that limit, giving uniform convergence; the uniform limit theorem makes the limit continuous. Thus the norm and the chosen regularity of the functions cooperate to keep every limit inside $C(K)$.

Changing to the L2 norm on $C([0,1])$ loses this property. For integers $n\ge 2$, the continuous functions $h_n(x)=\min (1,\max (0,n(x-1/2)))$ approximate the step function $h(x)=0$ for $x\le 1/2$ and $h(x)=1$ for $x>1/2$. Direct integration givesThe triangle inequality makes $(h_n)$ a Cauchy sequence, but no continuous function agrees with $h$ almost everywhere: continuity would force value zero on the whole left open interval and one on the whole right open interval. Hence this normed space is incomplete. The missing limit exists in L2 space, its completion of a normed space, showing exactly what the larger space adds.

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.

Given a metric space $(X,d)$ and $a\in X$, define a new metric by $d_a(x,x)=0$ and $d_a(x,y)=d(x,a)+d(a,y)$ for distinct points. It is the distance obtained by joining every point to a central vertex with its original distance to $a$ as edge length. Symmetry and positive definiteness are immediate. With three distinct points the triangle inequality follows from the extra nonnegative term $2d(z,a)$ in the route through an intermediate point; repeated-point cases are immediate.

Every $b\ne a$ is isolated, since $B_{d_a}(b,d(b,a))=\{b\}$. At the centre, however, $B_{d_a}(a,r)=B_d(a,r)$. Thus a map out of this space is continuous exactly when it is continuous at $a$ in the original metric. The closure of $B$ is $B$ together with $a$ if $a\notin B$ and $d(a,B)=0$; otherwise it is simply $B$. This construction isolates every point except the centre while preserving exactly its original neighbourhood filter.

Suppose $|f^{\prime \prime \prime }|\le M$ and each sampled value of $f$ has absolute error at most $\delta$. For a central finite difference with $h>0$, the triangle inequality and the sharp central-secant derivative error giveFor $M,\delta >0$, the derivative of this bound is $Mh/3-\delta /h^2$. It changes sign at $h_{\ast }=(3\delta /M{)}^{1/3}$, proving the unique minimum, of value $(3\delta {)}^{2/3}{M}^{1/3}/2$. This balance explains why reducing the step indefinitely amplifies sample absolute error even as truncation error decreases.

A strong stability preserving Runge-Kutta method writes its stages as convex combinations of previously computed states and suitable Forward Euler method steps. It transfers any convex-functional nonincrease property of the forward step, such as a norm bound, under a proportionally scaled time-step restriction.

For example, if $E_k(U)=U+kF(U)$ is nonexpansive in a norm, then $Y=E_k(U)$ and $U_{\mathrm{new}}=(U+E_k(Y))/2$ is also nonexpansive: the triangle inequality bounds the new difference by one half of the initial difference plus one half of the twice-advanced difference. Both are at most the initial difference. A Taylor expansion yields $U_{\mathrm{new}}=U+kF(U)+k^2{F}^{\prime }(U)F(U)/2+O(k^3)$, so this is a second-order Runge-Kutta method. The forward-step hypothesis must hold on the stage states; preservation of one chosen convex bound is not the same as unconditional B-stability.

For composable bounded linear operators, the operator norm satisfies $\Vert AB\Vert \le \Vert A\Vert \Vert B\Vert$. Indeed, $\Vert ABx\Vert \le \Vert A\Vert \Vert Bx\Vert \le \Vert A\Vert \Vert B\Vert \Vert x\Vert$, and taking the supremum over unit vectors proves the bound. It implies $\Vert [A,B]\Vert \le 2\Vert A\Vert \Vert B\Vert$ by the triangle inequality.