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A Hamiltonian path preserving the history subspace can be analysed using its restricted spectral gap. If this gap is $\Omega (T^{-2})$ and the derivative has operator norm $O(1)$, a runtime criterion proportional to $\Vert \dot{H}{\Vert }^2/(ϵ{\Delta }^3)$ permits computational history state preparation in time $O(T^6/ϵ)$. Other invariant sectors do not cause leakage from a state initialized in the history subspace.

Asymptotic regularization stops the gradient flow $x^{\prime }=-A^{\ast }(Ax-f)$ at $t=1/\alpha$, starting from zero. With the singular system of a compact operator convention $A{v}_j={\sigma }_j{u}_j$, its spectral filter givesThe scalar coefficient solves $c_j^{\prime }=-{\sigma }_j^2{c}_j+{\sigma }_j⟨f,u_j⟩$ with zero initial value. Since $1-e^{-s}\le \min (s,1)$ for $s\ge 0$, the operator norm is at most ${\alpha }^{-1/2}$. On the domain of the Moore–Penrose inverse of an operator, dominated convergence theorem of the squared spectral coefficients proves $R_{\alpha }f\to A^{†}f$. The noise-bias decomposition for linear regularization then gives noisy-data convergence when $\delta /\sqrt{\alpha }\to 0$.

For a $C^1$ flux $F$ and a weak solution of $\mathrm{div}F(Du)=0$, discrete integration by parts and the fundamental theorem of calculus along a line segment show that $w_h={\delta }_{\ell ,-h}u$ solves $\mathrm{div}(A_{h}D{w}_h)=0$, with $p_{-}=Du(x-h{e}_{\ell })$ and $p_{+}=Du(x)$. Coercive quadratic-form bounds and operator norm bounds for $DF$ on all segments between these gradients pass to $A_h$. This produces a divergence-form elliptic operator without initially assuming second partial derivatives of $u$.

A Banach limit is a positive linear functional on the real l-infinity sequence space that extends the ordinary limit on the convergent sequence space, has operator norm one, and is invariant under the left shift on bounded sequences. The Hahn-Banach theorem constructs one by separating the constant sequence $1$ from the subspace $(I-S){\ell }^{\infty }$, whose distance from $1$ is one. The functional is generally not unique.

If $X$ is a normed vector space and $Y$ is a Banach space, then the bounded linear maps $\mathcal{B}(X,Y)$ form a Banach space under the operator norm. Completeness of the domain is unnecessary.

The Born series is the Neumann series expansion of the Lippmann-Schwinger equation. Its terms describe successive orders of wave scattering; convergence is ensured, for example, by a scattering operator norm smaller than one.

For distinct interpolation sites and an invertible B-spline collocation matrix, the interpolant is $P=B{A}^{-1}R$, where $R$ samples data and $Ba={\sum }_i{a}_i{N}_i$. Sampling and basis synthesis have operator norms at most one in the supremum norm. For the reverse inequality, realize the signs of a maximal absolute row sum of $A^{-1}$ as values of a continuous function of norm one, and apply uniform-norm stability of a B-spline basis. This yields both displayed bounds.

If $H={\sum }_j{h}_j$ and all local terms commute, thenexactly. Approximating each fixed-size factor to error at most $\epsilon /m$ and applying the telescoping bound for products of operators gives total operator norm error at most $\epsilon$.

For proper lower semicontinuous convex functions on finite-dimensional Hilbert spaces, a nonempty saddle point set, and constant positive steps satisfying the displayed inequality, the primal-dual hybrid gradient method with extrapolation parameter one converges to a saddle point. The strict condition controls the bilinear coupling by the operator norm. It does not remove the need for existence of a saddle point. The fixed-step theorem and assumptions are recalled in Malitsky and Pock, A first-order primal-dual algorithm with linesearch, Section 1.

If $\Vert \tau f(t)\Vert \le C{\int }_0^t\Vert f(s)\Vert ds$, repeated integration produces the time-ordered simplex of volume $t^n/n!$. Thus the Neumann series ${\sum }_{n\ge 0}{\tau }^n$ converges on every finite interval, without requiring the single-step operator norm to be less than one.

For equal finite-dimensional closed subspaces of a Hilbert space $T,S$, positivity of $\cos {\theta }_{T,S}$ makes $P_S{|}_T$ invertible. The displayed reconstruction is the unique element of $T$ whose orthogonal projection onto $S$ matches that of $f$. It is the oblique projection onto $T$ along $S^{\perp }$. Its operator norm is the value of the secant function at the directed subspace angle, and its error is at most that secant function value times the best orthogonal projection error. The lower error bound follows from the Pythagorean identity. Zero-dimensional cases are handled directly.

If $A(t)=A+tE$ and $\lambda (t)$ is a differentiable function defining a simple eigenvalue, thenThe Cauchy-Schwarz inequality and the operator norm give

For a smooth solution of $\Delta u=f$, the corrected formula ${\Gamma }_{9}u=h^{2}f+(h^2/12){\Gamma }_{5}f$ has unscaled residual $O(h^6)$ and normalized local truncation error $O(h^4)$. The inverse interior matrix has Euclidean operator norm $O(h^{-2})$; on $O(h^{-2})$ grid points this gives unweighted Euclidean nodal error $O(h^3)$. Uniform derivatives up to the boundary are required for a grid-independent remainder bound.

For an operator on a tensor-product Hilbert space, averaging unitary conjugations on subsystem $B$ with normalized Haar measure gives ${\mathcal{E}}_A(O)=({\mathrm{Tr}}_{B}O/d_B)\otimes I_B$. It fixes precisely the operators acting trivially on $B$ and is contractive in operator norm. Writing $O-UO{U}^{†}=[O,U]U^{†}$ converts localization error into a commutator bound.

For $(Tg)(v)=\int k(v,w)g(w)dw$, the Cauchy-Schwarz inequality gives $\Vert Tg{\Vert }_2\le \Vert k{\Vert }_{L^2(v,w)}\Vert g{\Vert }_2$. The kernel norm is the Hilbert-Schmidt norm, which bounds the operator norm. Uniform kernel bounds apply also to spatially parametrized operators after integrating over position.

The interior consists exactly of strictly copositive matrices. If $x^{T}Ax$ has positive minimum $\delta$ on the compact nonnegative unit sphere, perturbations of operator norm less than $\delta$ preserve this positive lower bound there. Conversely, if a unit nonnegative vector has zero quadratic form, $A-\epsilon I$ leaves the copositive cone for every $\epsilon >0$. Thus such a matrix is not interior. The identity matrix is a convenient interior point.

The implicit iteration $(I+\tau K^{\ast }K)u_{k+1}=u_k+\tau K^{\ast }f$, starting from zero, multiplies each data singular vector by $[1-(1+\tau {\sigma }^2{)}^{-k}]/\sigma$. Its operator norm is at most $\sqrt{k\tau }$.

The J gate is $J(\alpha )=HP(\alpha )$, where $P(\alpha )=\mathrm{diag}(1,e^{i\alpha })$: the phase gate acts first and the Hadamard gate acts second. Unitary invariance of the operator norm reduces its angle error to the single nonzero diagonal difference of the phase gate. Its norm is $|e^{i\delta }-1|=2|\sin (\delta /2)|\le |\delta |$. The quantum circuit gate-error telescoping bound then makes an angle tolerance of $ϵ/k$ sufficient for $k$ such imperfect gates and exact other gates.

The left shift is a bounded linear operator of operator norm one on the l-infinity sequence space. For $y=x-Sx$, telescoping gives $N^{-1}{\sum }_{n=1}^N{y}_n=(x_1-x_{N+1})/N\to 0$. This identity is useful in constructing a Banach limit.

On the Hilbert space ${\ell }^2(\mathbb{N})$, the left shift deletes the first coordinate. It is the adjoint operator of the unilateral shift operator $S$. The identities $LS=I$ and $SLf=f-f_1{e}_1$ show that it is also the Moore–Penrose inverse of an operator for $S$; both maps have operator norm one.