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For a nonzero partial isometry $u$ in a C-star algebra, put $p=u{u}^{\ast }$ and $q=u^{\ast }u$. The closed linear space $pAq$ becomes a unital C-star algebra with the displayed operations and inherited norm, with unit $u$. The map $a\mapsto u^{\ast }a$ is an isometric -isomorphism onto the ordinary corner $qAq$, with inverse $b\mapsto ub$. The identities $u{u}^{\ast }a=a$ and $u^{\ast }ub=b$ verify the inverse maps, while contractivity of both multiplications gives isometry. Products and involutions transfer directly, giving all algebra and C-norm axioms. This construction turns a rectangular operator corner into a square algebra by using the partial isometry as its unit.

Regularization replaces sensitive inversion by a controlled approximation. For a bounded linear operator between Hilbert spaces, a linear regularization is a family of bounded linear operators $R_{\alpha }$ satisfying $R_{\alpha }y\to A^{†}y$ as $\alpha \downarrow 0$ for each datum in the domain of the Moore–Penrose inverse of an operator. Noisy-data convergence additionally requires a suitable regularization parameter choice.^[ref] Nonlinear variational regularization can instead use nonquadratic penalties.

The noise-bias decomposition for linear regularization separates two costs: amplifying measurement noise and changing the exact-data reconstruction. Truncated singular value decomposition discards weak modes; Tikhonov regularization attenuates them smoothly. A quadratic norm penalty expresses a preference for smaller reconstructions. Neither method manufactures a missing null space component: with complete averaging, the minimum-norm least-squares solution for $(0.5,0.5{)}^T$ is $(0.5,0.5{)}^T$, even if the original image was $(1,0{)}^T$. The preference selects a representative; extra observations or justified constraints are needed to identify the original.

For Tikhonov regularization, the Tikhonov stability bound limits transmitted measurement noise of norm at most $\delta$ to $\delta /(2\sqrt{\alpha })$. Taking $\alpha \to 0$ while $\delta /\sqrt{\alpha }\to 0$ gives a sufficient rule for convergent regularization of an inverse problem. Holding $\alpha$ fixed leaves approximation bias; letting it vanish too quickly can admit overwhelming measurement noise.

For a bipartite vector $|v⟩\in {\mathcal{H}}_R\otimes {\mathcal{H}}_A$ and a chosen orthonormal basis of $R$, the generally unnormalized relative state on $A$ is obtained by the partial inner product with one basis vector. It gives the expansion $|v⟩={\sum }_j|j⟩\otimes |{\eta }_j⟩$. The squared norm of a relative state is the probability of its corresponding basis outcome when $|v⟩$ is normalized. Relative states of eigenvectors of a Choi matrix construct the columns of its Kraus operators.

Let $S$ be a bounded multiplicatively closed subset of a unital Banach algebra. The displayed auxiliary norm $p$ is equivalent to the original norm, and each left multiplication $L_s$ is a contraction for $s\in S$. Taking the operator norm of the faithful left regular representation produces an equivalent algebra norm with $\Vert 1{\Vert }_S=1$ and $\Vert s{\Vert }_S\le 1$. If $M=\max (\Vert 1\Vert ,{\sup }_S\Vert s\Vert )$, then $\Vert a\Vert /M\le \Vert a{\Vert }_S\le M\Vert a\Vert$. The auxiliary norm is itself submultiplicative, since $p(ab)\le p(a)\Vert b\Vert \le p(a)p(b)$, but $p(1)=M$ can exceed one. The operator norm enforces the required unital normalization.

If an associative unital algebra has a complete norm and separately continuous multiplication, the Uniform boundedness principle applied to $L_x(y)=xy$, for $\Vert x\Vert \le 1$, yields $\Vert xy\Vert \le C\Vert x\Vert \Vert y\Vert$. The operator norm of $L_x$ then defines an equivalent algebra norm: $\Vert x\Vert /\Vert e\Vert \le \Vert L_x\Vert \le C\Vert x\Vert$, and $L_{xy}=L_x{L}_y$ proves submultiplicativity. It normalizes the identity to norm one.

A Reproducing kernel Hilbert space $\mathcal{H}$ is a Hilbert space of functions on a set $X$ in which evaluation at each fixed point is a continuous linear functional. The Riesz representation theorem therefore supplies an element $k_x\in \mathcal{H}$ such thatSetting $k(x,y)=⟨k_x,k_y{⟩}_{\mathcal{H}}$ defines its positive-semidefinite kernel. The reproducing property turns pointwise information into an inner product; the Cauchy-Schwarz inequality givesThus small changes in the norm control evaluation at each fixed point. In the linear kernel example $k(x,y)=x^{T}y$, the functions $f_w(x)=w^{T}x$ form the corresponding finite-dimensional space with the ordinary coefficient inner product. The Moore-Aronszajn theorem extends the correspondence to arbitrary positive-semidefinite kernels.^[ref]

The residual is the difference between predicted and observed data. Its norm measures data misfit and can determine an iteration stopping rule for regularization of an inverse problem.

For a closed unital subalgebra $B$ of a unital Banach algebra $A$ and $x\in B$, the smaller-algebra algebra resolvent set is relatively open and closed in the larger one. Openness follows from the Neumann series; closedness follows from continuity of inversion in $A$ and norm closedness of $B$. Every larger-algebra resolvent component is therefore either retained whole or absorbed into the smaller-algebra Banach algebra spectrum. The unbounded component is always retained, since inverses at large modulus are norm-convergent series in $x$.

Assume a finite diagonal Fourier extension estimate on the unit circle. For direction-separated rectangles of dimensions ${\delta }^{-2}\times {\delta }^{-1}$, choose disjoint frequency caps and translate their transforms to the rectangles by the Fourier modulation and translation identity. Randomize their signs. Disjointness gives an input $p$th-power norm at most $CM\delta$; the Khintchine inequality converts the averaged output norm into the square function. The circle cap Fourier lower bound gives ${\delta }^p\int ({\sum }_R{1}_R{)}^{p/2}{\lesssim }_{p}M\delta$. Each rectangle has area ${\delta }^{-3}$, giving the displayed form. Input cap disjointness is compatible with arbitrary spatial rectangle overlap.

The closed unit ball $B$ in the infinite-dimensional real Hilbert space ${\ell }^2$ admits a retraction onto its unit sphere. Write $S$ for the forward unilateral shift operator and set $F(x)=\sqrt{1-\Vert x{\Vert }^2}{e}_0+Sx$. Then $\Vert F(x)\Vert =1$ and $F(x)\ne x$: equality would force all coordinates of $x$ to coincide, hence vanish, contradicting $F(0)=e_0$.

For $d=x-F(x)$, setThe nonvanishing denominator makes $r$ a continuous map. Expanding its squared norm gives $\Vert r(x)\Vert =1$. On $\Vert x\Vert =1$, the fraction is one, so $r(x)=x$. Thus the finite-dimensional no-retraction theorem does not extend to this closed unit ball.

A Riemannian metric is a smoothly varying positive-definite inner product $g_p$ on each tangent space $T_{p}M$ of a smooth manifold. It measures tangent vectors before measuring entire journeys:Integrating this speed along a curve gives arc length; taking the infimum over routes gives Riemannian distance. A smooth manifold together with this measuring rule is a Riemannian manifold.

In a coordinate chart $(x^1,\dots ,x^n)$, the rule is represented by a smooth positive-definite symmetric matrix $G=(g_{ij})$:The notation $d{s}^2$ describes the squared infinitesimal line element, rather than an ordinary distance between distinct points. Changing coordinates changes the matrix entries; it preserves the measured norm of the same tangent vector.

For the plane with $d{s}^2=4d{x}^2+d{y}^2$, the vector $(1,0)$ has norm $2$ and $(0,1)$ has norm $1$. The coordinate change $X=2x,Y=y$ turns this into $d{s}^2=d{X}^2+d{Y}^2$: coefficients that look different can describe the same geometry. On a circular cylinder the induced metric $R^{2}d{\theta }^2+d{z}^2$ similarly becomes Euclidean when unrolled. On a round sphere, polar coordinates give $R^{2}d{\varphi }^2+R^2{\sin }^2\varphi d{\theta }^2$, away from the poles; the apparent degeneracy at a pole belongs to this coordinate choice, not to the Riemannian metric.

The point of the construction is that measurements make sense intrinsically, even without a chosen surrounding Euclidean space. It gives a common starting point for geodesics, curvature encoded by the Riemann curvature tensor, and the Laplace-Beltrami operator. ^[ref].

For an orthonormal family in a Hilbert space, the squared norm of every finite partial sum is the sum of squared coefficients. Square summability therefore makes the net of finite partial sums Cauchy, proving unconditional convergence by completeness. If the orthonormal family is complete, Bessel's inequality makes the coefficients of each vector square summable, and the residual after summing is orthogonal to the whole family, hence zero. This recovers the orthonormal expansion and Parseval identity.

For a linear operator bounded from $L^{p_i}$ to $L^{q_i}$ with norms $M_i$, $i=0,1$, interpolate reciprocals by $1/p=(1-\theta )/p_0+\theta /p_1$ and $1/q=(1-\theta )/q_0+\theta /q_1$. The intermediate operator norm is at most $M_0^{1-\theta }{M}_1^{\theta }$. The complex interpolation proof first treats simple functions; completeness and continuity on the endpoint sum spaces identify the extension with the original operator. On finite counting spaces it also gives norm bounds between sequence spaces.

For differences $D_i$ between corresponding stages and differences $F_i$ between their vector fields, a Runge--Kutta step satisfiesTo derive the identity, expand the squared norm of $d_{n+1}=d_n+h{\sum }_i{b}_i{F}_i$ and substitute $d_n=D_i-h{\sum }_j{a}_{ij}{F}_j$ in the linear term. Symmetrizing the double sum gives the displayed coefficients $m_{ij}$. For a dissipative vector field, the first sum is nonpositive when $b_i\ge 0$, and the second sum is nonnegative when the matrix $(m_{ij})$ is a positive semidefinite matrix. Thus algebraic stability implies B-stability.

A scalar-valued function is a function whose codomain is a field, typically the real numbers or complex numbers. Its output is a single scalar, while its input can belong to an arbitrary set. For example, the square of the norm of a vector is real-valued. In a relativistic application, being scalar-valued alone does not establish Lorentz invariance: the transformation of the input and of the complete quantity must also be specified.

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 $\Vert T_{\epsilon }(x)-Tx\Vert <\epsilon$.

The Schilder theorem gives a large deviation principle for small Brownian motion paths. For standard Brownian motion $W$ starting at zero on $[0,T]$, the laws of $\sqrt{\epsilon }W$ on continuous paths with the uniform norm have speed $1/\epsilon$ and good rate functionIn particular, the zero starting value and finite-energy path class are part of the rate. Typical Brownian motion paths are not themselves absolutely continuous; the rate describes the exponential cost of neighborhoods of prescribed paths.^[ref]

For paths constrained by $\varphi (T)=b$, the Cauchy-Schwarz inequality gives ${\int }_0^T|\dot{\varphi }{|}^{2}dt\ge b^2/T$, with equality for $\varphi (t)=bt/T$. Thus the least endpoint cost is $b^2/(2T)$, agreeing with the exponential scale of the normal distribution of $\sqrt{\epsilon }{W}_T$.

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.

Under a regular simple $p$-parameter null, the normalized score converges to $N_p(0,I_1)$, so its information-standardized squared norm converges to ${\chi }_p^2$.

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.