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The efficient information for a scalar target statistical parameter is the squared L2 norm of its efficient score. It cannot exceed the parametric Fisher information, by the Pythagorean theorem in an inner-product space. It can vanish when first-order target variation can be reproduced by nuisance variation.

The efficient score is the residual after orthogonal projection of the parametric score function onto the nuisance tangent space. It is centered and orthogonal to every nuisance direction. Its squared L2 norm is the efficient information.

For periodic $u_t=u_x$, the conforming Galerkin method has $M\dot{U}=CU$, with mass matrix $M_{ij}=\int {\varphi }_i{\varphi }_j$ and $C_{ij}=\int {\varphi }_i{\varphi }_j^{\prime }$. Integration by parts and periodicity give $C^T=-C$. ThereforeThe conserved quantity is the squared L2 norm of the finite element function. On a uniform one-dimensional mesh of spacing $h$, the piecewise-linear hat functions give $M_{ii}=2h/3$, $M_{i,i\pm 1}=h/6$, $C_{i,i+1}=1/2$ and $C_{i,i-1}=-1/2$.

For the focusing nonlinear Schrödinger equation on ${\mathbb{R}}^d$,the energy functional isFor a sufficiently smooth solution with decay permitting differentiation under the integral sign and integration by parts, it is constant in time. Indeed,The cancellation is the Hamiltonian conservation mechanism: the evolution rotates the energy gradient by multiplication by $i$. The focusing sign makes the nonlinear contribution negative, so conserved energy alone need not bound the L2 norm of the gradient.^[ref]

For less regular solutions the displayed computation is only formal. To inherit the identity, approximate by smooth solutions and justify convergence of both energy terms. Convergence in Sobolev space $H^1$ controls the nonlinear term through a suitable Sobolev embedding only in the applicable exponent range, or with additional integrability. An arbitrary weak solution must not be assumed to satisfy the identity solely because the smooth equation does.

An energy estimate tests a differential equation against the solution or one of its derivatives to control a norm by initial data and forcing. Coercive terms represent dissipation, while Young inequality and the Gronwall inequality control lower-order terms.

For the difference $w$ of two smooth heat equation solutions with $D>0$, identical PDE source term, initial conditions, and either periodic boundary conditions or Dirichlet boundary data, integration by parts givesThe initially zero nonnegative integral stays zero, proving uniqueness. The same computation bounds changes in the L2 norm when the initial conditions differ. Specifying data and a solution class makes this argument meaningful.^[ref]

An energy identity is an exact balance obtained by differentiating a chosen functional along a solution of a differential equation. The functional often is a squared L2 norm, $D$ collects dissipative terms, and $F$ collects forcing or reaction terms. Dropping nonnegative terms gives an energy estimate.

For $u_t=u_{xx}+g$ on $(0,1)$ with $u_x(0,t)=\alpha u(0,t)$ and $u(1,t)=0$, integration by parts provesThe boundary term is $[u{u}_x{]}_0^1=-\alpha u(0,t{)}^2$; the minus sign at the left endpoint is essential. For $\alpha \ge 0$ it contributes dissipation. If $ug\le a(t)u^2$ and the initial data vanish, the Gronwall inequality forces $u\equiv 0$.

An error norm measures a numerical error vector or error function in a specified norm. For nodal errors $e_j=U_j-u(x_j)$ on a one-dimensional uniform grid of spacing $h$, common choices areThe first controls the largest nodal error. The second approximates a continuum L2 norm on a fixed domain. For nonuniform grids, quadrature or finite-element mass weights replace the common factor $h$.

Mesh weighting matters in a refinement study. If $e_j=\epsilon$ on $N$ cells of a unit interval with $h=1/N$, then $\Vert e{\Vert }_{2,h}=|\epsilon |$, whereas the unweighted Euclidean norm is $\sqrt{N}|\epsilon |$. The latter grows solely because the number of entries grows. Compare the same norm on the same physical domain and, for an evolution problem, at the same physical time.

For nonzero $u\in H^1({\mathbb{R}}^d)$, the Weinstein functional is invariant under amplitude multiplication and spatial dilation. Apply symmetric decreasing rearrangement to a minimizing sequence, then use those two invariances to arrangewith nonnegative radial, radially nonincreasing $u_n$. Rearrangement preserves the relevant Lp norms and does not increase the gradient L2 norm.

These monotone radial profiles have uniform pointwise bounds $|u_n(x)|\le C_d|x{|}^{-d/2}$, since their value on a ball is bounded by its average squared mass. For every $p>2$, this givesLocal Rellich-Kondrachov compactness theorem and this tail bound yield strong convergence in $L^{2+4/d}$ after a subsequence; this exponent is below $2^{\ast }=2d/(d-2)$ for $d\ge 3$, the upper range in the Sobolev embedding theorem; put $2^{\ast }=\infty$ for $d=1,2$. The argument also covers $d=1$ because monotone rearrangement, rather than radial symmetry alone, controls the tails.

The Gagliardo-Nirenberg interpolation inequality bounds the infimum of the Weinstein functional away from zero. The strong nonlinear-norm limit is therefore nonzero. Weak lower semicontinuity of the two normalized quadratic norms shows that the limit attains the infimum, giving a nonnegative minimizer.

Let $D_h$ be the scaled Dirichlet discrete Laplacian and $V_h$ a bounded diagonal reaction matrix. If $0\le k/h_x^2\le 1/2$, the heat update $H_h=I+k{D}_h$ is contractive in both the maximum norm and the mesh-weighted L2 norm. For $\Vert V_h\Vert \le A$,This proves finite-time mesh-uniform stability without requiring the full update to have nonnegative entries or to be contractive. Negative reaction terms can destroy nonnegativity at the endpoint $k/h_x^2=1/2$, so it is the heat part alone that is treated as a nonnegative contraction.

For $u_t=u_{xx}+\alpha u_x$ on the whole line, apply centered differences in space and the Forward Euler method in time. With $r=\Delta t/(\Delta x{)}^2$ and $c=\alpha \Delta t/\Delta x$, the amplification factor isPutting $s={\sin }^2(\theta /2)$ givesFor $\Delta t>0$, the affine expression in braces is nonpositive on $[0,1]$ exactly when $r\le 1/2$ and $c^2\le 2r$. Hence von Neumann stability analysis givesThese conditions are weaker than requiring every stencil weight to be nonnegative; stability in the discrete L2 norm need not imply monotonicity.

A constant-coefficient translation-invariant grid update is diagonalized spatially by discrete Fourier modes. Its scalar or matrix multiplier is the displayed symbol. In a semidiscrete evolution, the analogous symbol is a continuous-time generator and its exponential propagates each mode. Discrete Parseval identity converts uniform modal bounds to discrete L2 norm bounds.

On $[0,L]$, the Fourier cosine series convention $f(x)=a_0/2+{\sum }_{n\ge 1}{a}_n\cos (n\pi x/L)$ uses the displayed coefficients, including $n=0$. Orthogonality of the cosine modes gives these formulas for square-integrable functions; the expansion is interpreted in the L2 norm until stronger convergence is established.

This symmetric finite difference method approximates a second derivative with $D_{4,h}u=u^{\prime \prime }-h^4{u}^{(6)}/90+O(h^6)$ for sufficiently smooth functions. Matching the constant, second-derivative and fourth-derivative coefficients of a symmetric five-node Taylor expansion determines the displayed weights uniquely; its sixth-derivative coefficient is nonzero. The Fourier symbol is $-4h^{-2}{\sin }^2(\theta /2)[1+{\sin }^2(\theta /2)/3]\le 0$. Thus the semidiscrete heat equation $\dot{U}=D_{4,h}U$ is contractive in the discrete L2 norm by the discrete Parseval identity. This says nothing by itself about stability of a subsequently chosen time update.

Functional analysis brings linear algebra and topology together to study vector spaces of functions and the linear operators acting on them. A function space such as $C([0,1])$ has infinitely many independent directions: the monomials $1,x,x^2,\dots$ are linearly independent. Finite linear combinations alone cannot describe all its continuous functions. A useful theory must also say when approximations converge, whether their limits belong to the chosen space, and which operations preserve that norm convergence. Normed vector spaces, Banach spaces, and bounded linear operators provide these three controls.^[ref]

A bounded linear operator $T:X\to Y$ between normed vector spaces satisfies $\Vert Tf{\Vert }_Y\le M\Vert f{\Vert }_X$ for every $f$, with one finite constant $M$. The least possible constant is its operator norm. By linearity, the same estimate controls approximation errors:This is exactly continuity for a linear map between normed vector spaces; neither space needs to be complete. It turns an input accuracy guarantee into an output accuracy guarantee, even when the input has infinitely many degrees of freedom.^[ref]

For a concrete linear functional on $C([0,1])$, take the integral $F(f)={\int }_0^{1}f(x)dx$. In the supremum norm,The constant function $f=1$ attains equality, so $\Vert F\Vert =1$. For $f(x)=x^2$, the output is $F(f)=1/3$; any approximation $g$ with $\Vert g-x^2{\Vert }_{\infty }\le 0.01$ therefore has $|F(g)-1/3|\le 0.01$. The Cauchy-Schwarz inequality also gives $|F(f)|\le \Vert f{\Vert }_2\Vert 1{\Vert }_2=\Vert f{\Vert }_2$ on this interval, again with equality at $1$. Thus integration remains a bounded linear functional under the L2 norm, although point evaluation behaves very differently.

The beauty of this viewpoint is that one estimate can justify an entire approximation procedure. Once a sequence converges in the chosen norm, every bounded linear operator carries its limit along. Completeness ensures that Cauchy approximations actually describe an element of the space. The choice of norm matters: it determines which errors are small and which measurements are stable. The examples below distinguish an average, which ignores sufficiently narrow peaks, from point evaluation, which detects their full height. For applications, this decides whether reducing an overall fitting error guarantees an accurate reported measurement. A small L2 norm of the error controls its integral on this interval, while it gives no such guarantee for a single sampled value.

For $i{\psi }_t=-{\psi }_{xx}/2-|\psi {|}^2\psi$, translation of a solution at velocity $u$ must be accompanied by the displayed phase. Direct differentiation cancels the terms proportional to ${\partial }_x{\psi }_0$, leaving the original equation. A standing profile of frequency $E=-{\kappa }^2/2$ thus gives $\kappa \mathrm{sech}(\kappa (x-ut-X_0))e^{iux-i(u^2/2+E)t+i{\theta }_0}$. Its centre travels uniformly, while its L2 norm is unchanged.

Draw a $k\times D$ random matrix $G$ whose entries are independent random variables with the standard normal distribution, and setThe factor $1/\sqrt{k}$ makes the expected value of the squared L2 norm equal to the original squared L2 norm. This Gaussian Johnson–Lindenstrauss transform is a linear map, generally not an orthogonal projection: its rows are not required to be orthogonal vectors. ^[ref]

For the fixed finite set $S$, let $0<\delta <1$ be an allowed failure probability, and defineWith $\log$ denoting the natural logarithm, a sufficient integer target dimension isThen one draw of this specified random matrix satisfies all the squared Euclidean distance bounds simultaneously with probability at least $1-\delta$. Since $c(\epsilon )\ge {\epsilon }^2/12$, the simpler sufficient bound $k\ge \lceil 12{\epsilon }^{-2}\log (n(n-1)/\delta )\rceil$ also works. Both are sufficient bounds for this argument; they do not claim the smallest possible target dimension.

This is a probability statement about a map drawn independently of the fixed finite set. It implies existence because the success probability is positive, but permits failed draws. The set cannot be chosen after inspecting the map with this same guarantee. In particular, no single draw preserves every possible finite set in the ambient Euclidean space when $k<D$.

The Generalized Korteweg–De Vries equation is the nonlinear partial differential equationFor the displayed real power, one commonly takes an integer $p\ge 2$; noninteger models need a specified real nonlinearity such as $|u{|}^{p-1}u$. The third spatial derivative produces dispersion, while the nonlinear term changes propagation with amplitude. The familiar Korteweg-De Vries equation corresponds, after coefficient rescaling, to $p=2$.

Its scaling is $u(t,x)\mapsto {\lambda }^{2/(p-1)}u({\lambda }^{3}t,\lambda x)$. Therefore the squared L2 norm scales by ${\lambda }^{4/(p-1)-1}$. The quintic case $p=5$ is mass-critical; the integer cases $p=2,3,4$ are mass-subcritical. This is the threshold behind global existence for the mass-subcritical generalized KdV equation.

For the integer powers $p=2,3,4$, the Generalized Korteweg–De Vries equation is mass-subcritical. Its Gagliardo-Nirenberg interpolation inequality isThe exponent $\alpha =(p-1)/2$ is strictly below two. Since KdV mass and energy conservation fixes $\Vert u{\Vert }_2$ and $E(u)$, the gradient size $X=\Vert u_x{\Vert }_2$ obeys $X^2/2\le |E(u_0)|+C_{M(u_0)}{X}^{\alpha }$. Young's inequality for products absorbs the lower power into $X^2/4$ and leaves a uniform bound for $X$.

Under the standard local $H^1(\mathbb{R})$ solution theory with a continuation time controlled by the H1 norm, this bound extends the solution globally. At $p=5$, the nonlinear estimate has gradient exponent two, so this absorption argument requires additional information such as a small-mass threshold. The scaling identifies this as an L2 norm threshold.

The Haar projection at level $J$ is the orthogonal projection onto the functions constant on level-$J$ dyadic intervals. On each cell it replaces $f$ by its cell average. This is the closest cellwise-constant approximation in the L2 norm, because choosing a constant $c$ on a cell minimizes ${\int }_I|f-c{|}^2$ at its cell average.

On $[0,1]$, take the step function with values $2,4,6,8$ on successive quarter-intervals. At level zero, its Haar projection is the constant $5$. At level one, it has values $3$ and $7$ on the two halves. At level two, it recovers the four original values exactly. With normalized Haar wavelets, these approximations areThe coefficients are L2 inner products, so they differ from the discrete vector coefficients by the factor $1/\sqrt{4}=1/2$. Their squared sum also has a direct meaning:The approximation errors in squared L2 norm of $P_{0}f,P_{1}f,P_{2}f$ are respectively $5,1,0$: each added detail removes exactly its own squared coefficient from the error.

The Haar projection can be expressed using level-$J$ Haar scaling functions, or through a coarse scaling family and the Haar wavelets at intermediate levels. On $[0,1]$ with lowest level zero, that coarse family is the single constant function $\varphi$. On the whole real line, use the integer translates of a coarse Haar scaling function instead; one global constant is not square-integrable. Refinement improves the representation while retaining the earlier coefficients.

The Haar projection replaces a function by its mean on each dyadic interval of length $2^{-j}$. For a function satisfying Lipschitz continuity with constant $L$, every value in a cell differs from its cell mean by at most $L{2}^{-j}$. This bound holds for all points with a consistent endpoint convention, and requires no global L2 norm assumption when the projection is defined by local averages.