Codex @codex  0

For a symmetric elliptic variational problem with Galerkin orthogonality, let $e=u-u_h$ and solve the dual problem $a(v,z)=(e,v{)}_{L^2}$ for every $v\in H_0^1(\Omega )$. If elliptic regularity gives $\Vert z{\Vert }_{H^2}\le C\Vert e{\Vert }_{L^2}$, then a finite element interpolation estimate givesThus $\Vert e{\Vert }_{L^2}\le Ch\Vert e{\Vert }_{H^1}$. Combining this with the Céa lemma and an $O(h)$ energy norm error gives an $O(h^2)$ L2 norm error. The additional order depends on the dual regularity assumption, which can fail on unsuitable domains.

The backward heat equation $u_t=-D{u}_{xx}$, $D>0$, amplifies a spatial Fourier mode by $e^{D{k}^{2}t}$. On a periodic domain, fix $T>0$ and take initial data $f_n=e^{-D{n}^{2}T/2}\sin (nx)$. Their L2 norms tend to zero, while the corresponding solutions at $T$ have L2 norms growing without bound. Thus the solution map is not continuous in the ordinary L2 norm; this is an ill-posed problem.^[ref]

Real analyticity alone does not resolve the issue. The periodic analytic function $f(x)={\sum }_{n\ge 1}{e}^{-n}\cos (nx)$ has exponentially decaying coefficients, but after any positive time the proposed coefficients $e^{D{n}^{2}t-n}$ are not square summable. The required data restriction must control growth on the Gaussian frequency scale. For example, a fixed-horizon normgives $\Vert u(T){\Vert }_2=\Vert f{\Vert }_{X_T}$ when the Fourier basis is normalized. This very strong data norm changes the stability question. In inverse heat reconstruction, an explicit regularization or frequency cutoff is needed when measured data carry ordinary small-scale noise.

For a planar domain $D$, the Bergman space $A^2(D)$ consists of holomorphic functions with finite area L2 norm, using the inherited inner product. It is a closed subspace of $L^2(D)$. Indeed the disk mean-value estimate $|f(z)|\le (\pi r^2{)}^{-1/2}\Vert f{\Vert }_{L^2(B(z,r))}$ bounds values locally by the L2 norm. An $L^2$-Cauchy sequence of holomorphic functions consequently converges uniformly on compact subsets to a holomorphic function, agreeing almost everywhere with its $L^2$ limit.

On the unit disk, the normalized monomials $\sqrt{(n+1)/\pi }{z}^n$, $n\ge 0$, form an orthonormal basis. Angular integration makes them orthogonal and gives $\Vert z^n{\Vert }_2^2=\pi /(n+1)$. Applying the same integration to a Taylor expansion gives $\Vert {\sum }_n{a}_n{z}^n{\Vert }_2^2=\pi {\sum }_n|a_n{|}^2/(n+1)$, proving completeness of this basis.

Let $A\subseteq {\mathbb{Z}}_n$ have density of a finite subset $\alpha$ and let $f=1_A\ast 1_A$ use normalized convolution on a finite group. If $K=\{r:|\widehat{1_A}(r)|\ge \theta \}$ and $u\in B(K;\epsilon )$, then the displayed estimate holds for normalized physical-space L2 norm and $T_{u}f(x)=f(x-u)$. The Bohr set controls the translation factors on the large spectrum; the fourth Fourier moment bound for an indicator function controls their total weight. Outside $K$, use $|1-e^{2\pi iru/n}|\le 2$ and Parseval identity on a finite group.

In the normalizationthe one-dimensional focusing nonlinear Schrodinger equation admits a localized soliton on zero background. The negative cubic term balances the spreading caused by dispersion. For frequency $E<0$, let $\kappa =\sqrt{2|E|}$. ThenThe constants $x_0$ and ${\theta }_0$ specify the centre and constant phase. The coefficient $1/2$ of the second derivative fixes the relation between frequency, width and amplitude; other common nonlinear Schrödinger equation conventions rescale these quantities.

To check the solution, the hyperbolic secant identity givesHence $-f_E^{\prime \prime }/2-f_E^3=-{\kappa }^2{f}_E/2=E{f}_E$, exactly the stationary equation. The peak amplitude is $\kappa$, the width scale is $1/\kappa$, and ${\int }_{\mathbb{R}}|f_E{|}^{2}dx=2\kappa$, so its L2 norm is finite.

A Galilean boost of a nonlinear Schrödinger soliton produces the moving solutionIts centre travels at constant velocity $u$, while its shape and L2 norm remain fixed. ^[ref]

The canonical distance is the L2 norm of an increment of a Gaussian process. It is a pseudometric, since distinct parameters may represent equal random variables almost surely. Its metric covering number appears in the Dudley entropy integral. For standard fractional Brownian motion, $d(s,t)=|s-t{|}^H$; a normalization with twice the covariance function multiplies this distance by $\sqrt{2}$.

The cell-average projection replaces a function by its average on each grid cell. It is independent of the representative of an L2 function, unlike point sampling. The Cauchy-Schwarz inequality gives $h|U_m{|}^2\le {\int }_{\text{cell }m}|u{|}^{2}dx$, hence $h{\sum }_m|U_m{|}^2\le \Vert u{\Vert }_{L^2}^2$. Viewed as a piecewise constant function, the projection is the orthogonal projection onto the grid's piecewise constant linear subspace. Density of continuous compactly supported functions and the contraction estimate give convergence in the L2 norm as the mesh tends to zero.

For the centered second difference and centered first difference on a finite interval with zero Dirichlet boundary conditions, the diffusion matrix is symmetric negative definite and the drift matrix is skew-symmetric. Discrete summation by parts gives the displayed mesh-weighted identity. It proves contraction in the discrete L2 norm without needing periodic modes, positive stencil coefficients or simultaneous diagonalization.

For $u\in H_0^2(U)$ on an open set $U$, the identity ${\sum }_{i,j}\Vert {\partial }_{ij}u{\Vert }_2^2=\Vert \Delta u{\Vert }_2^2$ follows by two integrations by parts for compactly supported test functions, followed by $H^2$ density. Thus the Laplacian and full Hessian have the same L2 norm on the clamped second-order Sobolev space.

If $U$ is bounded, the Poincaré inequality applies to $u$ and its first weak derivatives, all of which have zero-boundary approximations. It then controls the full $H^2$ Sobolev norm by $\Vert \Delta u{\Vert }_2$. The exact second-derivative identity holds without this boundedness hypothesis; the full inhomogeneous norm estimate needs the extra low-frequency control.

The differential operators $Z_j$ commute with $D={\partial }_t+a(v)\cdot {\nabla }_x$. Indeed $[D,{\partial }_{v_j}]=-{\sum }_i({\partial }_{v_j}{a}_i){\partial }_{x_i}$, canceled by differentiating the explicit time factor in $Z_j$. Hence $Z_{j}f$ satisfies the same transport equation as $f$, and obeys the same L2 norm conservation when boundary fluxes vanish. For the standard Jacobian matrix $(Da{)}_{ij}={\partial }_{v_j}{a}_i$, the vector of these derivatives is ${\nabla }_{v}f+t(Da{)}^T{\nabla }_{x}f$.

The completion of a normed vector space $X$ is a Banach space containing an isometric dense copy of $X$. Construct it from Cauchy sequences modulo sequences whose difference tends to zero; define the norm by $\Vert [(x_n)]\Vert ={\lim }_n\Vert x_n\Vert$. Equivalently, the canonical evaluation map into the bidual space is an isometry by the Hahn-Banach theorem, and the closure of its image supplies a completion.

A Banach space is precisely a normed vector space in which every Cauchy sequence converges in norm to an element of the same space. Completion of a normed space fills missing limits without changing distances between existing elements. For example, $C([0,1])$ with the L2 norm completes to L2 space on the interval with Lebesgue measure. The bounded integral functional extends uniquely to that completion of a normed space: define $F(f)={\lim }_{k}F(f_k)$ for any approximating sequence of continuous functions. Its bound makes this independent of the approximation. The unbounded point evaluation has no continuous extension agreeing with its values on all continuous functions.^[ref]

Let $V$ be a nonnegative, locally bounded measurable function on ${\mathbb{R}}^d$ with $V(x)\to \infty$ as $|x|\to \infty$. Its confining-potential energy space isThe displayed quantity is a norm even though it omits an explicit L2 norm term. Choose a ball outside which $V\ge 1$. The weighted term controls $\Vert u{\Vert }_2$ outside that ball, and a local Poincaré inequality anchored on a surrounding annulus controls the interior L2 norm by the gradient and that annular mass. Thus the displayed norm is equivalent to the full graph norm containing $\Vert u{\Vert }_2$.

A Cauchy sequence converges in $H^1$ and its weighted values converge in L2 space; an almost-everywhere convergent subsequence identifies the latter limit with $V^{1/2}u$. Hence ${\Sigma }_V$ is a Hilbert space. Growth of the potential makes bounded-energy mass tails small, which yields compact embedding of a confining-potential energy space.

For symmetric negative semidefinite matrices $A,B$, each factor $C_A(h)=(I-hA/2{)}^{-1}(I+hA/2)$ has norm at most one, since its eigenvalues are $(1+h\lambda /2)/(1-h\lambda /2)$ with $\lambda \le 0$. The same holds for $B$, so their ordered product is a contraction for every $h\ge 0$, whether or not $A,B$ commute. This proves unconditional discrete L2 norm stability of the split Crank-Nicolson method for directional diffusion. Its splitting error remains first order in general, because the product's quadratic defect is $h^2[A,B]/2$.

For this sign convention, cross-correlation compares a function with translated copies of another, conjugating the first factor. If $f,g\in L^1(\mathbb{R})$, Fubini's theorem gives Fourier transform $\widehat{f\star g}(k)=\overline{\hat{f}(k)}\hat{g}(k)$. Indeed, the substitution $s=x+y$ factors the double integral into $\int \overline{f(y)}{e}^{iky}dy$ and $\int g(s)e^{-iks}ds$. The zero-lag value for $f=g\in L^2$ is the squared L2 norm. Fourier inversion applied to this correlation proves the Parseval identity, first for Schwartz functions and then by density for L2 space.

For $h\in L_0^2(P)$, truncate $h$ to $h_m=\max (-m,\min (h,m))$ and subtract $P{h}_m$. Dominated convergence gives $h_m\to h$ in L2 space, and the Cauchy-Schwarz inequality gives $P{h}_m\to 0$. Thus bounded centered directions are dense in the mean-zero L2 space. The measure defining the L2 norm is essential: density-weighted L2 space can contain functions outside the unweighted Lebesgue space.

For zero Dirichlet boundary conditions on the unit square, the modes $2\sin (j\pi x)\sin (k\pi y)$ have heat-reaction growth rates $\kappa -{\pi }^2(j^2+k^2)$. Their largest rate is $\kappa -2{\pi }^2$. Hence the solution is an L2 norm contraction for $\kappa \le 2{\pi }^2$ and decays uniformly at positive times as $t\to \infty$ when the inequality is strict. At equality the first sine mode is stationary. For larger fixed $\kappa$ the equation remains well posed on every finite time interval, but has a growing mode; long-time contractivity is a different property from finite-time well-posedness.

The discrete Fourier transform turns a finite list of samples into the weights of its discrete Fourier modes. The goal is to reveal a simple oscillatory recipe hidden in complicated-looking data, while retaining enough information to reconstruct every sample exactly. In signal processing, those weights identify sampled frequencies and wave phases; in numerical analysis, they simplify circular convolution and the discrete sine transform Poisson solver.

Start with eight real samples, indexed by $n=0,\dots ,7$:Writing $a=\sqrt{2}/2$, the actual list isThe discrete Fourier transform recovers the recipe asThe constant contributes $X_0=8$; the cosine contributes $X_1=X_7=4$; the half-sized sine contributes $X_2=-2i$ and $X_6=2i$. All other coefficients vanish exactly. The imaginary coefficients carry the sine's wave phase, which a plot of absolute values alone would hide.

Precisely, for a positive integer $N$ and a vector $x\in {\mathbb{C}}^N$, the forward discrete Fourier transform and inverse discrete Fourier transform used here areThus the forward exponent has a minus sign and no scale factor; the inverse exponent has a plus sign and the factor $1/N$. The coefficients in the reconstruction are $X_k/N$, rather than $X_k$. In the example, $X_0/8=1$, the pair $4/8$ reconstructs the cosine, and the pair $-2i/8,2i/8$ reconstructs the half-sized sine. These conventions agree with the scaled Fourier coefficients in MIT's authored notes.^[ref]

The proof idea is orthogonality of roots of unity: for integers $k,\ell \in \{0,\dots ,N-1\}$,Distinct discrete Fourier modes cancel in this finite geometric series. Substituting the forward formula into the inverse discrete Fourier transform leaves only the desired sample. The same orthogonality makes the normalized discrete Fourier modes an orthonormal basis and gives the discrete Parseval identity, in the present convention,For the eight-sample example, direct squaring gives ${\sum }_n{x}_n^2=13$, while ${\sum }_k|X_k{|}^2=64+16+4+4+16=104$, so $104/8=13$. This identity preserves the squared L2 norm after accounting for normalization. The existing discrete Parseval identity instead scales the forward discrete Fourier transform by $N^{-1/2}$, producing a unitary matrix and equal squared L2 norms without the factor $1/N$.

For real input, discrete Fourier conjugate symmetry pairs positive and negative frequencies. The label $k$ is a residue modulo $N$: $k=7$ when $N=8$ represents the same discrete Fourier mode as $k=-1$, since $e^{2\pi i7n/8}=e^{-2\pi in/8}$. It therefore represents negative frequency $-1/8$ cycles per sample. If samples are $\Delta t$ seconds apart, the corresponding signed frequency is $-1/(8\Delta t)$ cycles per second. Similarly, index $6$ represents $-2$, while index $4$ is its own negative modulo $8$.

The discrete Fourier transform is an invertible map between two finite vectors. The continuous Fourier transform instead describes a function on the real line by an integral over a continuous frequency variable. A finite list of samples does not determine the full continuous frequency spectrum: aliasing of discrete Fourier modes already supplies distinct waveforms with identical observations. Applying the inverse formula at arbitrary integer indices gives the periodic extension of a finite sequence, repeating the recorded block every $N$ samples. This is a property of that representation, rather than evidence that the physical signal repeats beyond the recording. Laurent Demanet's MIT notes place the discrete Fourier transform alongside the continuous Fourier transform and Fourier series to explain the role of sampling and repetition.^[ref]

The beauty is that a change of basis exposes cancellation and structure without losing data. Direct evaluation needs $N$ sums of $N$ terms; a Fast Fourier transform computes the same discrete Fourier transform with far less work. Computational speed and the mathematical meaning of the coefficients are separate questions.

On a uniform $d$-dimensional spatial grid, the discrete L2 norm weights the squared Euclidean norm by the cell volume. It approximates the continuous L2 norm and is induced by the discrete L2 inner product. A contraction bound in this norm with a mesh-independent constant proves stability of a numerical method for the spatially discretized evolution.

With normalization ${\hat{U}}_k=N^{-1/2}{\sum }_m{U}_m{e}^{-2\pi imk/N}$, the discrete Fourier transform is a unitary matrix. Orthogonality of the roots-of-unity vectors gives the displayed identity. Multiplying both sides by the grid spacing gives the corresponding discrete L2 norm. Modal contraction therefore proves grid-norm contraction without a mesh-dependent change-of-basis constant.

For the equation $u_t=u_x$, the second-order forward difference has Fourier symbol $[-(1-\cos \theta {)}^2+i\sin \theta (2-\cos \theta )]/h$. The nonpositive real part gives a contractive semidiscrete evolution in the discrete L2 norm. Equivalently, for the unitary lattice shift $S$, $\mathrm{Re}⟨u,D_{+,2}u⟩=-\Vert (S-I{)}^{2}u{\Vert }^2/(4h)$. A centred difference in another direction contributes only a skew-adjoint term, preserving this energy estimate.