A complex-valued function is complex differentiable at $z_0$ ifexists. The same limit must hold for every approach, including curved paths. Equivalently,The error divided by $|h|$ tends to zero. Thus the derivative is a single complex multiplier, not an arbitrary real linear map of the plane. If the real coordinate map is differentiable, complex differentiability is equivalent to its satisfying the Cauchy-Riemann equations at that point.

The simplest failure is complex conjugation, $f(z)=\overline{z}$. At every $z_0$ its increment quotient is $\overline{h}/h$. For real $h=t\ne 0$ it is $1$, while for $h=it$ it is $-1$. More generally, $h=t{e}^{i\theta }$ gives $e^{-2i\theta }$. Although complex conjugation is a smooth real reflection and has every real directional derivative, these incompatible quotients rule out a complex derivative everywhere.

Agreement on two axes is necessary, but does not by itself establish a limit over the whole complex plane. The Cauchy-Riemann equations need the real differentiability condition above, or a sufficient regularity condition such as continuous partial derivatives near the point. There is a second distinction: $f(z)=|z{|}^2$ has complex derivative zero at zero because its quotient there is $\overline{h}\to 0$, but its Cauchy-Riemann equations fail at every nonzero point. It is complex differentiable at zero without being a holomorphic function on any neighbourhood of zero.

Two rational sphere maps are Möbius conjugate if $S=h\circ R\circ h^{-1}$ for a Möbius transformation $h$. This is precisely global holomorphic conjugacy between Riemann-sphere self-maps. It preserves degree, local degrees of a holomorphic map, exact periods and return periodic-orbit multipliers, and transports the Fatou set and Julia set by $h$. Since a Möbius map and its inverse are Lipschitz on the compact sphere, this last assertion follows directly from equicontinuity of the iterate families. An affine holomorphic conjugacy is the special case where $h$ fixes infinity.

For a random variable with finite fourth moment, apply Holder inequality with conjugate exponents $3/2$ and $3$ to $|X{|}^{2/3}|X{|}^{4/3}$. This gives $\mathbb{E}|X{|}^2\le (\mathbb{E}|X|{)}^{2/3}(\mathbb{E}|X{|}^4{)}^{1/3}$. Raising to the power $3/2$ proves the displayed Lp norm interpolation inequality.

A pure integral Hodge structure is a finite-rank free abelian group with a complex decomposition satisfying $\overline{H^{p,q}}=H^{q,p}$ and $p+q=w$. The integer $w$ is its weight. Its equivalent Hodge filtration is $F^p={⨁}_{r\ge p}{H}^{r,w-r}$.

For a finite family of irreducible separable polynomials $f_i(T,X)$ and a nonzero exclusion polynomial $h(T)$, this is the set of parameters $a\in K$ with $h(a)\ne 0$ for which every specialization remains irreducible of its original degree. A Hilbertian field has infinitely many points in each such set. Denominators, leading coefficients and discriminants can be included in $h$.

The rational numbers, and every number field, are Hilbertian fields. The classical integral form supplies infinitely many integral specializations avoiding a prescribed finite exceptional set. The theorem preserves irreducibility, rather than asserting that every specialization works: $X^2-T$ becomes reducible at square parameters.

For a unit vector $v$, the directional Hilbert transform integrates along lines parallel to $v$. Its Fourier multiplier is $-i\mathrm{sgn}(\xi \cdot v)$ under the usual transform convention. Applied to a rectangle whose long axis is $v$, it reduces to the one-dimensional logarithmic tail of an interval indicator function, separately on each transverse coordinate.

With convention $\mathcal{H}f(t)={\pi }^{-1}\mathrm{PV}\int f(\tau )/(t-\tau )d\tau$, the analytic function $F(z)=(e^{iz}-1)/(iz)$ has removable singularity at zero and decays on large upper semicircles. Its boundary real part is $\sin t/t$ and imaginary part is $(1-\cos t)/t$. The upper-half-plane contour indented above $t$ gives $\mathrm{PV}\int F(\tau )/(\tau -t)d\tau =i\pi F(t)$. Taking real parts after reversing the denominator proves the displayed transform, with value zero at $t=0$ by continuous extension.

For every nonempty closed convex set $C$ in a Hilbert space and every $x$ in that space, there is a unique $p\in C$ satisfyingClosedness and completeness let a limiting approximation attain the infimum; convexity prevents two different minimizers. The proof idea is to choose a sequence approaching the infimum. The parallelogram identity and the admissible midpoint make it a Cauchy sequence; applying the same identity to two putative minimizers proves uniqueness.

For a closed subspace of a Hilbert space, this nearest-point map is an orthogonal projection and is linear. For a general closed convex set it need not be linear: projecting the real line onto $[0,1]$ clips values below zero to zero and values above one to one. The theorem is therefore useful for both linear approximation and constrained nearest-point problems.^[ref]

Closedness is essential even in familiar examples. For the open interval $(0,1)\subseteq \mathbb{R}$ and $x=0$, the distance infimum is zero but no nearest point exists. The same obstruction appears when approximating $(1/k{)}_{k\ge 1}$ by finitely supported sequences: the infimum is zero, but the limit lies outside that subspace until we take its Hilbert space completion.

The Bargmann-Fock space is a Hilbert space of entire holomorphic functions with Gaussian-weighted area inner product. If $f(z)=\sum a_n{z}^n$, angular integration and the Gaussian integral give $\Vert f{\Vert }^2=\sum n!|a_n{|}^2$. Conversely any such square-summable coefficient sequence defines an entire function, since $\sum |a_n{z}^n|\le (\sum n!|a_n{|}^2{)}^{1/2}{e}^{|z{|}^2/2}$. This proves completeness and shows that $z^n/\sqrt{n!}$ is an orthonormal basis. The same bound shows boundedness of evaluation and gives reproducing kernel $e^{z\bar{w}}$.

The Hardy space of the circle is the closed subspace of L2 space whose negative Fourier coefficients vanish. The orthogonal projection onto it is denoted $P_{+}$. A continuous symbol $f$ defines a Toeplitz operator $T_f=P_{+}{M}_f{|}_{H^2}$. For continuous $f,g$, $T_f{T}_g-T_{fg}$ is compact: it is finite rank for Laurent polynomials, and uniform approximation extends the result.

The Hilbert tensor product is the completion of the algebraic tensor product of Hilbert spaces with $⟨u\otimes v,u^{\prime }\otimes v^{\prime }⟩=⟨u,u^{\prime }⟩⟨v,v^{\prime }⟩$, extended sesquilinearly. Orthonormal bases identify this inner product with the Euclidean inner product on coefficient arrays, proving positivity. In particular, $⟨u^{\otimes m},v^{\otimes m}⟩=⟨u,v{⟩}^m$ and $\Vert u^{\otimes m}\Vert =\Vert u{\Vert }^m$. These identities turn scalar power series into vector embeddings, as in the proof of the Grothendieck inequality.

A closed subspace of a Hilbert space reduces a bounded operator when both the operator and its adjoint leave the subspace invariant. Equivalently, the orthogonal projection onto it commutes with the operator: the adjoint invariance makes its orthogonal complement invariant too. For a star-closed algebra of operators, the closed span of the orbit of any vector is reducing for the whole algebra. This enables an orthogonal direct-sum decomposition into cyclic actions when constructing Borel functional calculus for a normal operator.

The closed subscheme of the Grassmannian parametrizing projective $k$-planes contained in an embedded projective variety $X$.

For a projective scheme $X$ over a field $k$, the Hilbert functor assigns to a $k$-scheme $T$ the set of closed subschemes $W\subseteq X{\times }_{k}T$ which are of finite presentation, proper and flat over $T$. Pullback of families makes it a contravariant functor. After choosing a very ample line bundle, requiring a fixed fibre Hilbert polynomial $P$ gives the subfunctor represented by ${\mathrm{Hilb}}^P(X)$. The unrestricted functor is represented by the disjoint union of these Hilbert schemes.

For a finitely generated algebra over an algebraically closed field $k$, radical ideals correspond, with inclusion reversed, to closed subsets of the maximal spectrum. If $f\notin \sqrt{I}$, the nonzero algebra $(A/I)[1/f]$ is still finitely generated over $k$. Its maximal residue field is $k$ by the Zariski lemma, so contracting a maximal ideal gives a maximal ideal of $A$ containing $I$ and avoiding $f$. Hence the intersection of all maximal ideals containing $I$ is $\sqrt{I}$. This proves the inverse correspondence, including the empty closed set and unit ideal.

For an ideal $I$ in $R[x]$, let $J_n$ contain the coefficients of $x^n$ in its elements of degree at most $n$. These are ideals of $R$ and form an ascending chain, since multiplying by $x$ embeds one into the next. If $R$ is a Noetherian ring, the chain stabilizes at $N$, and each $J_n$ for $0\le n\le N$ has finitely many generators. Choose corresponding polynomials in $I$. Their union generates $I$: for an element of degree $d$, subtract a combination of the chosen polynomials for $J_d$ if $d<N$, or the polynomials for $J_N$ multiplied by $x^{d-N}$ otherwise. This cancels its leading coefficient and lowers its degree. Induction completes the proof without assuming that $R$ is an integral domain.

Projected bulk Weyl tensor can contribute a radiation-scaling term to the brane Friedmann equation, even without ordinary brane photons. For an anti-de Sitter black-hole bulk the integration constant is related to its mass. This dark radiation on a cosmological brane changes the expansion rate tested by Big Bang nucleosynthesis and Cosmic microwave background anisotropy; it should not automatically be identified with a freely specifiable thermal particle species.

A simple motivating comparison is visible gas evolution from hydrogen peroxide, whose net chemical equation isThe products are water and oxygen. The Royal Society of Chemistry's teaching resource uses small test tubes and droppers, hydrogen peroxide labelled 5–10 volumes, solutions of metal salts including copper(II) sulfate, and approximately $1\mathrm{mol}{\mathrm{L}}^{-1}$ sodium hydroxide. Its qualitative procedure tests additions singly and in pairs to tubes containing about $1\mathrm{cm}$ depth of hydrogen peroxide. It reports the expected classroom result that the additions individually are not effective under those conditions, whereas the solids produced by mixing metal-salt solution with sodium hydroxide give effective heterogeneous catalysis. This is the resource's expected observation, not a newly measured ranking.^[ref]

A focused adaptation compares four matched tubes: no addition, copper(II) sulfate alone, sodium hydroxide alone, and the pair. Use equal initial hydrogen peroxide amounts, identical tubes and the same starting temperature. Record stock concentrations and added volumes; add water to the other tubes to match the total added liquid. Additions can first be checked without hydrogen peroxide to distinguish precipitation from gas evolution. Start a timer at contact, mix consistently, and observe gas evolution over the same interval. These matching controls are proposed improvements to the qualitative resource, not claimed details of its original trials. Follow its supervised chemical-handling requirements; the stated strengths describe its teaching protocol rather than an unrestricted recipe.

The immediate readout is relative bubbling, together with whether a solid appears. More vigorous bubbling in the paired tube supports faster conversion under the comparison conditions; it does not identify the solid's exact composition or establish a unique reaction mechanism. Bubble counts are not a calibrated reaction rate: bubble sizes, gas retention and warming can differ. For a quantitative extension, use a freely moving gas syringe to record collected volume $V(t)$ and compare early slopes $dV/dt$ at matched temperature and pressure; check for leaks and repeat runs. This extension is a measurement proposal, not source-reported gas traces.

The paired treatment also changes pH and produces a surface, so the four tubes alone cannot assign the whole effect to one microscopic catalytic step. A filtered-liquid comparison and a separated, washed-solid comparison would help distinguish dissolved and solid contributions, although washing itself can change the active material. More gas at one time establishes neither a changed equilibrium constant nor indefinite catalyst reuse. The open gas-evolution experiment is especially unsuitable as a demonstration that two closed mixtures reach the same chemical equilibrium: the escaping oxygen changes the constraints. The reversible model above supplies that separate explanation. No numerical experimental rate constants, barrier heights or gas volumes are asserted here.

For the Hénon map $(x,y)\mapsto (y,\mu -bx-y^2)$ with $b>-1$, fixed points exist at $y_{\pm }=-(1+b)/2\pm \sqrt{\mu +(1+b{)}^2/4}$. Their multipliers satisfy ${\lambda }^2+2y\lambda +b=0$. The Jury stability criterion gives asymptotic stability only for the upper branch, with $-1<b<1$ and $-(1+b{)}^2/4<\mu <3(1+b{)}^2/4$. The two-cycle has alternating values $y,z$ satisfying $y+z=1+b$ and $(y-z{)}^2/4=\mu -3(1+b{)}^2/4$. Its second-iterate trace is $4[(1+b{)}^2-\mu ]-2b$ and determinant $b^2$, so it is attracting for $-1<b<1$ until $\mu =(5b^2+6b+5)/4$. The fixed-point fold and first flip are nondegenerate away from endpoints and persist under small smooth perturbations.