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Here $C^{\infty }([0,1])$ means the real- or complex-valued functions continuous on $[0,1]$, smooth on $(0,1)$, with every interior derivative having a continuous extension to the endpoints; endpoint values of derivatives are these extensions. On this space of smooth functions on a compact interval, defineThese finite-valued seminorms increase with $k$ and separate points. In fact every $p_k$ is a norm, since it includes $p_0(f)=\Vert f{\Vert }_{\infty }$. The terminology seminorm includes norms; it does not assert that every measurement has a nontrivial kernel of a seminorm. Smooth convergence means $p_k(f_n-f)\to 0$ for every fixed $k$, equivalently uniform convergence of every derivative separately. It does not demand one tolerance or one convergence index that works for all orders simultaneously. The maximum convention is equivalent to the derivative supremum norm convention using a finite sum, since $p_k(f)\le {\sum }_{j=0}^k\Vert f^{(j)}{\Vert }_{\infty }\le (k+1)p_k(f)$.

For a worked example, $f(t)=t^3$ gives $p_0(f)=1$, $p_1(f)=3$, $p_2(f)=6$, and $p_k(f)=6$ for $k\ge 3$. The linear functional $\ell (f)=f^{\prime \prime }(1/2)$ obeys $|\ell (f)|\le p_2(f)$ and is continuous. Similarly, differentiation is a continuous linear operator $D:f\mapsto f^{\prime }$ for smooth convergence, because $p_k(Df)\le p_{k+1}(f)$.

The space of smooth functions on a compact interval is complete for this seminorm topology. A Cauchy sequence has uniform limits $g_j$ for its derivatives of each order. Passing to the limit in the fundamental theorem of calculus identitygives $g_j(t)-g_j(0)={\int }_0^t{g}_{j+1}(s)ds$, so $g_j^{\prime }=g_{j+1}$ and $g_0$ belongs to the same space. This is the completeness of the smooth-function metric argument; the metric there and the increasing-$p_k$ metric above induce the same smooth topology and the same Cauchy sequences.

Suppose $f(a+h)=f(a)+Lh+r(h)$ and $|r(h)|/\Vert h{\Vert }_1\to 0$. If $c\Vert h{\Vert }_1\le \Vert h{\Vert }_2\le C\Vert h{\Vert }_1$, then $|r(h)|/\Vert h{\Vert }_2\le c^{-1}|r(h)|/\Vert h{\Vert }_1\to 0$. The boundedness of $L$ is also preserved, and the same argument in reverse proves equivalence of differentiability. The derivative is the same linear map, not a norm-dependent replacement.

A finite-dimensional commutative real algebra with a Euclidean multiplicative norm has dimension at most two. Automatic real bilinearity from a multiplicative norm supplies bilinearity if it was not part of the initial description. On its unit sphere, squaring has derivative $v\mapsto 2xv$, injective because there are no zero divisors. The only equal squares are antipodal pairs, so the map induces an injective immersion ${\mathbb{RP}}^{n-1}\to S^{n-1}$. Compactness and the inverse function theorem make it a diffeomorphism when $n\ge 2$. For $n\ge 3$, the fundamental groups $\mathbb{Z}/2$ and zero contradict this. The real and complex number algebras attain dimensions one and two.

The directional derivative tests a function along a chosen straight line, including lines that follow neither coordinate axis. For a fixed vector $v$ and a point $a$ in an open subset of ${\mathbb{R}}^n$, defineif the limit exists. With a unit vector $v$, the parameter $t$ measures signed distance in the input Euclidean space, so the directional derivative is change per unit distance. For a general $v$, it is change per unit parameter; the norm of $v$ also sets the rate of travel.^[ref]

For a differentiable real-valued function, the multivariable chain rule givesThus one gradient predicts the local response in every direction. This formula requires differentiability, rather than merely separate partial derivatives.^[ref]

Return to $f(x,y)=x^2+xy+2y^2$ at $a=(1,1)$. Its gradient is $(3,5)$. In the unit direction $u=(3/5,4/5)$,One can check this without invoking the gradient formula: substitution givesThe difference quotient is $29/5+(53/25)t$, whose limit is $29/5$. This calculation separates the exact change along a finite path from its first-order rate at the starting point.

Changing the travel speed changes the parameter rate. Taking $v=(3,4)=5u$ gives $D_{v}f(a)=29$, because the input travels five distance units per parameter unit. For a differentiable function, $D_{cv}f(a)=c{D}_{v}f(a)$ for any real $c$: negative $c$ reverses the orientation as well as scaling the rate. Normalizing $v$ before comparing different slopes avoids confusing direction with speed.

The Cauchy-Schwarz inequality bounds the rate in any unit direction by $\Vert \nabla f(a)\Vert$. Here the largest rate is $\sqrt{34}\approx 5.831$, attained in the direction $(3,5)/\sqrt{34}$; the chosen $(3/5,4/5)$ gives the slightly smaller rate $5.8$. The perpendicular unit direction $(-5,3)/\sqrt{34}$ gives zero. That means no first-order height change, rather than no change at all: along this direction the example has $f(a+tu)=4+(28/34)t^2$.

This is useful for following a height or temperature scalar field, choosing a local search direction in mathematical optimization, and distinguishing a coordinate sensitivity from a response along an actual path. For a curved path, derivative along a curve uses its instantaneous tangent vector. Even existence of all directional derivatives does not guarantee one common linear approximation; directional derivatives do not imply differentiability gives a concrete counterexample.

For a smooth map between Riemannian manifolds, its Dirichlet energy is half the integral of the squared norm of its differential, using the source and target metrics. When the source is two-dimensional, a conformal change of its metric leaves this energy unchanged. For a J-holomorphic curve into a symplectic manifold with a compatible almost complex structure, the Energy identity for a J-holomorphic curve identifies it with the pulled-back symplectic area. The norm of a differential here is the full tensor norm, summing its squared values on an orthonormal source frame.

For the Kac collision operator, expanding the squared Hilbert space norm gives $⟨F,(I-Q)F⟩=(4\pi (\genfrac{}{}{0ex}{}{N}{2}){)}^{-1}{\sum }_{i<j}{\int }_0^{2\pi }{\int }_{{\mathbb{R}}^N}|F(R_{ij,\theta }v)-F(v){|}^{2}dvd\theta$. The nonnegative form measures failure of invariance under the pair rotations. It vanishes precisely on the radial kernel of the Kac collision operator.

The discrete L2 inner product is the cell-volume-weighted Hermitian inner product on a uniform grid. Its associated norm is the discrete L2 norm. It provides the finite-dimensional analogue of spatial integration by parts and continuous energy method identities.

For a real number $x$, define $\Vert x{\Vert }_{\mathbb{R}/\mathbb{Z}}={\min }_{k\in \mathbb{Z}}|x-k|$. It lies in $[0,1/2]$ and obeys $\Vert \ell x\Vert \le |\ell |\Vert x\Vert$ for an integer $\ell$. It is a distance on the circle, rather than a norm on the real vector space.

For $p\ge 2$ and $p^{\prime }=p/(p-1)$, interpolate the two-coordinate map $(x,y)\mapsto (x+y,x-y)$ between ${\ell }^1\to {\ell }^{\infty }$ norm one and ${\ell }^2\to {\ell }^2$ norm $\sqrt{2}$. This proves the scalar inequality with factor $2^{1/p}$. Integrating that estimate and applying Minkowski inequality to $|f{|}^{p^{\prime }}+|g{|}^{p^{\prime }}$ in $L^{p/p^{\prime }}$ gives the displayed function inequality.

The duality mapping takes a vector to its supporting dual functionals for half its squared norm. Equivalently $p\in \mathcal{J}(x)$ when $\Vert p{\Vert }_{\ast }=\Vert x\Vert$ and $⟨x,p⟩=\Vert x{\Vert }^2$. This follows from the dual-norm inequality and equality in the conjugate relation for squared norms. In a Hilbert space, the Riesz representation theorem identifies it with the identity; in a general Banach space it can be set-valued.

An equatorial coordinate system labels a viewing direction on the celestial sphere by right ascension $\alpha$, increasing eastward around the reference celestial equator, and declination $\delta$, increasing northward from that celestial equator. These two angles locate a direction, without specifying the object's Euclidean distance. In the traditional convention, zero right ascension points toward the reference equinox, the northward crossing of the Sun's annual path through the celestial equator. An hour circle through the direction supplies the north-south arc on which declination is measured. ^[ref].

For a concrete construction, choose Cartesian coordinates with $x$ pointing toward the reference equinox, $z$ toward the north reference celestial pole, and positive $y$ given by the cross product of the positive $z$ and $x$ directions. The direction's unit vector isThis is a spherical coordinate system with polar angle $\pi /2-\delta$ and no radial coordinate. The trigonometric functions require consistent angular units: convert right ascension in hours to radians before evaluating them. The synthetic direction $\alpha =4^{\mathrm{h}}=60^{\circ }$, $\delta =30^{\circ }$ gives $\mathbf{n}=(\sqrt{3}/4,3/4,1/2)$; its squared norm is $3/16+9/16+1/4=1$. A change of reference axes applies a rotation to this unit vector before recovering the two angles. Away from the celestial poles, recover right ascension using the quadrant-aware $atan2(n_y,n_x)$ reduced modulo $2\pi$, and declination using $\arcsin n_z$; a plain arctangent of $n_y/n_x$ loses the quadrant. Adding a common offset to every right ascension describes only the special case of a rotation about the same polar axis.

The reference specification matters. For an equatorial coordinate system tied to the Earth, precession changes the reference celestial equator and reference equinox, and nutation distinguishes their mean and true versions. The date specifying those axes is distinct from the astronomical epoch at which a catalogue's source directions apply: advancing a star's proper motion and changing reference axes are separate operations. A modern catalogue may instead use fixed reference frame axes, aligned approximately with the traditional J2000.0 axes, without a moving reference equinox. Thus a pair of numbers labelled merely "J2000" does not settle every convention. State the reference frame, the astronomical epoch, and whether a direction is catalogued or apparent. ^[ref].

During one night's elementary fixed-direction approximation, a star's right ascension and declination remain constant while its altitude, azimuth and hour angle change because the Earth turns. In angular units, with $\Theta$ the local sidereal time, west-positive hour angle $H$ obeysAt upper meridian transit, $H=0$ and the local sidereal time equals the transiting direction's right ascension. Pair apparent sidereal time with apparent right ascension referred to the true celestial equator and reference equinox of date; mean sidereal time refers to the mean reference equinox. Raw fixed-frame catalogue coordinates require conversion before precision pointing. ^[ref].

This separation makes the equatorial coordinate system useful: a catalogue can identify a target independently of the observer's current astronomical horizon, while local sidereal time tells the observer which hour circle is on the celestial meridian. Precision observations also account for proper motion, stellar parallax, aberration of light and atmospheric refraction; the fixed-direction construction above omits those effects. A reference-axis rotation preserves angles between directions, whereas changing the observer or correcting atmospheric refraction is a different operation. ^[ref].

With signed active covariate columns $X_A^s=(s_j{x}_j{)}_{j\in A}$ and their positive-definite Gram matrix $G_A=(X_A^s{)}^T{X}_A^s$, the displayed direction has unit norm and $(X_A^s{)}^{T}u=\alpha 1$. Thus it makes equal angles with unit-length signed active covariates, and movement along it reduces their signed residual correlations at the common rate $\alpha$. The regression coefficient direction is $d_A=\mathrm{diag}(s_A)w$, with inactive components zero.

For a L2-bounded continuous martingale, the terminal and maximal norms satisfyThe first inequality follows from almost sure convergence. Apply the Doob L2 maximal inequality on $[0,T]$ and then the monotone convergence theorem as $T\to \infty$ for the second. These are equivalent norms on the space modulo indistinguishability of stochastic processes; neither inequality needs a zero initial value.

Global sign-even functions have zero coefficients on odd-size Walsh characters. Every nonconstant remaining character has generator eigenvalue at most $-2$, yielding the displayed improvement from gap one to gap two. Combining this with the convexity of a norm proves the sharp Rademacher second-moment inequality.

Conjugate the collocation matrix by the diagonal matrix with entries $(-1{)}^i$. The resulting negative-off-diagonal matrix has a nonnegative inverse, as a convergent Neumann series shows. The absolute inverse row sums are therefore the entries of its inverse applied to the all-ones vector. They solve $4u_i-u_{i-1}-u_{i+1}=6$ with $u_0=u_{n+1}=0$, whose solution is displayed. The maximum occurs at the central row or central two rows and equals the inverse operator norm on ${\ell }^{\infty }$. Consequently the associated spline interpolation operator has norm at most three independently of dimension.

Let $\Omega$ be a bounded connected Lipschitz domain, $\alpha >0$, $g\ge 0$ bounded, and $T:L^1\to L^1$ bounded and positivity preserving with $T1\ne 0$. Minimize $\alpha \mathrm{TV}(u)+D_g(u)$ over nonnegative BV space functions. The logarithmic Jensen inequality bounds $\Vert Tu{\Vert }_1$ and variation on sublevels; mean control for positive imaging operators then bounds the full $BV$ norm. Bounded-variation compactness gives strong $L^1$ convergence, including nonnegativity. The scalar fidelity has bounded derivative on $s\ge 0$, so continuity of $T$ makes its integral continuous in that limit. Lower semicontinuity of variation proves existence by the direct method in the calculus of variations. If $g>0$ almost everywhere and $T$ is injective, strict convexity of the fidelity gives uniqueness. For $0<g<1$, zero is the unique minimizer even without injectivity, since zero energy forces both $Tu=0$ and a constant nonnegative $u$.

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.

Inside any three-dimensional domain containing a ball, let $u_{\epsilon }(x)={\epsilon }^{1/2}\varphi ((x-x_0)/\epsilon )$ for a fixed smooth compactly supported nonconstant function. The right-hand side $f_{\epsilon }=\Delta u_{\epsilon }$ has constant $L^2$ norm, whereas $\Vert \nabla u_{\epsilon }{\Vert }_{\infty }={\epsilon }^{-1/2}\Vert \nabla \varphi {\Vert }_{\infty }$ diverges. The functions have zero boundary trace, giving a scaling counterexample to a uniform $L^2$-forcing gradient estimate.

This integral domain embeds in $\mathbb{C}$ but has the inequivalent factorizations $6=2\cdot 3=(1+\sqrt{-5})(1-\sqrt{-5})$. The multiplicative norm $N(a+b\sqrt{-5})=a^2+5b^2$ takes neither the value two nor three, proving that the four displayed factors are irreducible elements. Its only units are $\pm 1$, so those two factorizations cannot be related by units and reordering.