For $T_{ab}=(\rho +p)u_a{u}_b+p{g}_{ab}$ and nonzero null $k$, contraction gives $T_{ab}{k}^a{k}^b=(\rho +p)(u\cdot k{)}^2$. A timelike fluid velocity has nonzero scalar product with every nonzero null vector, so the null energy condition is exactly the displayed inequality. Individual nonnegativity of pressure or density is unnecessary for null focusing. A cosmological constant term has zero null contraction.

The volume ratio of a $d$-dimensional normed vector space is $\mathrm{vr}(E)=(\mathrm{vol}(B_E)/\mathrm{vol}(\mathcal{E}){)}^{1/d}$, where $\mathcal{E}$ is its John ellipsoid. It is unchanged by invertible linear changes of coordinates. In John position, polar integration gives $\mathrm{vr}(E{)}^d={\int }_{S^{d-1}}\Vert u{\Vert }_E^{-d}d\sigma (u)$, with normalized spherical measure. Consequently $\sigma \{\Vert u{\Vert }_E\le t\}\le (\mathrm{vr}(E)t{)}^d$.

Given $f$ in a normed vector space and a vector subspace $U$, an element $p_{\ast }\in U$ is a best approximation if $\Vert f-p_{\ast }\Vert ={\inf }_{p\in U}\Vert f-p\Vert$. Existence is automatic for a finite-dimensional vector subspace: a minimizing sequence can be bounded by the triangle inequality, and a subsequence converges by compactness in finite dimension. An arbitrary infinite-dimensional vector subspace need not contain a minimizer. The norm matters: best uniform approximation uses the supremum norm, while $L^1$ approximation minimizes integrated absolute error.

A normed vector space is strictly convex when distinct unit vectors have their midpoint strictly inside the unit ball. The Lp space is strictly convex for $1<p<\infty$: strict convexity of $|z{|}^p$ gives a strict integrated midpoint inequality for distinct unit functions. The endpoint spaces $L^1$ and $L^{\infty }$ need not have this property.

A normed vector space is uniformly convex if, for each $\epsilon >0$, there is $\delta >0$ such that unit-ball vectors $x,y$ with $\Vert x-y\Vert \ge \epsilon$ satisfy $\Vert (x+y)/2\Vert \le 1-\delta$. Completeness is not part of this definition; a complete example is a uniformly convex Banach space.

In a nonzero normed vector space, a unit vector $x$ is approached from outside the unit ball by $(1+\epsilon )x$, so no boundary point is interior. It is approached from inside by $(1-\epsilon )x$, so every boundary point is in the closure of the open ball. The zero space satisfies the same identities trivially. These statements need the radial structure of a normed space; a two-point discrete metric space with interpoint distance one disproves both analogous claims for general metric balls.

In a cyclic group of odd prime order, the normalized ordered three-term arithmetic progression count is ${\sum }_r\widehat{1_A}(r{)}^2\widehat{1_A}(-2r)$. The zero frequency contributes ${\alpha }^3$. If the count is less than ${\alpha }^3/2$, its nonzero-frequency contribution has magnitude greater than ${\alpha }^3/2$. The Parseval identity on a finite group bounds that magnitude by $\alpha {\max }_{r\ne 0}|\widehat{1_A-\alpha }(r)|$, proving the displayed estimate. The count includes progressions of zero common difference.

If the normalization of an integral scheme is flat, then it is an isomorphism. On an affine chart, let $A\subseteq B\subseteq \mathrm{Frac}(A)$ be the integral closure. Integrality gives surjectivity on spectra, so flatness makes $B$ faithfully flat. For $b=u/v\in B$ with $u,v\in A$ and $v\ne 0$, faithful flatness gives $vB\cap A=vA$. Thus $u=va$ for some $a\in A$, and $b=a$. Consequently $B=A$. This proof does not need finiteness of the normalization.

For an algebraically closed field $k$ and a finitely generated field extension $K/k$ of transcendence degree one, choose $t$ with $K/k(t)$ finite. The integral closures of the two standard coordinate rings of the projective line in $K$ are finite modules and glue to a finite normalization of an algebraic curve of the projective line. A one-dimensional normal local ring is a discrete valuation ring; over the perfect field $k$ the resulting projective curve is smooth. Extension of rational maps gives uniqueness of this smooth projective model.

On $x_3^2=x_1^2+x_2^2$ with $x_3>0$, the gradient of the defining function gives a normal proportional to $(-x_1,-x_2,x_3)$. Its Euclidean norm is $\sqrt{2}{x}_3$, proving the displayed normalization. This upward normal lies on the boundary of the Lorentz cone, so self-duality of a second-order cone makes its pairing with every point of the closed upper cone nonnegative.

The core of a subgroup $H\le G$ isIt is the largest normal subgroup of $G$ contained in $H$. If $H$ has finite index $k$, the action on $G/H$ gives $[G:{\mathrm{Core}}_G(H)]\le k!$.

A subnormal series is a finite chain of subgroups in which each is a normal subgroup of the next. A member need not be normal in the whole group. Requiring the factors to be cyclic groups defines a polycyclic group; allowing finite factors as well defines a poly-(cyclic or finite) group.

A proper normal subgroup maximal among proper normal subgroups. Its quotient is a nontrivial simple group. Maximality among normal subgroups does not imply maximality among all subgroups.

// Target: algebra.bigb

For a finite group with Sylow $p$-subgroup $P$, a normal $p$-complement is a normal subgroup of order $|G|/|P|$. It consists exactly of the elements of order prime to $p$: every such element has trivial image in the $p$-group quotient, and every element of the complement has prime-to-$p$ order. Thus a normal $p$-complement, if it exists, is unique and a characteristic subgroup.

If finite normal subgroups $H,K$ of a group have coprime orders, their intersection is trivial by Lagrange's theorem. For $h\in H$ and $k\in K$, the commutator $hk{h}^{-1}{k}^{-1}$ belongs to both subgroups: write it as $h(k{h}^{-1}{k}^{-1})$ to use normality of $H$, and as $(hk{h}^{-1})k^{-1}$ to use normality of $K$. Hence it is the identity and $hk=kh$. The product subgroup is therefore isomorphic to the direct product of groups $H\times K$.

A group complement to a normal subgroup $N◃G$ is a subgroup $H$ with $G=NH$ and $N\cap H=1$. Projection identifies $H$ with $G/N$, and conjugation gives $G=N⋊H$. This is a subgroup notion, not the set-theoretic complement of $N$.

The p-residual is the smallest normal subgroup with p-group quotient, equivalently the subgroup generated by all elements of order prime to $p$. It is characteristic and satisfies $O^p(O^p(G))=O^p(G)$: an extension of two finite p-groups is a p-group. A finite group has a normal p-complement exactly when its p-residual has order prime to $p$.

If $A,B$ are disjoint closed subsets of a normal space, there is a continuous function $f:X\to [0,1]$ with $f{|}_A=0$ and $f{|}_B=1$.

If $F$ is closed, $U$ is open, and $F\subset U$ in a normal topological space, separate $F$ from the closed complement $X\setminus U$ by disjoint open sets $W,V$. Then $\overline{W}\subset X\setminus V\subset U$. In particular, first separate two disjoint closed sets by disjoint open sets and then shrink each neighborhood this way: their closures are disjoint. No additional metric or compactness assumption is needed.

Normal shock tables tabulate upstream-to-downstream property ratios for a normal shock wave, typically as functions of upstream Mach number and heat capacity ratio. They encode the Rankine-Hugoniot conditions for a perfect gas rather than defining the discontinuity itself.