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For a finite extension $K/F$, the automorphism group $G={\mathrm{Aut}}_F(K)$ has order dividing $[K:F]$. Indeed, Artin's theorem gives $[K:K^G]=|G|$, while $F\subseteq K^G$ and the tower law gives the divisibility.

A polynomial over a field is separable when it has no repeated root in a splitting field. Equivalently, it is coprime to its formal derivative.

In characteristic $p$, the coefficient $i$ in a formal derivative vanishes exactly when $p$ divides $i$.

In a field of characteristic $p>0$, the Frobenius map $x\mapsto x^p$ is an injective field endomorphism becauseConsequently every element of an extension field has at most one $p$th root.

Galaxy dynamics studies the gravitational motions and evolution of stars, gas, dark matter, and satellite systems in galaxies.

A galaxy rotation curve gives circular speed as a function of radius. Its decomposition into stellar, gaseous, and dark-matter contributions is generally nonunique.

The star formation rate is the mass converted into stars per unit time. Infrared luminosity traces dust-reprocessed light from young stars, while molecular-gas tracers locate their cold fuel.

The Milky Way is the barred spiral galaxy containing the Solar System.

A Banach space isomorphism is a bounded linear bijection with bounded inverse.

This is the sequence space $c_0$ of sequences vanishing at infinity.

The Banach space $c_0$ consists of scalar sequences tending to zero with the supremum norm.

For conjugate exponents $p,q$,with the analogous inequality ${\sum }_n|x_n{y}_n|\le \Vert x{\Vert }_p\Vert y{\Vert }_q$ for sequences.

The continuous dual space $X^{\ast }$ of a topological vector space is the vector space of its continuous linear functionals. Unlike the algebraic dual, it requires that a small change in an input produce a small change in its scalar measurement. For a normed vector space, this is equivalent to a finite boundfor all vectors $x$ and some constant $C\ge 0$. The least such constant is the operator norm of $\phi$.

This is the continuity-restricted subspace of the algebraic dual space; in infinite dimension the two can differ.

For the linear functional $\phi (x,y)=2x-y$ on ${\mathbb{R}}^2$ with its Euclidean norm, the Cauchy-Schwarz inequality givesEquality holds on the unit vector $(2,-1)/\sqrt{5}$, so its operator norm is exactly $\sqrt{5}$. The same estimate controls measurement error: $|\phi (v+h)-\phi (v)|\le \sqrt{5}\Vert h{\Vert }_2$. The linear functional diagram shows which changes are entirely invisible to this measurement; its kernel of a linear map consists of the changes with zero response. All linear functionals on a finite-dimensional real or complex normed vector space are continuous, but discontinuous functional detected by vanishing sequence tails shows why the restriction matters in infinite dimension.

Translations of a compactly supported bump can form an infinite family separated by a fixed sup-norm distance, disproving compactness.

C0(X) consists of continuous functions whose values become arbitrarily small outside compact sets.

For real $s$, the $L^2$-based Sobolev space $H^s({\mathbb{R}}^n)$ consists of tempered distributions satisfying

The theorem supplies solvability for a coercive weak formulation.

A bounded bilinear form $a$ on a real Hilbert space $H$ that is a coercive bilinear form represents every bounded linear functional uniquely: for each $\ell \in H^{\prime }$, there is a unique $u\in H$ with $a(u,v)=\ell (v)$ for every $v\in H$.

For a proof, the Riesz representation theorem gives a bounded linear operator $T$ and a vector $g$ with $a(u,v)=⟨Tu,v⟩$ and $\ell (v)=⟨g,v⟩$. Coercivity and the Cauchy-Schwarz inequality give $\Vert Tu\Vert \ge \alpha \Vert u\Vert$, so $T$ is injective and its image is closed: convergence of $T{u}_n$ makes $u_n$ a Cauchy sequence, whose limit supplies the image limit. If $v$ belongs to the orthogonal complement of the image, then $a(u,v)=0$ for every $u$; setting $u=v$ and using coercivity forces $v=0$. Thus the image is also dense and equals $H$. There is consequently a unique $u$ with $Tu=g$, and $\Vert u\Vert \le \Vert \ell \Vert /\alpha$. The complex version uses a bounded sesquilinear form with $\mathrm{Re}a(u,u)\ge \alpha \Vert u{\Vert }^2$.

The closed unit ball of a dual Banach space is compact in the weak-star topology.

Every bounded linear functional on a linear subspace of a real normed space extends to the whole space without increasing its norm.

The numerical range is $W(A)=\{⟨Ax,x⟩:x\in D(A),\Vert x\Vert =1\}$.

A bounded operator is compact when it maps the unit ball to a relatively compact set.