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A fan in toric geometry is a finite collection of rational strongly convex cones in toric geometry in a real vector space with a lattice, closed under taking faces of a polyhedral cone, such that the intersection of two cones is a face of each. A fan determines a toric variety.

A fan $\Sigma$ in $N_{\mathbb{R}}$ is a finite collection of strictly convex rational polyhedral cones closed under taking faces, such that the intersection of two cones is a face of each.

For a real $n$-dimensional vector space of continuous functions on a compact interval, choose sites maximizing the absolute evaluation determinant in any basis. Linear independence ensures some nonzero evaluation determinant, so a maximizer exists and has distinct sites. Replacing row $i$ by evaluation at $t$ multiplies the determinant by the cardinal function ${\ell }_i(t)$. Maximality implies $|{\ell }_i(t)|\le 1$, hence the Lebesgue constant of interpolation is at most $n$. This gives a useful interpolation set without asserting that determinant maximization minimizes the Lebesgue constant of interpolation.

For a finite real convex function on an entire real vector space, apply the convex domination form of the Hahn-Banach theorem to $\varphi (y+z)-\varphi (y)$ and the zero functional on $\{0\}$. It supplies a linear functional ${\ell }_y$ dominated by this translated function. The affine function $a_y(x)=\varphi (y)+{\ell }_y(x-y)$ lies below $\varphi$ everywhere and equals it at $y$. Taking the supremum proves the formula. These are algebraic affine minorants; continuity requires additional topological hypotheses.

Let $K$ be complete for a Non-Archimedean absolute value and let $V$ be a finite-dimensional vector space with a norm satisfying $\Vert av\Vert =|a|\Vert v\Vert$ and the ultrametric inequality. All such norms are equivalent to the maximum of coordinates in any basis. The upper bound is immediate. For the lower bound, induct on dimension. The span $W$ of all but the last basis vector is complete by induction, hence closed for the given norm. Thus $\delta ={\inf }_{w\in W}\Vert e_n-w\Vert >0$. For $v=w+a{e}_n$, $\Vert v\Vert \ge |a|\delta$; also $\Vert w\Vert \le \max (\Vert v\Vert ,|a|\Vert e_n\Vert )$. The induction bound on coordinates of $w$ now bounds all coordinates by a constant times $\Vert v\Vert$. This proof needs no local compactness.

For an explicit basis extension in ${\mathbb{R}}^2$, start with $A=\{u\}$, where $u=(1,1)$. Its linear span is the line $y=x$. Adding $w=(1,-1)$ preserves linear independence, because $w$ is outside that line, and gives a basis. Every vector has the finite linear combinationThe vector $2u=(2,2)$ cannot be added while preserving linear independence: it is already in the linear span. Many choices outside the line work, so maximality does not select a unique basis.

For a finite-dimensional vector space with a supplied finite spanning set, a greedy algorithm scans the list, retaining each vector precisely when it is outside the linear span of those already retained. Starting with a linearly independent $A$, the same scan adjoins only the needed vectors. Retained vectors remain linearly independent, and every skipped vector is in the final linear span, so the result is a basis containing $A$. Finite reasoning needs no axiom of choice.

For an arbitrary vector space, this finite stopping argument is unavailable. Accepting Zorn's lemma asserts existence of a maximal element once every chain in a partial order has an upper bound; it supplies no general computational recipe. Over ZF, Zorn's lemma, the axiom of choice and the well-ordering theorem are equivalent. Particular infinite vector spaces can still have explicit bases, such as $1,x,x^2,\dots$ for the vector space of univariate polynomials over a specified field; infinity alone does not force a nonconstructive argument.^[ref]

A finitely supported sequence has only finitely many nonzero coordinates. Their vector space is commonly denoted $c_{00}$.

A flag is a strictly nested chain of vector subspaces of a fixed vector space. A complete flag contains a subspace of every possible dimension. For an action by linear maps, an invariant flag means that each of its vector subspaces is an invariant subspace. A basis adapted to an invariant complete flag makes all these maps upper triangular, connecting complete flags to simultaneous triangularization of a Lie algebra representation.

For a vector space $M$ over a field $k$, its formal power series module is $M[[t]]={\underleftarrow{\lim }}_{n}M{\otimes }_{k}k[t]/(t^n)$. It is complete for the filtration by $t^{n}M[[t]]$. The ordinary tensor product $M{\otimes }_{k}k[[t]]$ embeds in it and consists of series whose coefficients span a finite-dimensional vector subspace; it equals $M[[t]]$ if $M$ is finite-dimensional. The inclusion is proper for infinite-dimensional $M$, as shown by a series with linearly independent coefficients.

For a positively graded vector space $V$, the free graded Lie algebra $\mathbb{L}(V)$ receives a linear map from $V$ and extends every graded linear map $V\to L$ uniquely to a Lie homomorphism. Over $\mathbb{Q}$, it is the Lie subalgebra generated by $V$ inside the tensor algebra $T(V)$, using $[u,v]=uv-(-1{)}^{|u||v|}vu$. Its universal enveloping algebra is $T(V)$: both have the same universal property for maps from $V$ into associative algebras. The primitives of the tensor Hopf algebra with primitive generators are exactly $\mathbb{L}(V)$.

A free module is a module with a basis of a module $(e_i{)}_{i\in I}$: every element has exactly one expression $m={\sum }_{i\in I}{r}_i{e}_i$ with $r_i\in R$ and only finitely many $r_i$ nonzero. Equivalently, the coefficient map ${⨁}_{i\in I}R\to M$ is a module isomorphism. Existence means generation; uniqueness means linear independence in a module. For finite $I$ of size $n$, this says $M\cong R^n$, a finite free module. See Neil Strickland's Rings, Modules and Linear Algebra, section 11.

The integer lattice ${\mathbb{Z}}^2$ in the opening diagram is a free module over $\mathbb{Z}$ with basis of a module $e_1=(1,0)$, $e_2=(0,1)$. For example, $(3,2)=3e_1+2e_2$, and comparing the two coordinates makes the coefficients unique. A different basis of a module is $u=(1,0)$, $v=(1,1)$, because $(a,b)=(a-b)u+bv$; in particular $(3,2)=u+2v$.

By contrast, $w_1=(2,0)$ and $w_2=(0,1)$ are linearly independent in a module, but generate only the submodule $2\mathbb{Z}\times \mathbb{Z}$. They miss $(1,0)$ because $2a=1$ has no integer solution. They would form a basis over the rational numbers. Even generation need not let one extract a basis of a module: the elements $2,3$ generate $\mathbb{Z}$, since $1=3-2$, but neither singleton generates it, and $3\cdot 2-2\cdot 3=0$ is a nontrivial linear relation between them. These are concrete failures of the usual vector space argument that divides by any nonzero coefficient.

The cyclic module $\mathbb{Z}/6\mathbb{Z}$ has no basis of a module over $\mathbb{Z}$: every element has the nontrivial relation $6m=0$, so no nonempty subset is linearly independent in a module, while the empty subset cannot generate this nonzero module. Specifying the scalar ring is essential: over $R=\mathbb{Z}/6\mathbb{Z}$, the same additive object is a free module with the one-element basis of a module $\{\overline{1}\}$, since each coefficient is itself a unique class in $R$.

A monoid object $A$ with a morphism $\epsilon :A\to I$ such that $\epsilon m$ and a suitable coevaluation make $(A,A)$ a dual pair in a monoidal category. In a category of vector spaces this is a Frobenius algebra. This condition alone is not the usual separability condition.

Functional analysis brings linear algebra and topology together to study vector spaces of functions and the linear operators acting on them. A function space such as $C([0,1])$ has infinitely many independent directions: the monomials $1,x,x^2,\dots$ are linearly independent. Finite linear combinations alone cannot describe all its continuous functions. A useful theory must also say when approximations converge, whether their limits belong to the chosen space, and which operations preserve that norm convergence. Normed vector spaces, Banach spaces, and bounded linear operators provide these three controls.^[ref]

A bounded linear operator $T:X\to Y$ between normed vector spaces satisfies $\Vert Tf{\Vert }_Y\le M\Vert f{\Vert }_X$ for every $f$, with one finite constant $M$. The least possible constant is its operator norm. By linearity, the same estimate controls approximation errors:This is exactly continuity for a linear map between normed vector spaces; neither space needs to be complete. It turns an input accuracy guarantee into an output accuracy guarantee, even when the input has infinitely many degrees of freedom.^[ref]

For a concrete linear functional on $C([0,1])$, take the integral $F(f)={\int }_0^{1}f(x)dx$. In the supremum norm,The constant function $f=1$ attains equality, so $\Vert F\Vert =1$. For $f(x)=x^2$, the output is $F(f)=1/3$; any approximation $g$ with $\Vert g-x^2{\Vert }_{\infty }\le 0.01$ therefore has $|F(g)-1/3|\le 0.01$. The Cauchy-Schwarz inequality also gives $|F(f)|\le \Vert f{\Vert }_2\Vert 1{\Vert }_2=\Vert f{\Vert }_2$ on this interval, again with equality at $1$. Thus integration remains a bounded linear functional under the L2 norm, although point evaluation behaves very differently.

The beauty of this viewpoint is that one estimate can justify an entire approximation procedure. Once a sequence converges in the chosen norm, every bounded linear operator carries its limit along. Completeness ensures that Cauchy approximations actually describe an element of the space. The choice of norm matters: it determines which errors are small and which measurements are stable. The examples below distinguish an average, which ignores sufficiently narrow peaks, from point evaluation, which detects their full height. For applications, this decides whether reducing an overall fitting error guarantees an accurate reported measurement. A small L2 norm of the error controls its integral on this interval, while it gives no such guarantee for a single sampled value.

A function space is a collection of functions with common domain and codomain, often supplied with a vector space structure and a norm or topology. Examples include L2 spaces and Sobolev spaces.

The group of maps $x\mapsto Ax+b$ of a finite-dimensional vector space, with $A$ invertible. Identifying the underlying points with a regular elementary abelian group identifies its normalizer in the symmetric group with this semidirect product.

The general linear group $GL(V)$ is the group of invertible linear maps from a vector space $V$ to itself, with composition as its operation.

Let $H(n)$ be the real vector space of Hermitian matrices. The polar decomposition of an invertible complex matrix gives a diffeomorphismIts inverse sends $g$ toRight multiplication by $U(n)$ changes only $u$, so the orbit space is smoothly $H(n)$ and the quotient map is a globally trivial principal bundle.

The endomorphisms of a vector space form a Lie algebra under the commutator $[A,B]=AB-BA$. It is the Lie algebra of the general linear group; the Jacobi identity follows by expanding associative products.

The global dimension of a ring is the supremum of the projective dimensions of all its modules, allowing infinity. It measures the longest projective resolution needed by any module. For a field it is zero, since every module is a vector space and hence free.

The group algebra $k[G]$ is the vector space with basis $G$ and multiplication obtained by extending the group law $k$-bilinearly.