nLab basis of a vector space

Definition

Definition

For $\mathbb{K}$ a field and $V$ a $\mathbb{K}$-vector space, hence a $\mathbb{K}$-module, a linear basis for $V$ is

1. a set $B$ (of basis elements)

2. a $\mathbb{K}$-linear isomorphism

   $$\underset{B}{\oplus} \mathbb{K} \xrightarrow{\;\; \simeq \;\;} V$$

to $V$ from the $B$-indexed direct sum (the coproduct in $\mathbb{K}$Vect) of copies of $\mathbb{K}$ (canonically regarded as a vector space over itself), also known as the free $\mathbb{K}$-linear span of $B$, hence the vector space of free $\mathbb{K}$-linear combinations of elements of $B$.

Hence if a basis for $V$ exists it means in particular that $V$ is a free module over $\mathbb{K}$.

Remark

(finiteness of linear combinations)
Beware that a certain finiteness-condition is hidden in Def. : Since a linear combination is defined to be a sum of finitely many vectors, a basis of a vector space must be such that every vector in the space is the (unique) combination of finitely many basis elements – even if there are infinitely many elements in the basis.

More abstractly this is to do with the appearance of the coproduct $\amalg_b = \oplus_B$ (direct sum) in Def. instead of the product $\prod_B$. Many vector spaces in practice arise as (subspaces) of products $\prod_W \mathbb{K}$ (function spaces $W \to \mathbb{K}$), but if $W$ here is not a finite set then it is not going to be a basis set.

(On the other hand, if $W$ is a finite set, then we have a biproduct $\prod_W \,\simeq\, \coprod_W \,\simeq\, \oplus_W$ in the additive category $\mathbb{K}$Vect, see there.)

Related to this are the following phenomena:

Remark

(basis and dimension of finitely-generated spaces)
For every finitely generated vector space $V$ (Def. ) it is straightforward to construct a linear basis, and to see that the cardinality of all bases is the same finite natural number $dim(V) \in \mathbb{N}$, called the dimension of the vector space (whence a finite-dimensional vector space).

On the other hand:

Remark

(Hamel-bases of infinite-dimensional vector spaces)
While the definition applies also to not-necessarily finitely generated vector spaces – such as for instance the space of (continuous) functions from a non-finite (topological) space to the (topological) ground field – it turns out to be subtle and somewhat ill-behaved in this generality.

In fact, in practice infinite-dimensional vector spaces tend to appear and to be understood with extra structure (typically that of topological vector spaces such as Banach spaces or Hilbert spaces) in which cases there are more appropriate notions of linear bases for them (such as that of Schauder bases, which allow infinite-linear combinations subject to a condition of convergence of a sequence).

In order to distinguish the plain notion of basis (Def. ) from these more refined notions, one also speaks of Hamel bases here.

(This is in honor of Hamel 1905 pp. 460 who considered this notion for the special case of $V = \mathbb{R}$ the real numbers regarded as a rational vector space, hence over the ground field $\mathbb{K} = \mathbb{Q}$ of rational numbers).

Hence, in principle, also a linear basis of a finitely generated vector space is thus a Hamel basis, but rarely called this way unless in the context of infinite-dimensional vector spaces.

Remark

(basis theorem and dimension)
For an infinitely-generated vector space it is not in general possible to construct a (Hamel-)basis, but the existence of such a basis is nevertheless implied, in classical mathematics, by Zorn's lemma (essentially a form of the axiom of choice): This is the content of the basis theorem.

With this classical context understood, it follows that every vector vector space admits a linear basis (even if non-constructible in general) and that each basis is of the same cardinality, then called the dimension of the vector space.

Background definitions

Definition

(generated vector space)
For $V$ a vector space, a subset $G \subset V$ of its underlying set is called a generating set or spanning set if every element of $V$ can be expressed as a linear combination of elements of $G$ in $V$, hence if the linear span of $G$ inside $V$ is all of $V$.

The vector space $V$ is called finitely generated if it admits a generating set (spanning set) which is a finite set.

Related concepts

• orthogonal basis

• basis in functional analysis

• Schauder basis

• dual basis

• basis

• mutually unbiased bases

• Gelfand-Tsetlin basis (in representation theory)

• seminormal basis (in representation theory of the symmetric group)

References

Lecture notes with much conceptual exposition:

• Karen E Smith, Bases for infinite-dimensional vector spaces [pdf, pdf]

Lecture notes with the proofs concisely spelled out:

• Christoph Schweigert, Basis und Dimension, §2.4 in: Lineare Algebra, lecture notes, Hamburg (2022) [pdf]

See also:

• Wikipedia, Basis (linear algebra)

The original discussion (for $\mathbb{R}$ regarded as a rational vector space) after which Hamel bases are named:

• Georg Hamel, pp. 460 of: Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: $f(x + y) = f(x) + f(y)$, Mathematische Annalen 60 (1905) 459–462 [doi:10.1007/BF01457624]