every vector space has a basis

This result, trivial in the finite case, is in fact rather surprising when one thinks of infiniteMathworldPlanetmath dimensionial vector spacesMathworldPlanetmath, and the definition of a basis: just try to imagine a basis of the vector space of all continuous mappings f:. The theorem is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the axiom of choiceMathworldPlanetmath family of axioms and theorems. Here we will only prove that Zorn’s lemma implies that every vector space has a basis.


Let X be any vector space over any field F and assume Zorn’s lemma. Then if L is a linearly independent subset of X, there exists a basis of X containing L. In particular, X does have a basis at all.


Let 𝒜 be the set of linearly independent subsets of X containing L (in particular, 𝒜 is not empty), then 𝒜 is partially ordered by inclusion. For each chain C𝒜, define C^=C. Clearly, C^ is an upper bound of C. Next we show that C^𝒜. Let V:={v1,,vn}C^ be a finite collectionMathworldPlanetmath of vectors. Then there exist sets C1,,CnC such that viCi for all 1in. Since C is a chain, there is a number k with 1kn such that Ck=i=1nCi and thus VCk, that is V is linearly independentMathworldPlanetmath. Therefore, C^ is an element of 𝒜.

According to Zorn’s lemma 𝒜 has a maximal elementMathworldPlanetmath, M, which is linearly independent. We show now that M is a basis. Let M be the span of M. Assume there exists an xXM. Let {x1,,xn}M be a finite collection of vectors and a1,,an+1F elements such that


If an+1 was necessarily zero, so would be the other ai, 1in, making {x}M linearly independent in contradictionMathworldPlanetmathPlanetmath to the maximality of M. If an+10, we would have


contradicting xM. Thus such an x does not exist and X=M, so M is a generating set and hence a basis.

Taking L=, we see that X does have a basis at all. ∎

Title every vector space has a basis
Canonical name EveryVectorSpaceHasABasis
Date of creation 2013-03-22 13:04:48
Last modified on 2013-03-22 13:04:48
Owner GrafZahl (9234)
Last modified by GrafZahl (9234)
Numerical id 14
Author GrafZahl (9234)
Entry type Theorem
Classification msc 15A03
Synonym every vector space has a Hamel basisMathworldPlanetmath
Related topic ZornsLemma
Related topic AxiomOfChoice
Related topic ZermelosWellOrderingTheorem
Related topic HaudorffsMaximumPrinciple
Related topic KuratowskisLemma