every vector space has a basis
This result, trivial in the finite case, is in fact rather surprising
when one thinks of infinite dimensionial vector spaces
, and the
definition of a basis: just try to imagine a basis of the vector space
of all continuous mappings . The theorem is
equivalent
to the axiom of choice
family of axioms and theorems. Here
we will only prove that Zorn’s lemma implies that every vector space
has a basis.
Theorem.
Let be any vector space over any field and assume Zorn’s lemma. Then if is a linearly independent subset of , there exists a basis of containing . In particular, does have a basis at all.
Proof.
Let be the set of linearly independent subsets of
containing (in particular, is not empty), then
is partially ordered by inclusion. For each chain ,
define . Clearly, is an upper bound of . Next we
show that . Let
be a finite collection of vectors. Then there exist sets such that for all . Since is
a chain, there is a number with such that
and thus , that is is
linearly independent
. Therefore, is an element of .
According to Zorn’s lemma has a maximal element, , which
is linearly independent. We show now that is a basis. Let
be the span of . Assume there exists an . Let
be a finite collection of vectors and
elements such that
If was necessarily zero, so would be the other , ,
making linearly independent in contradiction to the
maximality of . If , we would have
contradicting . Thus such an does not exist and , so is a generating set and hence a basis.
Taking , we see that 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 basis![]() |
Related topic | ZornsLemma |
Related topic | AxiomOfChoice |
Related topic | ZermelosWellOrderingTheorem |
Related topic | HaudorffsMaximumPrinciple |
Related topic | KuratowskisLemma |