This is a quick note to prove that two bases of an infinite dimensional vector space have the same cardinality. We freely use the axiom of choice and other standard facts about cardinalities of infinite sets. We will in fact prove the following:
Theorem: Let be a vector space with basis
with
an infinite set. Let
be a linearly independent subset of
. (e.g. a basis of a subspace). Then
.
To prove this, WLOG is a basis of
(by extending
to a basis of
if necessary). For all
, write
Let be the set of pairs
with
. Then
has finite fibres, since the sum above is finite, and
is surjective, since the
lie in the span of the
with
in the image of
, but also the
generate
. Since
is assumed infinite, this is enough to prove that
, as required.