A Hamel basis for the real numbers 
, considered as a vector space
over the rationals 
, is a set of real numbers 
such that every real number
has a unique representation of the
form
 |
where 
is rational and 
depends on 
.
The axiom of choice is equivalent to the statement: "Every vector space has a vector
space basis." In particular, the axiom of
choice guarantees the existence of a Hamel basis for 
over 
.
