An argument often used in favor of the axiom of choice is that it is equivalent to every infinite dimensional vector space having a hamel basis. However the article on wikipedia says that those basis are usually not very useful when they're uncountable, and that other concepts such as "orthogonal basis" are more important in these cases.
So why does it matter whether infinite dimensional vector spaces have hamel basis?
Best Answer
One interesting element is that if two vector spaces over the same field have the same dimension, they are isomorphic. So identifying the cardinality of a Hamel basis is a way to identify isomorphic vector spaces.