Basis existence theorem

Basis existence theorem. Assuming the axiom of choice, every vector space $V$ has a Hamel basis $B\subseteq V$ : every $v\in V$ has a unique finitely supported coefficient family $(c_b)_{b\in B}$ such that $v=\sum_{b\in B}c_b b$ .

Remarks

The standard proof applies Zorn's lemma to the partially ordered set of linearly independent subsets of $V$ . In set theory without choice, the assertion that every vector space has a basis is not provable.