Basis existence theorem

Basis existence theorem: Every vector space $V$ has a subset $B\subseteq V$ such that every $v\in V$ can be written uniquely as a finite linear combination of elements of $B$.

This theorem guarantees that vector spaces admit coordinate descriptions once a basis is chosen, even when $V$ is infinite-dimensional. Standard proofs use Zorn’s lemma (equivalently, the axiom of choice), and the result underlies many structural statements about linear maps .