Existence of a basis

Theorem (existence of a Hamel basis). Assuming the axiom of choice, every vector space $X$ has a Hamel basis.

Derivation

The empty set is linearly independent, so the extension theorem, proved using Zorn's lemma, extends it to a basis of $X$ . For $X=\{0\}$ , this basis is the empty set.

Examples

$\mathbb{R}^n$ has the standard basis.
Infinite-dimensional examples (like all sequences) have a Hamel basis, but it typically cannot be written down explicitly.