Existence of a basis

Corollary (Existence of a Hamel basis). If $X$ is a vector space with $X\neq\{0\}$, then $X$ has a basis , i.e., a Hamel basis.

Connection to the extension theorem. Pick any nonzero $x\in X$. Then $\{x\}$ is linearly independent, so the extension theorem produces a basis containing $\{x\}$.

Remark. If $X=\{0\}$ is the trivial vector space, it is standard to declare the empty set to be a basis of $X$ (so that “every vector space has a basis” holds uniformly).

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.