symmetric monoidal (∞,1)-category of spectra
For $k$ a field and $V$ a $k$-vector space, a basis for $V$ is a basis of a free module for $V$ regarded as a free module over $k$. In functional analysis, a basis in this sense is called a Hamel basis.
The basis theorem asserts that, with the axiom of choice, every vector space admits a basis, hence that every module over a field is a free module.