
Finding a Hamel base.
Let X be a nonempty set, and let $\displaystyle \mathcal{\mathcal{H}}$ be the set of functions $\displaystyle \mathcal{H}=\left\{ f\,:\, X\rightarrow\mathbb{C}:\, f(x)=0\text{ for all but a finite number of points } x\in X\right\}$ equipped with the usual pointwise vector operations. For each $\displaystyle f\in\mathcal{H}$, define the subset S(f) of X by $\displaystyle S(f)=\left\{ x\in X\::\: f(x)\neq0\right\}$
Exhibit an explicit Hamel base for $\displaystyle \mathcal{H}$, showing that it satisfies all the requirements of such a base.