Finding a Hamel base.

• Feb 19th 2010, 08:00 AM
Cairo
Finding a Hamel base.
Let X be a non-empty set, and let $\mathcal{\mathcal{H}}$ be the set of functions $\mathcal{H}=\left\{ f\,:\, X\rightarrow\mathbb{C}:\, f(x)=0\text{ for all but a finite number of points } x\in X\right\}$ equipped with the usual pointwise vector operations. For each $f\in\mathcal{H}$, define the subset S(f) of X by $S(f)=\left\{ x\in X\::\: f(x)\neq0\right\}$
Exhibit an explicit Hamel base for $\mathcal{H}$, showing that it satisfies all the requirements of such a base.