I am trying to understand how to find a spanning set of all real-valued functions such that this set is defined on a finite domain. I know how to do this with polynomials defined on ℝ, but I can't seem to get anywhere with a finite domain and such a blank slate for the type of function that needs to be engendered by the spanning set.
I would like, for example, to find the spanning set of F({3,4,5}).
How would I go about this?
Thank you for your help.
Precision:
F(D) is the set of all real-valued functions defined on D. In this case D = {3,4,5}.
In general, a vector space has infinitely many spanning sets. If $D$ is a finite set and if, for each $d\in D$, you define$$\begin{array}{rccc}\chi_d\colon&D&\longrightarrow&\mathbb R\\&x&\mapsto&\begin{cases}1&\text{ if }x=d\\0&\text{ otherwise,}\end{cases}\end{array}$$then $\{\chi_d\mid d\in D\}$ spans $F(D)$.