I'm curious how can I define inner product between two functions whose domains are finite sets. I know function's inner product is defined as $\langle f, g \rangle = \int_a^b f \cdot g \space dx\space\space $ But domains of $f$ and $g$ are to be $[a, b]$ in this case, which are (continuous) infinite sets.
To get more examples, what is $\langle f, g \rangle\space$ if $\space f(x)=x\space$ and $\space g(x)=x+3 \space \space$ for $x\in\{1, 2, 3\}$ (-> finite domain)
I think it would be $(1, 2, 3)\cdot(4,5,6)=32\space$ because the range of $f\space $ is $\{1, 2, 3\}\space$ and it is $\{4,5,6\}$ for $g$. But I can't be sure. Does anybody know this?
Thank you.