In the top answer to the question here I am wondering why they are using the cross product and not the dot product in the following argument. I would ask on the question, but it is from 2012 so it is quite old... They state:
The way I would do it is first to show that $\{f_n\times g_m\}$ is orthonormal: indeed, $$ \langle f_n\times g_m, f_s\times g_t\rangle = > \langle f_n,f_s\rangle\,\langle g_m,g_t\rangle > =\delta_{n,s}\,\delta_{m,t} = \delta_{(n,m),(s,t)}. $$
I am confused because I think we would want to show that $\{f_n\cdot g_m\}$ is orthonormal wouldn't we?
Would it also be true to say this... To show that $\{f_n\cdot g_m\}$ is orthonormal: $$ \langle f_n\cdot g_m, f_s\cdot g_t\rangle = \langle f_n,f_s\rangle\,\langle g_m,g_t\rangle =\delta_{n,s}\,\delta_{m,t} = \delta_{(n,m),(s,t)}. $$
These denote the same, namely $(x, y) \mapsto f_n(x)\cdot g_m(y)$.