Today I was working on solving a partial differential equation in the unit square, $[0,1] \times [0,1].$ Using the method of separation of variables on the spatial problem gave solutions of the form $f_i(x) g_j(y)$, where $\{f_i\}$ and $\{g_j\}$ each form complete orthonormal families of functions in $L^2([0,1])$. I would like to be able to claim that the collection of all functions of the form $f_i g_j$ form a complete orthonormal family of functions in $L^2([0,1]^2)$, with possible renormalization. I know this to be true of the trigonometric and exponential families of functions, which leads me to hope that it will hold in general, or at least work with the families of functions I'm dealing with (a mix of Bessel and trigonometric functions).
I've spent a bit thinking about how this might be proven. Since $C([0,1]^2)$ is dense in $L^2([0,1]^2)$ and completeness of an orthonormal family $\{x_i\}$ in a Hilbert space H is equivalent to the density of linear combinations of elements of $\{x_i\}$ in H, it would be sufficient to show that linear combinations of our $f_i g_j$ are dense in $C([0,1]^2)$ under the supremum norm. The proofs I have seen of the completeness of the trigonometric and exponential systems use these observations and the Stone-Weierstrass theorem. However, this approach seems to not fit my needs, since as far as I know one of the families of functions I'm working with do not form an algebra. I've thought about trying to use the compactness and separation properties of $[0,1]^2$ to construct a suitable approximation, but my attempts with this approach have ran into problems with the stringent convergence requirements of the sup norm.
While trying to figure this out, I've become curious about the general theory surrounding cartesian products and complete orthonormal families. In particular, I'd love to know the answer to the following question:
Under what conditions on $X$ and $Y$ does $\{f_n\}$ and $\{g_m\}$ being complete orthogonal families of functions in $L^2(X)$ and $L^2(Y)$ guarantee that $\{f_n g_m\}$ is a complete orthogonal family of functions in $L^2(X \times Y)$?
If possible, I'd appreciate references for whatever results are out there. Thanks in advance!
If $\{ e_n \}$ is an orthonormal set in a Hilbert space $H$, then the Parseval identity holds for some $h\in H$ iff $f$ is in the closed linear span $M$ of the $e_n$. So, in order to prove that $\{ f_n g_m \}$ is a complete orthonormal basis, it is enough to show that the Parseval identity holds for all $h$ in a dense subspace of $L^2([0,1]^2)$. One such subspace is the set of all $h \in C([0,1]^2)$. To prove this, note that \begin{align} \iint |f(x,y)|^2dxdy&=\int \sum_n\left|\int f(x,y)f_n(x)dx\right|^2 dy \\ &= \sum_n \int \left|\int f(x,y)f_n(x)dx\right|^2 dy \\ &= \sum_n \sum_m \left|\int\int f(x,y)f_n(x)dx g_m(y)dy\right|^2 \\ &=\sum_{n,m}|\langle f,f_ng_m\rangle|^2. \end{align} Therefore, the Parseval identity holds for all $f\in C([0,1]^2)$ with respect to the orthonormal set $\{ f_n(x)g_m(y)\}$. So this orthonormal set is complete in $L^2([0,1]^2)$.