Quite often (see, for example, this PDF, 50 KB) when discussing the Born-Oppenheimer approximation the following assertion is made: any well-behaved function of two independent variables $f(x,y)$ can always be expanded over the complete set of functions $\{ g_{i}(x) \}$ and $\{ h_{j}(y) \}$ in the following way $$ f(x,y) = \sum\limits_{i} \sum\limits_{j} c_{ij} g_{i}(x) h_{j}(y) \, , \quad (1) $$ or by defining $$ c_{j}(x) = \sum\limits_{i} c_{ij} g_{i}(x) \, , $$ as $$ f(x,y) = \sum\limits_{j} c_{j}(x) h_{j}(y) \, . $$
Sometimes an argument in favor of the statement above is that it is equivalent to expanding a function of one variable $f(y)$ over the complete set of functions $\{ h_{j}(y) \}$ $$ f(y) = \sum\limits_{j} c_{j} h_{j}(y) \, , $$ with the difference that coefficients $c_{j}$ in the former case carry the $x$-dependence.
For me it seems like this argument is a bit far-fetched. So my question is: how do we know that (1) is true? Is it a theorem, or an axiom, or something?
Answer
Start with $f(x,y)$. Fix a certain $x$, let's call it $x_0$. Obviously then $f(x_0, y)$ is a function of just one variable, so it has an expansion in the complete set, $$f(x_0, y) = \sum_j c_j(x_0) h_j(y)$$
While the expansion coefficients now depend on $x_0$, since we get a different $y$-dependent function for each value of $x_0$ that we fix, the ability to expand doesn't depend on that. So we might as well call it just $x$ again,
$$f(x,y) = \sum_j c_j(x) h_j(y)$$
Next step: Since $c_j(x)$ is a single-variable function of $x$, we can expand it: $$c_j(x) = \sum_i c_{ij} g_i(x)$$
Putting it all together, $$f(x,y) = \sum_{i}\sum_j c_{ij} g_i(x) h_j(y)$$
EDIT: In the true fashion of a physicist, I did assume that the functions are sufficiently well behaved to do all these steps. Technically we have to worry about various notions of convergence of functions, but especially for square-integrable Hilbert spaces we should be fine :)
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