Given a bivariate function $f(x, y)$ with $x \in [-a,a]$ and $y \in [-b, b]$, what is the necessary and sufficient condition under which we can write $f(x, y) = \sum g\_k(x)h\_k(y)$ for all $(x,y)$ in the domain of $f$ (the series could be infinite, in that case the convergence is just pointwise convergence)? Are there any established results or good references related to this kind of problems? 



series representation of bivariate functions
For example, $C^\infty([-a,a])\hat \otimes C^\infty([-b,b]) = C^\infty([-a,a]\times [-b,b])$
for the completed tensor product, and all tensor product topologies between the projective and the inductive one coincide, because the spaces are nuclear.
See for example,


* MR2296978 Trèves, François: Topological vector spaces, distributions and kernels. Unabridged republication of the 1967 original. Dover Publications, Inc., Mineola, NY, 2006. xvi+565 pp.


Also, $C^0([-a,a]) \hat{\hat \otimes} C^0([-b,b]) = C^0([-a,a]\times [-b,b])$, but now only for the inductive tensor product norm. For $\mathbb R$ instead of a compact interval, this only holds for functions which vanish at infinity. 


Similarly for $L^2$, with the $\ell^2$-tensor norm. 


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Elements in the algebraic tensor product are exactly the functions $\sum g\_k(x)f\_k(y)$ (finite sum), so in the completed tensor product they are convergent series (at least for the Frechet case).


