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?

1$\begingroup$ You should specify where are your functions defined, and in what sense does the series converge. Otherwise, one cannot give "necessary and sufficient conditions". $\endgroup$– Alexandre EremenkoJan 17, 2014 at 3:58

$\begingroup$ Thank you for the reminder. Already updated the question. $\endgroup$– ChaoJan 17, 2014 at 4:52
2 Answers
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.
Edit:
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).

$\begingroup$ ... and can something in the completed tensor product be written as a sum of a series of "elementary tensors" $g_k(x)h_k(x)$ ? $\endgroup$ Jan 17, 2014 at 14:17

$\begingroup$ ... yes: For the (completed) projective tensor product this is a result of Grothendieck and for the injective tensor product this is also true (and motivated this question). $\endgroup$ Jan 18, 2014 at 12:26
Since you explicitly ask formseries, perhaps it is more relevant to phrase your questions in terms of Schauder bases. Most of the function spaces of interest in analysis have such bases (the original Schauder bases for spaces of continuous functions, Haar type systems for $L^p$ spaces, classsical families of special functions for spaces of test functions and distributions). As indicated above, the bivariate versions usually have representations as tensor products. The circle is closed by a series of results showing how to tensor use such bases to get ones on tensor products. This was originally done by Gelbaum and de Lamadrid (Pac. J. Math. 11 (1961), 12811186) for the case of Banach spaces and the result can easily be extended to locally covex spaces (e.g, in "Bases and complete systems for analytic functions", Studia Math. 33(1969), 157164).
So the answer to your question is yes in most concrete cases (with the usually proviso about pointwise convergence in the case of $L^p$spaces).