The question below was posted on [Mathematics Stack Exchange](https://math.stackexchange.com/questions/3040258/is-there-any-two-variable-function-which-has-no-representation-of-the-form-sum). It received no answer, and I do not expect any direct answer to it here. However, the question seems to me a natural one. Thus I wonder whether it has been posed before and, if so, whether there is any history to it. The question is:


Is there any function $f:\Bbb R^2\to\Bbb R$ which has no representation in the following form?


$$f(x,y)=\sum\_{n=1}^{\infty} g\_n(x)h\_n(y)\quad(x,y\in \Bbb R).$$As a candidate for such a function, I suggest $f(x,y)=\min\{x,y\}$.



Is this a known question about the expression of a function on $\Bbb R^2$ as an infinite sum of products?
In ["Representation of functions of two variables as sums of rectangular functions, I"](http://matwbn.icm.edu.pl/ksiazki/fm/fm85/fm85118.pdf), Roy O. Davies shows that under the continuum hypothesis every function has a representation of this form.


More precisely, he shows that we can get a representation with the additional property that for all $x,y$, the sum has only finitely many non-zero terms.
Conversely, he proves that if there is a representation with this additional property for $(x,y) \mapsto e^{xy}$, then the continuum hypothesis holds.


In his survey [Set Theoretic Real Analysis](https://www.math.wvu.edu/%7Ekcies/prepF/56STA/STAsurvey.html)(p. 18), Krzysztof Ciesielski mentions this problem (and a lot of interesting similar results), ~~it seems that it is open whether or not the continuum hypothesis is equivalent to the existence of weak representations~~. It is not open, cf. Santi Spadaro's comment below, or Péter Komjáth's answer.


