Complete set of multi-variable functions

[ShayanJ]

We know that in the space of functions, its possible to find a complete set so that you can write for an arbitrary function f, $f(x)=\sum_n a_n \phi_n(x)$ and use the orthonormality relations between $\phi$s to find the coefficients.
But is it possible to find a set of functions $\phi_n(x,y)$ such that you can write for an arbitrary function of x and y $f(x,y)=\sum_n c_n \phi_n(x,y)$? Is there any orthonormality relation that let's you find the coefficients?
Thanks

 

[S.G. Janssens]

But is it possible to find a set of functions $\phi_n(x,y)$ such that you can write for an arbitrary function of $x$ and $y$ $f(x,y)=\sum_n c_n \phi_n(x,y)$ ? Is there any orthonormality relation that let's you find the coefficients?

For an arbitrary function, the answer is "no".

However, if you consider, for example, the Hilbert space $L^2([a,b] \times [a,b]; \mathbb{C})$ (with the inner product equal to the iterated integral) then the answer is "yes". In that case an orthonormal basis is obtained by taking products of the functions that form any orthonormal basis of $L^2([a,b]; \mathbb{C})$. This is proven in most introductory functional analysis books.

 

[micromass]

We know that in the space of functions

What does "The space of functions" even mean here?

 

[ShayanJ]

What does "The space of functions" even mean here?

I got my answer from Krylov's post. I'll be more careful in the future.