Characterization of separable functionals

Question

Under what conditions is a functional $f(\vec{a},\vec{b})$ separable with rank $r$? That is, when can it be expressed as \begin{equation} f(\vec{a},\vec{b})=\sum_{i=1}^r A_i(\vec{a})B_i(\vec{b}), \end{equation}

for some $A_i, B_i$. The rank $r$ can potentially be infinite (in which case convergence is assumed in the equation above)? Note we have assumed nothing about continuity of $f$,$A_i$, $B_i$.

I have a suspicion from an illustrative example: $f(\vec{a},\vec{b})=\text{sgn}(\vec{a}\cdot\vec{b})$ is not separable for any $r$. Is the fact that is has a discontinuity whose position depends on both arguments what determines its inseparability?

Note: $f$,$A_i$, $B_i$, are all real scalar valued functions (i.e. functionals), and their arguments $\vec{a},\vec{b}$ are real vectors of arbitrary possibly unequal dimension (though they happen to be of equal dimension in my $\text{sgn}$ example).

Any references would be appreciated. Thank you.

Answer 1

A function $f:A\times B\to C$ corresponds to the function $\tilde f:A\to C^B$ in the "typographical" isomorphism $C^{A\times B}\sim (C^B)^A$ defined by $\tilde f(a)(b):=f(a,b)$ for all $a$ and $b$. If $A$ and $B$ are vector spaces on the field $C$, $ f$ writes in the form $f(a,b)=\sum_{i=1}^r u_i(a)v_i(b)$, that is $\tilde f(a)=\sum_{i=1}^r u_i(a)v_i$, if and only if $\tilde f$ takes values into an $r$-dimensional linear subspace $V\subset C^B$.

In your example (with $A=B$ a Hilbert space) the family of functions $\{b\mapsto \operatorname{sgn}(a\cdot b)\}_{a\in A}$ spans an infinite dimensional subspace of $C^A$. Indeed, for instance, if $S\subset A$ is such that no elements of $S$ are collinear (e.g. a hemisphere) then $\{b\mapsto \operatorname{sgn}(a\cdot b)\}_{a\in S}$ is a linearly independent family. The latter fact is apparent looking at the discontinuity set of these functions, as you were saying.

Answer 2

Here is a modest contribution to the cases $r=1$ and $r=\infty$.

For the first case, you are asking when a function $f$ of two variables separates multiplicatively as a product of two functions of one variable. There are two such criteria: in the absence of smoothness, one has the condition $$f(a_0,b_1)f(a_1,b_0)=f(a_0,b_0)f(a_1,b_1)$$ for all choices of arguments. For a smooth function of two real variables, one has the pde $ff_{12}=f_1f_2$ whereby suffixes denote partial derivatives (Of course one needs to state suitable conditions on the function and its domain to get a precise result).

Your formulation in the infinite case is too vague to allow a definitive answer (you would need to specify a space of functions and the corresponding topology, presumably one of the standard locally convex function spaces) but in general it will often be "always". For this, see the topic of tensor products of locally convex spaces, in particular the projective tensor product (due to Grothendieck).