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?

Any references would be appreciated. Thank you.

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.