Can $||f(s)g(t)-p(s)q(t)||_{L_2}<\epsilon$ be implied by $||f-p||_{L_2}$ and $||g-q||_{L_2}$ small enough?

Question

Assume $f,g,p,q\in L_2([0,1])=\{h: \int_{0}^1 h^2(x)dx<\infty\}$ and $f(s)g(t),p(s)q(t)\in L_2([0,1]^2)$. If I want to show $$||f(s)g(t)-p(s)q(t)||_{L_2}<\epsilon,$$ is it enough to show $||f-p||_{L_2}$ and $||g-q||_{L_2}$ are small enough? (of course the first norm is in $L_2([0,1]^2)$, the later two are in $L_2([0,1])$)

Answer 1

I don't believe so. In the special case $g=q$ you have $\|fg-pq\|^2 = \|f-p\|^2 \|g\|^2$ which can be arbitrarily large by increasing $\|g\|^2$, even when $\|f-p\|$ and $\|g-q\|=0$ are fixed.

Answer 2

It doesn't work for scalars. Let $p=q = n$, $f=p+{1 \over n}, q=p+{1 \over n}$, then $fg-pq = 2+{1 \over n^2}$.