If f+g in L2 then so are f and g

Question

Is it true that if f+g is a function in L2(R) then f and g must be in L2(R)? Does it also hold true for the multiplication of f and g?

Answer 1

Nope. $h+ (-h) \equiv 0 \in L^2(\mathbb R)$ for any $h$ and $1_{(-\infty, 0)} \cdot 1_{(0, \infty)} \equiv 0 \in L^2(\mathbb R)$.