When do we not have $|fg|=|f||g|$?

Question

Let $f,g:\mathbb{R}\mapsto\mathbb{R}$ be two functions. Is it ever the case that $|fg|\ne|f|\cdot|g|$?

What if instead of functions and the standard norm we consider something else, is there a situation where the equality is not true?

Answer 1

It is always the case for your particular choice of Euclidean norm. However, considering operator norms like the ones for matrices, they are only submultiplicative, i.e. $$||AB|| \leq ||A||| |B||$$ does only hold. Explicitely, one such norm is induced by the Euclidean norm $|\cdot|$ by setting $$||A|| := \sup_{|x| = 1} |Ax|$$

Answer 2

For the one-dimensional case, we must have $|fg|=|f|\cdot|g|$ because $|fg|(x)=|(fg)(x)|=|f(x)\cdot g(x)|=|f(x)|\cdot|g(x)|=|f|(x)\cdot|g|(x)$.

Consider now that $f,g:{\bf{R}}^{n}\rightarrow{\bf{R}}^{n}$, then $|f\cdot g|\ne|f|\cdot|g|$, here $\cdot$ is the dot product, because $|f(x)\cdot g(x)|\ne|f(x)|\cdot|g(x)|$ in general, note that $u\cdot v=|u||v|\cos\theta$ for vectors $u,v$ in the two-dimensional case.