We know that for the absolute value $\; \vert \bullet \vert\;$ it holds: $\;\vert xy \vert \ge xy\;\; \forall x,y \in \mathbb R$
Now if I consider $\; f,g:\mathbb R \supset (a,b) \rightarrow \mathbb R^n\;$ would the above inequality be still valid? To be more specific, is it true that $\;\vert fg \vert \ge f g\;$?
EDIT: $\;fg\;$ is the inner product.
Thanks in advance!