Is the p-norm on $l^p$ submultiplicative?

Question

Consider a set $X$ and $l^p(X)$ for $p > 1$. In my notes, it is claimed that $$\Vert fg \Vert_p \leq \Vert f \Vert_p \Vert g \Vert_p$$

However, I can't prove it. I tried to apply Minkowski-like inequalities but it did not work out. I'm beginning to suspect this is not even true?

Answer 1

If $\|f\|_p \leq 1$ then $|f(x)| \leq 1$ for all $x$. Hence $|\sum f(x)g(x)|^{p} \leq \sum |g(x)|^{p}$. From this we get $\| fg\|_p \leq \|g\|_p$. Now for a general $f \neq 0$ apply this result to $\frac f {\|f\|_p}$. That gives the desired inequality.