Norm of multiplication and multiplication of norms

Question

It is well known that $\|u \cdot v \|_2 \le \|u \|_2 \cdot \| v \|_2 $ for all $u, v \in \mathbb{R}^d$.

Is the following true for all $p \in \mathbb{R}$:

$$\|u \cdot v \|_p \le \|u \|_p \cdot \| v \|_p$$
for all $u, v \in \mathbb{R}^d$?

Answer 1

No, this is not true. Take $p=3$ and $u=v = (1,1)$.

Then the right side is $2$, while the left-side is $2^{1/3}2^{1/3}$.

For a result somewhat in that direction see Hölder's inequality.