$\begingroup$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 $u, v \in \mathbb{R}^d$?
1 Answer
$\begingroup$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.
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