Sup Norm $|| x||_∞$

Question

If $X,Y \in \mathbb R^P \ with \ p\in \mathbb N $

Is true that $|X \cdot Y| \leq || x||_∞ || y||_∞ ? $

I know that

$|| x||_∞= Sup( |x_1|,|x_2|,...,|x_n|) $

I don't know how to start this excercise, thank you for your help!!

Answer 1

A counterexample would be $p=2$ and $$X=Y=\begin{pmatrix}1 \\ 1 \end{pmatrix}$$ since then $X\cdot Y=2$ and $\|x\|_\infty\|y\|_\infty=1$. So it is not necessarily true that $|X\cdot Y|\leq\|x\|_\infty\|y\|_\infty$.