In the beginning of Chapter 5 of his real analysis book, Folland defines the product norm of $X \times Y$ (where $X$ and $Y$ are normed vector spaces) as $$||(x,y)|| = \max(||x||,||y||).$$
This is fine and dandy, but then he goes on to say "Sometimes other norms equivalent to this one, such as $||(x,y)|| = ||x||+||y||$ or $||(x,y)|| = (||x||^2 + ||y||^2)^{1/2}$ are used instead."
But clearly the maximum of two numbers isn't always the sum of the two numbers or their root mean square. So in what sense is he calling them equivalent?
Two norms $\|.\|_1$ and $\|.\|_2$ on a vector space are called equivalent if there exist positive numbers $c$ and $d$ such that $c\|x\|_1 \leq \|x\|_2 \leq d \|x\|_1$ for all $x$. This is true iff the two norms induce then same topology on the space. In this sense the norms mentioned by Folland are equivalent.
The equivalence of those norms follows from the elmentary fact that for $a,b \geq 0$ we have $\max (a,b) \leq a+b \leq \sqrt 2 \sqrt {a^{2}+b^{2}} \leq 2\max (a,b)$.