How would I show that $\|\cdot\|_3$ and $\|\cdot\|_\infty$ are equivalent norms on $\mathbb R^2$?
I understand that to say two norms are equivalent, then there exist two real constants, $m,M$ such that,
And that if we were to sketch the norms $\|\cdot\|_\infty=\|\cdot\|_3=1$ then we could stretch or shrink them to fit into each other, again, by the constants $m,M$.
However, I am not to sure what $\|\cdot\|_3$ exactly looks like, and haven't had much luck with a graphing calculator, and so I am not entirely sure how to go about rigorously finding the constants, having not worked with $\|\cdot\|_3$ at all before and not being able to sketch it.
Can anybody help me to discern its shape as well as with finding $m,M$?