Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 2 / 12E / b

For each positive integer $N$, let $x^{(N)}$ have its first $N$ coordinates equal to one and all later coordinates zero. This is an element of the l-p sequence space for every finite $p$, andNo constant can bound $\Vert x{\Vert }_p$ by a fixed multiple of $\Vert x{\Vert }_{\infty }$ for all $N$. Thus $\Vert \cdot {\Vert }_p$ and $\Vert \cdot {\Vert }_{\infty }$ are not Lipschitz equivalent.

For $1\le p<q<\infty$, the same finitely supported sequence givesOne of the two inequalities required for equivalent norms therefore fails. Hence