Norms on $\mathbb{R}^n$ which are not $L^p$

Question

I am aware that all norms in a finite dimension vector space are equivalent regardless of how weird they are. However I am wondering what are some concrete examples of norms on $\mathbb{R}^n$ which are not weighted $L^P$ norms? So I would like examples beyond

$$||v|| := \bigg(\sum_{i=1}^n \lambda_i|v_i|^p \bigg)^{1/p}, \quad p\geq 1$$

Where $v = (v_1,...,v_n)^T$ and $\lambda_i > 0$. Thanks in advance.

Answer 1

Every norm is generated by a centrally symmetric, convex, compact set, so there is a lot.

A very nice concrete family of examples generated from any function from a wide class are Luxemburg norms. See https://regularize.wordpress.com/2018/05/24/building-norms-from-increasing-and-convex-functions-the-luxemburg-norm/

Answer 2

If $v=(a_0,\ldots,a_{n-1})\in\mathbb R^n$, then let $P_v(x)=a_0+a_1x+\cdots+a_{n-1}x^{n-1}$. Now, consider the norms:

• $\lVert v\rVert=\sqrt{\int_0^1P_v^2(x)\,\mathrm dx}$
• $\lVert v\rVert=\max_{x\in[0,1]}\bigl\lvert P_v(x)\bigr\rvert$
• $\lVert v\rVert=\sum_{k=0}^n\bigl\lvert P_v(k)\bigr\rvert$