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.

  • $\begingroup$ If $v= (v_1,..., v_n)$ then define $|| v || = \sup_{j} |v_j| $ $\endgroup$ – Mustafa Said May 9 '19 at 18:07

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/

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  • $\begingroup$ (+1) I was also about to mention the Minkowski functional. Nice link, too! $\endgroup$ – Sangchul Lee May 9 '19 at 18:15

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:

  • $\lVert v\rVert=\sqrt{\int_0^1P_v^2(x)\,\mathrm dx}$
  • $\lVert v\rVert=\max_{x\in[0,1]}\bigl\lvert P_v(x)\bigr\rvert$
  • $\lVert v\rVert=\sum_{k=0}^n\bigl\lvert P_v(k)\bigr\rvert$
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  • 1
    $\begingroup$ This is a good idea, as it just uses norms defined on the vector space of $n-1$ degree polynomials. Thanks! $\endgroup$ – Dayton May 9 '19 at 18:12

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