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

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.

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

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