all norms I've seen on $\mathbb{R}^n$ are either the sup norm or the $l_p$ norms. Are these the only norms that exist or are there others?
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2$|x| = \sup_{t \in [0,1]} |x_1+x_2t+x_3 t^2+\cdots+ x_n t^{n-1}|$. – copper.hat Oct 23 '20 at 01:56
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@copper.hat Is there a name for that norm? – angryavian Oct 23 '20 at 02:01
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Consider a trivial norm ($0$ if $\mathbf 0$, $1$ otherwise) – J. W. Tanner Oct 23 '20 at 02:03
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3any symmetric convex body in $\mathbb{R}^n$ defines a norm on $\mathbb{R}^n$ for which that convex body is the unit ball, and every norm arises this way, see e.g. this answer. – user125932 Oct 23 '20 at 02:05
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2I call it Fido. – copper.hat Oct 23 '20 at 02:06
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Just kidding, I don't think it has a well known name. – copper.hat Oct 23 '20 at 02:06
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@J.W.Tanner that's not absolutely homogeneous though. Cause $|ax|=|a||x|$ and if x is non zero we would have $|ax|=1$ but also $|ax|=|a|$ – MIO Oct 23 '20 at 02:07
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@user125932 thank you so much – MIO Oct 23 '20 at 02:08
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Thanks, @aldodecristo, I was wondering about that – J. W. Tanner Oct 23 '20 at 02:14
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1@J.W.Tanner Actually I did some research and it can be a norm as long as $mathbb{R}$ is endowed with the trivial valuation (the generalization of absolute value) – MIO Oct 23 '20 at 02:17
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There are infinitely many norms but they all generate the same topology. – Ivo Terek Oct 23 '20 at 03:57