Is true that if $N$ is a norm in $\mathbb{R}^n$ then $N(\sigma(x))=N(x)$ for all $\sigma$ (permutation of coordinates)?
Invariance of the norm by permutations (I)
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linear-algebra
functional-analysis
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0For a plethora of counterexamples, take any norm $N$ for which $N(\sigma(x)) = N(x)$ for all $\sigma$, and any vector space automorphism $\phi$ of $\mathbf{R}^n$ whose matrix with respect to the standard basis is not a permutation matrix of $1$s and $-1$s, e.g. not something like $$\begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}$$ and define a norm by $$||v|| = N(\phi(v))$$ – 2017-01-19
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1One should also bear in mind that $M(x) := N(\sigma(x))$ is again a (different) norm. – 2017-01-19
1 Answers
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No. For example $\Vert(x,y)\Vert = |x| + 2 |y|$ defines a norm on $\mathbb{R}^2$.