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2013年1月24日 · We can define the norm of a complex number in other ways, provided they satisfy the following properties. Positive homogeneity. Triangle inequality. Zero norm iff zero vector. We could define a $3$-norm where you sum up all the components cubed and take the cubic root. The infinite norm simply takes the maximum component's absolute value as the ...

$\begingroup$ This definition of the "0-norm" isn't very useful because (1) it doesn't satisfy the properties of a norm and (2) $0^{0}$ is conventionally defined to be 1. $\endgroup$ – Brian Borchers

2021年10月17日 · The unit circle, also the unit circle in the $\infty$ norm, which is a square; finally, the unit circle in the $1$ norm, which is a square rotated $45^\circ.$ Anyway, get some graph paper and draw some pictures. $\endgroup$ –

2014年4月16日 · Modulus is the term specifically used for complex numbers (scalars), and reduces to the concept of absolute value when referring to real numbers. When viewing a complex number as a real pair in the complex plane, then modulus corresponds to the (euclidian) norm on $\mathbb{R}^2$.

Does someone know how to prove that the dual norm of the $\mathcal l_{p}$ norm is the $\mathcal l_{q}$ norm? I read about norms and it was stated without proof in a book. I read about norms and it was stated without proof in a book.

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The norm you describe in your post, $||\epsilon||=\max|\epsilon_i|$ is a particular norm that can be placed on $\mathbb{R}^n$; there are many norms that can be defined on $\mathbb{R}^n$. The notion of norm on a vector space can be done with any field that is contained in $\mathbb{C}$, by restricting the modulus to that field. Added.

So every vector norm has an associated operator norm, for which sometimes simplified expressions as exist. The Frobenius norm (i.e. the sum of singular values) is a matrix norm (it fulfills the norm axioms), but not an operator norm, since no vector norm exists so that the above definition for the operator norm matches the Frobenius norm.

There are infinitely many norms that satisfy your requirements, and some of them are not invariant under right-multiplication of orthogonal matrix.