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 ...

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.

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.

Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

I know the definitions of the $1$ and $2$ norm, and, numerically the inequality seems obvious, although I don't know where to start rigorously. Thank you. analysis

Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

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

2019年9月13日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

2014年7月7日 · Hence $\cup_{ p \ge 1 } B_p(0,1)$ is bounded, balanced, convex and contains $0$ in its interior, and so is the unit ball of some norm, in this case, $\|\cdot \|_\infty$. The point here is that there is a natural motivation in terms of the unit balls which 'converge' (in the above sense) to the $\|\cdot \|_\infty$ unit ball.

2017年12月17日 · I've read the Uniform Norm Wikipedia page, but my most of it went over my head. What is the sup-norm in simple and / or intuitive terms? Are there any good examples which illustrate it?