Norm and modulus, a little help.

Question

a little help with Analysis 2. I know what the definition of a norm is, and I know that in a multidimensional space R^2 I can define infinite "equivalent" functions that respect the properties of Norm. So I will have infinite equivalent norms from a topological point of view, and I will only change the "distance" that these represent.

Now, in an attempt to ask me why this is so, I have attempted to draw a parallel with R, where the "norm" is the modulus or absolute value. I saw in a textbook that |x|= max{x,-x} and the parallel with the infinite norm came naturally to me ||v||∞ :=max{|x_i|} but I believe (and correct me if I am wrong) that they are not the same thing. The modulus is a special case of a norm in R considering the vector space R over the field R. But I can't figure out when I can consider R a vector space, and how to make a parallel between these two elements.

In R I have the absolute value of x which is a positive value, in R^n I have the sum of absolute values of the components with ||v||_1 or other positive values according to the norm I use. How do I relate these two elements?

How do I understand why on R^n I have an infinite number of functions that demonstrate norm property and on R only absolute value?

Thanks in advance and sorry for the confusion.

You can change the norm on the one-dimensional space $\mathbb R$. — hardmath, Jul 02 '21 at 16:19
The absolute value map $x\mapsto |x|$ is not the only norm on $\Bbb R(\Bbb R)$. A scaled absolute value map would also be a norm, ie, $|x|:=a|x|$ for some $a\ge 0$ is also a norm on $\Bbb R(\Bbb R)$. For $a=1$, it is the usual absolute value (Euclidean) norm. But it is true that all norms on $\Bbb R$ are equivalent. More generally, on a finite dimensional vector space $V$, all norms are equivalent. — Prasun Biswas, Jul 02 '21 at 16:21

Norm and modulus, a little help.

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