Equivalence of Norms in R^n .... D&K Corollary 1.8.10 .... ....

[Math Amateur]

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 1: Continuity ... ...

I need help with an aspect of Corollary 1.8.10 ... ...

Duistermaat and Kolk's Corollary 1.8.10 and the preceding notes and results read as follows:View attachment 7743In the notes after Theorem 1.8.8 above we read the following:

" ... ... A useful application of this theorem is to show the equivalence of all norms on the finite-dimensional vector space \(\displaystyle \mathbb{R}^n\). ... ... "My question is as follows:

How/why does Corollary 1.8.10 show the "equivalence" of all norms on the finite-dimensional vector space \(\displaystyle \mathbb{R}^n\). ... ... in what sense is this meant?

Peter

 

[Evgeny.Makarov]

How/why does Corollary 1.8.10 show the "equivalence" of all norms on the finite-dimensional vector space \(\displaystyle \mathbb{R}^n\).

Corollary 1.8.10 states the definition of equivalent norms. Two norms $\mu(x)$ and $\nu(x)$ are called equivalent if there exist positive numbers $c_1$ and $c_2$ such that $c_1\mu(x)\le\nu(x)\le c_2\mu(x)$.

 

[Math Amateur]

Corollary 1.8.10 states the definition of equivalent norms. Two norms $\mu(x)$ and $\nu(x)$ are called equivalent if there exist positive numbers $c_1$ and $c_2$ such that $c_1\mu(x)\le\nu(x)\le c_2\mu(x)$.

Hi Evgeny ... thanks for the help ...

Can you go further and state that equivalent norms means that the theorems proved in \(\displaystyle \mathbb{R}^n\) are essentially unchanged when the norm is changed to an equivalent one ...

Peter