- #1

Math Amateur

Gold Member

MHB

- 3,998

- 48

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 1.8.12 ... ...

Duistermaat and Kolk's Corollary 1.8.12 and the preceding definition of \(\displaystyle \text{Aut} \mathbb{R}^n\) read as follows:View attachment 7747

I can see how D&K arrive at the result:

\(\displaystyle c_1 \mid \mid x \mid \mid \le \mid \mid Ax \mid \mid \le c_2 \mid \mid x \mid \mid\)... BUT ... how, exactly, did D&K derive the result ...\(\displaystyle c_2^{ -1 } \mid \mid x \mid \mid \le \mid \mid A^{ -1 } x \mid \mid \le c_1^{ -1 } \mid \mid x \mid \mid

\)

Hope that someone can help ...

Peter=========================================================================================

***NOTE***

It may help MHB members reading the above post to have access to the results preceding Corollary 1.8.12 ... so I am providing the same ... as follows:https://www.physicsforums.com/attachments/7745

View attachment 7746Hope that helps ...

Peter

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

I need help with an aspect of Corollary 1.8.12 ... ...

Duistermaat and Kolk's Corollary 1.8.12 and the preceding definition of \(\displaystyle \text{Aut} \mathbb{R}^n\) read as follows:View attachment 7747

I can see how D&K arrive at the result:

\(\displaystyle c_1 \mid \mid x \mid \mid \le \mid \mid Ax \mid \mid \le c_2 \mid \mid x \mid \mid\)... BUT ... how, exactly, did D&K derive the result ...\(\displaystyle c_2^{ -1 } \mid \mid x \mid \mid \le \mid \mid A^{ -1 } x \mid \mid \le c_1^{ -1 } \mid \mid x \mid \mid

\)

Hope that someone can help ...

Peter=========================================================================================

***NOTE***

It may help MHB members reading the above post to have access to the results preceding Corollary 1.8.12 ... so I am providing the same ... as follows:https://www.physicsforums.com/attachments/7745

View attachment 7746Hope that helps ...

Peter

Last edited: