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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 $$\text{Aut} \mathbb{R}^n$$ read as follows:View attachment 7747
I can see how D&K arrive at the result:
$$c_1 \mid \mid x \mid \mid \le \mid \mid Ax \mid \mid \le c_2 \mid \mid x \mid \mid$$... BUT ... how, exactly, did D&K derive the result ...$$c_2^{ -1 } \mid \mid x \mid \mid \le \mid \mid A^{ -1 } x \mid \mid \le c_1^{ -1 } \mid \mid x \mid \mid
$$
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 $$\text{Aut} \mathbb{R}^n$$ read as follows:View attachment 7747
I can see how D&K arrive at the result:
$$c_1 \mid \mid x \mid \mid \le \mid \mid Ax \mid \mid \le c_2 \mid \mid x \mid \mid$$... BUT ... how, exactly, did D&K derive the result ...$$c_2^{ -1 } \mid \mid x \mid \mid \le \mid \mid A^{ -1 } x \mid \mid \le c_1^{ -1 } \mid \mid x \mid \mid
$$
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: