MHB Norms in R^n .... and Aut(R^n) .... .... D&K Corollary 1.8.12 .... ....

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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
 
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Hi, Peter.

The trick is to write $x=AA^{-1}x$. Now set $y=A^{-1}x$ and see if you can use the bounds you know to be true on $Ay.$
 
GJA said:
Hi, Peter.

The trick is to write $x=AA^{-1}x$. Now set $y=A^{-1}x$ and see if you can use the bounds you know to be true on $Ay.$
Thanks ...

Just reflecting on this ...

Peter
 
Peter said:
Thanks ...

Just reflecting on this ...

Peter
Sorry GJA ... despite your help, I am not making progress ...

In particular having trouble determining the bounds on Ay ...

Can you please help further ...

Peter
 
We have $x=AA^{-1}x$. Setting $y=A^{-1}x$ and using the known inequality we obtain
$$\|x\|=\|AA^{-1}x\|=\|Ay\|\leq c_{2}\|y\|=c_{2}\|A^{-1}x\|.$$
Dividing through by $c_{2}$ we get
$$c_{2}^{-1}\|x\|\leq \|A^{-1}x\|,$$
as desired. A similar argument will give the second inequality.
 
GJA said:
We have $x=AA^{-1}x$. Setting $y=A^{-1}x$ and using the known inequality we obtain
$$\|x\|=\|AA^{-1}x\|=\|Ay\|\leq c_{2}\|y\|=c_{2}\|A^{-1}x\|.$$
Dividing through by $c_{2}$ we get
$$c_{2}^{-1}\|x\|\leq \|A^{-1}x\|,$$
as desired. A similar argument will give the second inequality.
Thanks for your guidance and assistance GJA ... I appreciate your help ...

With your help I am beginning to understand analysis in $$\mathbb{R}^n$$ ...

Thanks again ...

Peter
 
Any time, Peter!
 
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