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

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In summary, Peter was struggling with an aspect of Corollary 1.8.12 from Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk. GJA helped by providing the results preceding the corollary.
  • #1
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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 \(\displaystyle \text{Aut} \mathbb{R}^n\) 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
 
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  • #2
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 $Ay.$
 
  • #3
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
 
  • #4
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
 
  • #5
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.
 
  • #6
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 \(\displaystyle \mathbb{R}^n\) ...

Thanks again ...

Peter
 
  • #7
Any time, Peter!
 

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

1. What are norms in R^n?

Norms in R^n refer to a mathematical way of measuring the length or magnitude of a vector in n-dimensional space. They are essentially a generalization of the absolute value function in one-dimensional space.

2. Can you provide an example of a norm in R^n?

One example of a norm in R^n is the Euclidean norm, also known as the 2-norm. It is calculated by taking the square root of the sum of the squared values of each component of the vector.

3. What is Aut(R^n)?

Aut(R^n) stands for the automorphism group of R^n, which is the set of all invertible linear transformations from R^n to itself. In simpler terms, it is the collection of all operations that preserve the structure of R^n.

4. What is the D&K Corollary 1.8.12?

The D&K Corollary 1.8.12, also known as the Fundamental Theorem of Norms, states that any two norms on a finite-dimensional vector space are equivalent. This means that they induce the same topology on the vector space.

5. How is the D&K Corollary 1.8.12 useful?

The D&K Corollary 1.8.12 is useful because it allows us to work with different norms in a vector space without changing the underlying structure or properties of the space. This gives us more flexibility and options in solving mathematical problems.

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