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I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ...
I am currently focused on Chapter 9: "Differentiation on $$\mathbb{R}^n$$"
I need some help with the proof of Proposition 9.2.3 ...
Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows:
https://www.physicsforums.com/attachments/7889
View attachment 7890
In the above proof we read the following:
" ... ... If $$\mathbf{x} \neq \mathbf{0}$$ then $$\| \mathbf{x} \|^{-1} \mathbf{x}$$ has a norm $$1$$, hence
$$\| \mathbf{x} \|^{-1} \| T \mathbf{x} \| = \| T ( \| \mathbf{x} \|^{-1} \mathbf{x} ) \| \le 1$$ ... ... "
Now I know that $$T( c \mathbf{x} ) = c T( \mathbf{x} ) $$... BUT ...... how do we know that this works "under the norm sign" ...... that is, how do we know ...$$\| \mathbf{x} \|^{-1} \| T \mathbf{x} \| = \| T ( \| \mathbf{x} \|^{-1} \mathbf{x} ) \| $$
... and further ... how do we know that ...
$$\| T ( \| \mathbf{x} \|^{-1} \mathbf{x} ) \| \le 1 $$Help will be appreciated ...
Peter
I am currently focused on Chapter 9: "Differentiation on $$\mathbb{R}^n$$"
I need some help with the proof of Proposition 9.2.3 ...
Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows:
https://www.physicsforums.com/attachments/7889
View attachment 7890
In the above proof we read the following:
" ... ... If $$\mathbf{x} \neq \mathbf{0}$$ then $$\| \mathbf{x} \|^{-1} \mathbf{x}$$ has a norm $$1$$, hence
$$\| \mathbf{x} \|^{-1} \| T \mathbf{x} \| = \| T ( \| \mathbf{x} \|^{-1} \mathbf{x} ) \| \le 1$$ ... ... "
Now I know that $$T( c \mathbf{x} ) = c T( \mathbf{x} ) $$... BUT ...... how do we know that this works "under the norm sign" ...... that is, how do we know ...$$\| \mathbf{x} \|^{-1} \| T \mathbf{x} \| = \| T ( \| \mathbf{x} \|^{-1} \mathbf{x} ) \| $$
... and further ... how do we know that ...
$$\| T ( \| \mathbf{x} \|^{-1} \mathbf{x} ) \| \le 1 $$Help will be appreciated ...
Peter