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I am currently focused on Chapter 9: "Differentiation on \(\displaystyle \mathbb{R}^n\)"

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 \(\displaystyle \mathbf{x} \neq \mathbf{0}\) then \(\displaystyle \| \mathbf{x} \|^{-1} \mathbf{x}\) has a norm \(\displaystyle 1\), hence

\(\displaystyle \| \mathbf{x} \|^{-1} \| T \mathbf{x} \| = \| T ( \| \mathbf{x} \|^{-1} \mathbf{x} ) \| \le 1\) ... ... "

Now I know that \(\displaystyle T( c \mathbf{x} ) = c T( \mathbf{x} ) \)... BUT ...... how do we know that this works "under the norm sign" ...... that is, how do we know ...\(\displaystyle \| \mathbf{x} \|^{-1} \| T \mathbf{x} \| = \| T ( \| \mathbf{x} \|^{-1} \mathbf{x} ) \| \)

... and further ... how do we know that ...

\(\displaystyle \| T ( \| \mathbf{x} \|^{-1} \mathbf{x} ) \| \le 1 \)Help will be appreciated ...

Peter