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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:

View attachment 7904

https://www.physicsforums.com/attachments/7905

Near the end of the above proof we read the following:

" ... ... To see that \(\displaystyle \| T \|\) defines a norm, note that homogeneity follows directly from the definition ... ... "

My question is as follows:

How, exactly, do we demonstrate rigorously that homogeneity follows directly from the definition ... that is how do we show that

\(\displaystyle \| t T \| = \ \mid t \mid \| T \|\) ... ... for \(\displaystyle t \in \mathbb{R}\) ... ... ?

Help will be appreciated ...

Peter