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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ...

I need some help in fully understanding Lemma 8.4 ...

Lemma 8.4 reads as follows:

View attachment 9371

View attachment 9372

In the above proof of Lemma 8.4 by Browder we read the following:

" ... ... On the other hand since \(\displaystyle \sum_{ j = 1 }^m ( a_k^j )^2 = \ \mid T e_k \mid \ \le \| T \|^2\) for every \(\displaystyle k, 1 \le k \le n\) ... ... "

My question is as follows:

Can someone please demonstrate rigorously that \(\displaystyle \sum_{ j = 1 }^m ( a_k^j )^2 = \ \mid T e_k \mid \ \le \| T \|^2\) ...

(... ... it seems plausible that \(\displaystyle \sum_{ j = 1 }^m ( a_k^j )^2 = \ \mid T e_k \mid \ \le \| T \|^2\) but how do we demonstrate it rigorously ... ... )

Help will be much appreciated ...

Peter

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ...

I need some help in fully understanding Lemma 8.4 ...

Lemma 8.4 reads as follows:

View attachment 9371

View attachment 9372

In the above proof of Lemma 8.4 by Browder we read the following:

" ... ... On the other hand since \(\displaystyle \sum_{ j = 1 }^m ( a_k^j )^2 = \ \mid T e_k \mid \ \le \| T \|^2\) for every \(\displaystyle k, 1 \le k \le n\) ... ... "

My question is as follows:

Can someone please demonstrate rigorously that \(\displaystyle \sum_{ j = 1 }^m ( a_k^j )^2 = \ \mid T e_k \mid \ \le \| T \|^2\) ...

(... ... it seems plausible that \(\displaystyle \sum_{ j = 1 }^m ( a_k^j )^2 = \ \mid T e_k \mid \ \le \| T \|^2\) but how do we demonstrate it rigorously ... ... )

Help will be much appreciated ...

Peter