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I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ...

I need some help with an aspect of the proof of Theorem 11.4.1 ...

Garling's statement and proof of Theorem 11.4.1 reads as follows:

View attachment 7921In the above proof by Garling we read the following:

" ... ... Let \(\displaystyle f_j = x_j - \sum_{ i = 1 }^{ j-1 } \langle x_j , e_i \rangle e_i\). Since\(\displaystyle x_j \notin W_{ j-1 }, f_j \neq 0\).

Let \(\displaystyle e_j = \frac{ f_j }{ \| f_j \| } \). Then \(\displaystyle \| e_j \| = 1\) and

\(\displaystyle \text{ span } ( e_1, \ ... \ ... \ e_j ) = \text{ span } ( W_{ j - 1 } , e_j ) = \text{ span }( W_{ j - 1 } , x_j ) = W_j \)

... ... "

Can someone please demonstrate rigorously how/why \(\displaystyle f_j = x_j - \sum_{ i = 1 }^{ j-1 } \langle x_j , e_i \rangle e_i \)

and

\(\displaystyle e_j = \frac{ f_j }{ \| f_j \| }\)imply that \(\displaystyle \text{ span } ( e_1, \ ... \ ... \ e_j ) = \text{ span } ( W_{ j - 1 } , e_j ) = \text{ span }( W_{ j - 1 } , x_j ) = W_j\)

Help will be much appreciated ...

Peter