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I am focused on Chapter 1: Continuity ... ...

I need help with an aspect of the proof of the Cauchy-Schwarz Inequality ...

Duistermaat and Kolk"s proof of the Cauchy-Schwarz Inequality reads as follows:View attachment 7639In the above proof we read the following:

" ... ... Now\(\displaystyle x = \frac{ \langle x, y \rangle }{ \mid \mid y \mid \mid^2} y + ( x - \frac{ \langle x, y \rangle }{ \mid \mid y \mid \mid^2} y ) \)is a decomposition into two mutually orthogonal vectors ... ... "I have tried to demonstrate that the two vectors \(\displaystyle \frac{ \langle x, y \rangle }{ \mid \mid y \mid \mid^2} y\) and \(\displaystyle ( x - \frac{ \langle x, y \rangle }{ \mid \mid y \mid \mid^2} y )\) are in fact orthogonal by showing that the inner product of these two vectors is zero ... but I failed to make any meaningful progress ...Can someone please demonstrate that these two vectors are in fact orthogonal ...

Help will be much appreciated ...

Peter