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I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ...

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

I need some help to fully understand the proof of Corollary 11.3.2 ... (Corollary to the Cauchy-Schwarz Inequality ... )

Garling's statement and proof of Corollary 11.3.2 reads as follows:View attachment 8960

View attachment 8961In the above text from Garling we read the following:

" ... ... Equality holds if and only if \(\displaystyle \mathscr{R} \ \langle x, y \rangle = \| x \| . \| y \|\), which is equivalent to the condition stated. ... ... "

Can someone please rigorously demonstrate that the condition that ... \(\displaystyle \mathscr{R} \ \langle x, y \rangle = \| x \| . \| y \| \) ...

... is equivalent to ... either \(\displaystyle y = 0\) or \(\displaystyle x = \alpha y\) with \(\displaystyle \alpha \ge 0\) ... ...Help will be appreciated ...

Peter

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

I need some help to fully understand the proof of Corollary 11.3.2 ... (Corollary to the Cauchy-Schwarz Inequality ... )

Garling's statement and proof of Corollary 11.3.2 reads as follows:View attachment 8960

View attachment 8961In the above text from Garling we read the following:

" ... ... Equality holds if and only if \(\displaystyle \mathscr{R} \ \langle x, y \rangle = \| x \| . \| y \|\), which is equivalent to the condition stated. ... ... "

Can someone please rigorously demonstrate that the condition that ... \(\displaystyle \mathscr{R} \ \langle x, y \rangle = \| x \| . \| y \| \) ...

... is equivalent to ... either \(\displaystyle y = 0\) or \(\displaystyle x = \alpha y\) with \(\displaystyle \alpha \ge 0\) ... ...Help will be appreciated ...

Peter