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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 $$\mathscr{R} \ \langle x, y \rangle = \| x \| . \| y \|$$, which is equivalent to the condition stated. ... ... "
Can someone please rigorously demonstrate that the condition that ... $$\mathscr{R} \ \langle x, y \rangle = \| x \| . \| y \| $$ ...
... is equivalent to ... either $$y = 0$$ or $$x = \alpha y$$ with $$\alpha \ge 0$$ ... ...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 $$\mathscr{R} \ \langle x, y \rangle = \| x \| . \| y \|$$, which is equivalent to the condition stated. ... ... "
Can someone please rigorously demonstrate that the condition that ... $$\mathscr{R} \ \langle x, y \rangle = \| x \| . \| y \| $$ ...
... is equivalent to ... either $$y = 0$$ or $$x = \alpha y$$ with $$\alpha \ge 0$$ ... ...Help will be appreciated ...
Peter