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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 Proposition 11.3.1 ...

Garling's statement and proof of Proposition 11.3.1 reads as follows:View attachment 8956I need help with exactly how Garling concluded that with his substitution for \(\displaystyle \lambda\) ... we have

\(\displaystyle \lambda^2 \langle x, y \rangle = \frac{ \langle x, y \rangle^2 }{ \mid \langle x, y \rangle \mid^2 } \frac{ \| x \|^2 }{ \| y \|^2 } \| y \|^2 = \| x \|^2\) ... ... My problem is what sign (plus or minus) and value do we give to \(\displaystyle \langle x, y \rangle^2\) ... Garling seems to treat \(\displaystyle \langle x, y \rangle^2\) as if it were equal to \(\displaystyle \mid \langle x, y \rangle \mid^2\) ... and cancels with the denominator ... or so it seems ...?

But ... \(\displaystyle \langle x, y \rangle\) is a complex number, say \(\displaystyle z\) ... and so we are dealing with a complex number \(\displaystyle z^2 = \langle x, y \rangle^2\) ... and, of course, \(\displaystyle z^2\) is neither positive or negative ... ... ? ... so how do we end up with

\(\displaystyle \lambda^2 \langle x, y \rangle = \| x \|^2\)Hope that someone can help ...

Peter=========================================================================================It may help readers of the above post to have access to Garling's introduction to inner product spaces where he gives the relevant definitions and notation ... so I am providing access to the relevant text as follows:

View attachment 8957

View attachment 8958

Hope that helps ...

Peter

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

I need some help to fully understand the proof of Proposition 11.3.1 ...

Garling's statement and proof of Proposition 11.3.1 reads as follows:View attachment 8956I need help with exactly how Garling concluded that with his substitution for \(\displaystyle \lambda\) ... we have

\(\displaystyle \lambda^2 \langle x, y \rangle = \frac{ \langle x, y \rangle^2 }{ \mid \langle x, y \rangle \mid^2 } \frac{ \| x \|^2 }{ \| y \|^2 } \| y \|^2 = \| x \|^2\) ... ... My problem is what sign (plus or minus) and value do we give to \(\displaystyle \langle x, y \rangle^2\) ... Garling seems to treat \(\displaystyle \langle x, y \rangle^2\) as if it were equal to \(\displaystyle \mid \langle x, y \rangle \mid^2\) ... and cancels with the denominator ... or so it seems ...?

But ... \(\displaystyle \langle x, y \rangle\) is a complex number, say \(\displaystyle z\) ... and so we are dealing with a complex number \(\displaystyle z^2 = \langle x, y \rangle^2\) ... and, of course, \(\displaystyle z^2\) is neither positive or negative ... ... ? ... so how do we end up with

\(\displaystyle \lambda^2 \langle x, y \rangle = \| x \|^2\)Hope that someone can help ...

Peter=========================================================================================It may help readers of the above post to have access to Garling's introduction to inner product spaces where he gives the relevant definitions and notation ... so I am providing access to the relevant text as follows:

View attachment 8957

View attachment 8958

Hope that helps ...

Peter