- #1

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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 11.4.2 ...Garling's statement and proof of Corollary 11.4.2 reads as follows:View attachment 8971In the third sentence of the above proof by Garling we read that:

\(\displaystyle \text{span}( e_{k + 1}, \ ... \ ... , e_d ) \subseteq W^{ \bot }\) Although this claim looks very plausible I am unable to formulate a rigorous demonstration of its truth ...

Can someone provide a rigorous argument showing that \(\displaystyle \text{span}( e_{k + 1}, \ ... \ ... , e_d ) \subseteq W^{ \bot }\) ... ... ?

Help will be appreciated ...

Peter

==========================================================================================

Since I suspect a rigorous proof of the above claim of interest will involve Theorem 11.4.1 ( Gram-Schmidt Orthonormalization ... ) ... I am providing text of the same ... as follows:

View attachment 8972

View attachment 8973

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 Corollary 11.4.2 ...Garling's statement and proof of Corollary 11.4.2 reads as follows:View attachment 8971In the third sentence of the above proof by Garling we read that:

\(\displaystyle \text{span}( e_{k + 1}, \ ... \ ... , e_d ) \subseteq W^{ \bot }\) Although this claim looks very plausible I am unable to formulate a rigorous demonstration of its truth ...

Can someone provide a rigorous argument showing that \(\displaystyle \text{span}( e_{k + 1}, \ ... \ ... , e_d ) \subseteq W^{ \bot }\) ... ... ?

Help will be appreciated ...

Peter

==========================================================================================

Since I suspect a rigorous proof of the above claim of interest will involve Theorem 11.4.1 ( Gram-Schmidt Orthonormalization ... ) ... I am providing text of the same ... as follows:

View attachment 8972

View attachment 8973

Hope that helps ...

Peter