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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 11.4.3 ...Garling's statement and proof of Corollary 11.4.3 reads as follows:

View attachment 8975

View attachment 8976

In the last sentence of the proof Garling asserts that if \(\displaystyle w_j = x_j\) for \(\displaystyle 1 \leq j \leq k\) then we have \(\displaystyle w = P_W(x)\) ...

I cannot formulate an explicit formal and rigorous proof of this statement ... can someone please help me with this ...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 11.4.3 ...Garling's statement and proof of Corollary 11.4.3 reads as follows:

View attachment 8975

View attachment 8976

In the last sentence of the proof Garling asserts that if \(\displaystyle w_j = x_j\) for \(\displaystyle 1 \leq j \leq k\) then we have \(\displaystyle w = P_W(x)\) ...

I cannot formulate an explicit formal and rigorous proof of this statement ... can someone please help me with this ...Help will be appreciated ...

Peter