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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 Theorem 11.1.13 ...

Garling's statement and proof of Theorem 11.1.13 (together with some other relevant text) reads as follows:

View attachment 8947

View attachment 8948I am having trouble determining exactly what is going on in the above proof by Garling ...

Can someone make clear exactly what Garling is doing in this proof ... ?

I have two specific questions ... which read as follows:

In the above proof by Garling we read the following:

" ... ... Thus if we define \(\displaystyle d( q(a), q(b) ) = p(a,b)\), this is well-defined ... "

What is Garling doing here ... what does he mean by the above statement ... ?

In the above proof by Garling we read the following:

" ... ... Finally if \(\displaystyle d( q(a), q(b) ) = 0\) then \(\displaystyle p(a, b) = 0\) so that \(\displaystyle a \sim b\) and \(\displaystyle q(a) = q(b)\) ... "

Can someone please explain what is going on here ...

Hope someone can help ..

Peter

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

I need some help to fully understand the proof of Theorem 11.1.13 ...

Garling's statement and proof of Theorem 11.1.13 (together with some other relevant text) reads as follows:

View attachment 8947

View attachment 8948I am having trouble determining exactly what is going on in the above proof by Garling ...

Can someone make clear exactly what Garling is doing in this proof ... ?

I have two specific questions ... which read as follows:

**Question 1**

In the above proof by Garling we read the following:

" ... ... Thus if we define \(\displaystyle d( q(a), q(b) ) = p(a,b)\), this is well-defined ... "

What is Garling doing here ... what does he mean by the above statement ... ?

**Question 2**

In the above proof by Garling we read the following:

" ... ... Finally if \(\displaystyle d( q(a), q(b) ) = 0\) then \(\displaystyle p(a, b) = 0\) so that \(\displaystyle a \sim b\) and \(\displaystyle q(a) = q(b)\) ... "

Can someone please explain what is going on here ...

Hope someone can help ..

Peter

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