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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 in order to understand Garling's proof of Proposition 11.3.5 - 5 ...

Garling's statement and proof of Proposition 11.3.5 reads as follows:View attachment 8965I can follow Garling's proof of Proposition 11.3.5 - 5 (except that when he refers to (iii) and (iv) ... he means 3 and 4 ...)

... ... BUT ... the proof assumes \(\displaystyle (A^{ \bot } )^{ \bot \bot } = (A^{ \bot \bot } )^{ \bot } = A^{ \bot \bot \bot }\) ... ...How do we know that this is the case ... that is, true ... ?

Peter

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

I need some help in order to understand Garling's proof of Proposition 11.3.5 - 5 ...

Garling's statement and proof of Proposition 11.3.5 reads as follows:View attachment 8965I can follow Garling's proof of Proposition 11.3.5 - 5 (except that when he refers to (iii) and (iv) ... he means 3 and 4 ...)

... ... BUT ... the proof assumes \(\displaystyle (A^{ \bot } )^{ \bot \bot } = (A^{ \bot \bot } )^{ \bot } = A^{ \bot \bot \bot }\) ... ...How do we know that this is the case ... that is, true ... ?

Peter