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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 the meaning and the point or reason for some remarks by Garling made after Proposition 11.3.5 ...

The remarks by Garling made after Proposition 11.3.5 ... read as follows:View attachment 8966I understand the "mechanics" of the equations/expressions in Garling's remarks but do not know the reason or the point of his remarks ... can someone please explain the reasons behind or the point of Garling's remarks ... further what does he mean by "decomposition" ... ... Help will be appreciated ...

Peter==========================================================================================

The above post refers to Proposition 11.3.5 ... so I am providing text of the same in order for readers to be able to understand the context of my question ...View attachment 8967Hope that helps ...

Peter

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

I need some help in order to understand the meaning and the point or reason for some remarks by Garling made after Proposition 11.3.5 ...

The remarks by Garling made after Proposition 11.3.5 ... read as follows:View attachment 8966I understand the "mechanics" of the equations/expressions in Garling's remarks but do not know the reason or the point of his remarks ... can someone please explain the reasons behind or the point of Garling's remarks ... further what does he mean by "decomposition" ... ... Help will be appreciated ...

Peter==========================================================================================

The above post refers to Proposition 11.3.5 ... so I am providing text of the same in order for readers to be able to understand the context of my question ...View attachment 8967Hope that helps ...

Peter