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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 Example 11.5.2 on a linear isometry ... ... I wish to prove the mapping given is a linear isometry ... but I am not sure I understand the context of the example/problem ...

The start of Section 11.5 defining isometries plus example 11.5.2 ... ... reads as follows:

View attachment 8978In Example 11.5.2 we are given \(\displaystyle f: \mathbb{R}^2 \to \mathbb{C}\) ... where \(\displaystyle f(x,y) = x + iy\) ...

I wish to show that \(\displaystyle f\) is a linear isometry ... but how do I proceed ...

Basically I am unsure how to go about considering \(\displaystyle \mathbb{C}\) as a real vector space ... how do we go about this ... ?

Is it just a matter of considering the scalars as real numbers?Help will be appreciated ...

Peter

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

I need some help in order to understand Example 11.5.2 on a linear isometry ... ... I wish to prove the mapping given is a linear isometry ... but I am not sure I understand the context of the example/problem ...

The start of Section 11.5 defining isometries plus example 11.5.2 ... ... reads as follows:

View attachment 8978In Example 11.5.2 we are given \(\displaystyle f: \mathbb{R}^2 \to \mathbb{C}\) ... where \(\displaystyle f(x,y) = x + iy\) ...

I wish to show that \(\displaystyle f\) is a linear isometry ... but how do I proceed ...

Basically I am unsure how to go about considering \(\displaystyle \mathbb{C}\) as a real vector space ... how do we go about this ... ?

Is it just a matter of considering the scalars as real numbers?Help will be appreciated ...

Peter