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I am reading Reinhold Remmert's book "Theory of Complex Functions" ...I am focused on Chapter 0: Complex Numbers and Continuous Functions ... and in particular on Section 1.2:\(\displaystyle \mathbb{R}\)-linear and \(\displaystyle \mathbb{C}\)-linear mappings of \(\displaystyle \mathbb{C}\) into \(\displaystyle \mathbb{C}\) ... ... I need help in order to fully understand one of Remmert's results regarding \(\displaystyle \mathbb{C}\)-linear mappings of \(\displaystyle \mathbb{C}\) into \(\displaystyle \mathbb{C}\) ... ... Remmert's section on \(\displaystyle \mathbb{R}\)-linear and \(\displaystyle \mathbb{C}\)-linear mappings of \(\displaystyle \mathbb{C}\) into \(\displaystyle \mathbb{C}\) reads as follows:

View attachment 8543

View attachment 8544In the above text by Remmert we read the following: (... fairly near the start of the text ...)

" ... ... An \(\displaystyle \mathbb{R}\)-linear mapping \(\displaystyle T : \mathbb{C} \to \mathbb{C}\) is then \(\displaystyle \mathbb{C}\)-linear when \(\displaystyle T(i) = i T(1)\); in this case it has the form \(\displaystyle T(z) = T(1) z\). ... ... "My questions are as follows:

How/why exactly is an \(\displaystyle \mathbb{R}\)-linear mapping \(\displaystyle T : \mathbb{C} \to \mathbb{C}\) also \(\displaystyle \mathbb{C}\)-linear when \(\displaystyle T(i) = i T(1)\) ... ... ?

Why/how exactly does a \(\displaystyle \mathbb{C}\)-linear mapping have the form \(\displaystyle T(z) = T(1) z\) ... ...

Hope someone can help ...

Peter

View attachment 8543

View attachment 8544In the above text by Remmert we read the following: (... fairly near the start of the text ...)

" ... ... An \(\displaystyle \mathbb{R}\)-linear mapping \(\displaystyle T : \mathbb{C} \to \mathbb{C}\) is then \(\displaystyle \mathbb{C}\)-linear when \(\displaystyle T(i) = i T(1)\); in this case it has the form \(\displaystyle T(z) = T(1) z\). ... ... "My questions are as follows:

**Question 1**

How/why exactly is an \(\displaystyle \mathbb{R}\)-linear mapping \(\displaystyle T : \mathbb{C} \to \mathbb{C}\) also \(\displaystyle \mathbb{C}\)-linear when \(\displaystyle T(i) = i T(1)\) ... ... ?

**Question 2**Why/how exactly does a \(\displaystyle \mathbb{C}\)-linear mapping have the form \(\displaystyle T(z) = T(1) z\) ... ...

Hope someone can help ...

Peter

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