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**R-Linear and C-Linear Mappings...**

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 get a clear idea of the definition and nature of \(\displaystyle \mathbb{R}\)-linear and \(\displaystyle \mathbb{C}\)-linear mappings of \mathbb{C} into \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 8534

View attachment 8535It seems to me (this may be unfair and I just may be missing the point! ... ) that Remmert has not clearly defined \(\displaystyle \mathbb{R}\)-linear and \(\displaystyle \mathbb{C}\)-linear mappings ...

Can someone please give a first-principles definition of \(\displaystyle \mathbb{R}\)-linear and \(\displaystyle \mathbb{C}\)-linear mappings ... and some simple examples ...Hope someone can help ... ... help will be appreciated ...

Peter

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