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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:$$ \mathbb{R}$$-linear and $$\mathbb{C}$$-linear mappings of $$\mathbb{C}$$ into $$\mathbb{C}$$ ... ... I need help in order to get a clear idea of the definition and nature of $$\mathbb{R}$$-linear and $$\mathbb{C}$$-linear mappings of \mathbb{C} into \mathbb{C} ... ...
Remmert's section on $$\mathbb{R}$$-linear and $$\mathbb{C}$$-linear mappings of $$\mathbb{C}$$ into $$\mathbb{C}$$ 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 $$\mathbb{R}$$-linear and $$\mathbb{C}$$-linear mappings ...
Can someone please give a first-principles definition of $$\mathbb{R}$$-linear and $$\mathbb{C}$$-linear mappings ... and some simple examples ...Hope someone can help ... ... help will be appreciated ...
Peter
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:$$ \mathbb{R}$$-linear and $$\mathbb{C}$$-linear mappings of $$\mathbb{C}$$ into $$\mathbb{C}$$ ... ... I need help in order to get a clear idea of the definition and nature of $$\mathbb{R}$$-linear and $$\mathbb{C}$$-linear mappings of \mathbb{C} into \mathbb{C} ... ...
Remmert's section on $$\mathbb{R}$$-linear and $$\mathbb{C}$$-linear mappings of $$\mathbb{C}$$ into $$\mathbb{C}$$ 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 $$\mathbb{R}$$-linear and $$\mathbb{C}$$-linear mappings ...
Can someone please give a first-principles definition of $$\mathbb{R}$$-linear and $$\mathbb{C}$$-linear mappings ... and some simple examples ...Hope someone can help ... ... help will be appreciated ...
Peter
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