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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:##\mathbb{R}##-linear and ##\mathbb{C}##]-linear mappings of ##\mathbb{C}## into ##\mathbb{C}## ... ...
I need help in order to fully understand one of Remmert's results regarding ##\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:
In the above text by Remmert we read the following: (... fairly near the start of the text ...)
" ... ... An ## \mathbb{R}##-linear mapping ##T : \mathbb{C} \to \mathbb{C}## is then ##\mathbb{C}##-linear when ##T(i) = i T(1)##; in this case it has the form ##T(z) = T(1) z##. ... ... "My questions are as follows:Question 1
How/why exactly is an ## \mathbb{R}##-linear mapping ##T : \mathbb{C} \to \mathbb{C}## also ##\mathbb{C}##-linear when ##T(i) = i T(1)## ... ... ?
Question 2
Why/how exactly does a ##\mathbb{C}##-linear mapping have the form ##T(z) = T(1) z## ... ...
Hope someone can help ...
Peter
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 fully understand one of Remmert's results regarding ##\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:
In the above text by Remmert we read the following: (... fairly near the start of the text ...)
" ... ... An ## \mathbb{R}##-linear mapping ##T : \mathbb{C} \to \mathbb{C}## is then ##\mathbb{C}##-linear when ##T(i) = i T(1)##; in this case it has the form ##T(z) = T(1) z##. ... ... "My questions are as follows:Question 1
How/why exactly is an ## \mathbb{R}##-linear mapping ##T : \mathbb{C} \to \mathbb{C}## also ##\mathbb{C}##-linear when ##T(i) = i T(1)## ... ... ?
Question 2
Why/how exactly does a ##\mathbb{C}##-linear mapping have the form ##T(z) = T(1) z## ... ...
Hope someone can help ...
Peter
Attachments
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Remmert - 1 - R-linear and C-linear Mappings, Ch. 0, Section 1.2 ... PART 1 .png32.2 KB · Views: 336
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Remmert - 2 - R-linear and C-linear Mappings, Ch. 0, Section 1.2 ... PART 2 .png32.7 KB · Views: 328
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