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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 1.4: Angle-Preserving Mappings ... ...
I need help in order to fully understand a remark of Remmert's regarding injective $$\mathbb{R}$$-linear mappings The relevant part of Remmert's section on Angle-Preserving Mappings reads as follows:View attachment 8550In the above text from Remmert we read the following:
" ... ... we look at $$\mathbb{R}$$-linear injective (consequently also bijective) mappings $$T : \mathbb{C}\to \mathbb{C}$$ ... ... " Can someone please explain how/why exactly $$\mathbb{R}$$-linear injective mappings are necessarily surjective ... ... ?
Peter
I am focused on Chapter 0: Complex Numbers and Continuous Functions ... and in particular on Section 1.4: Angle-Preserving Mappings ... ...
I need help in order to fully understand a remark of Remmert's regarding injective $$\mathbb{R}$$-linear mappings The relevant part of Remmert's section on Angle-Preserving Mappings reads as follows:View attachment 8550In the above text from Remmert we read the following:
" ... ... we look at $$\mathbb{R}$$-linear injective (consequently also bijective) mappings $$T : \mathbb{C}\to \mathbb{C}$$ ... ... " Can someone please explain how/why exactly $$\mathbb{R}$$-linear injective mappings are necessarily surjective ... ... ?
Peter