R-Linear and C-Linear Mappings.... Another Question .... Remmert, Section 1.2, Ch. 0 .... ....

[Math Amateur]

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: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

 

[jvicens]

Hello Peter,

I am happy to assist you with understanding Remmert's results regarding \mathbb{C}-linear mappings of \mathbb{C} into \mathbb{C}. Let me address your questions one by one.

Question 1:

An \mathbb{R}-linear mapping T : \mathbb{C} \to \mathbb{C} is a function that satisfies the following properties:

1. T(a+b) = T(a) + T(b) for all a,b \in \mathbb{C}
2. T(ka) = kT(a) for all k \in \mathbb{R} and a \in \mathbb{C}

Now, in order for T to be \mathbb{C}-linear, it needs to satisfy the following properties:

1. T(a+b) = T(a) + T(b) for all a,b \in \mathbb{C}
2. T(ka) = kT(a) for all k \in \mathbb{C} and a \in \mathbb{C}

Since i \in \mathbb{C}, we can rewrite the second property as T(ia) = iT(a). Now, if we substitute a = 1, we get T(i) = iT(1). This means that T(i) = i T(1) is a necessary condition for T to be \mathbb{C}-linear.

Question 2:

A \mathbb{C}-linear mapping T : \mathbb{C} \to \mathbb{C} has the form T(z) = T(1)z because of the following reasoning. Since T is \mathbb{C}-linear, we know that T(ia) = iT(a) for all a \in \mathbb{C}. Now, let z = a+bi \in \mathbb{C}. Then, T(z) = T(a+bi) = T(a) + iT(b) = T(1)a + iT(1)b = T(1)(a+bi) = T(1)z. Therefore, we can see that T(z) = T(1)z for all z \in \mathbb{C}.

I hope this helps clarify your understanding of Remmert's results. Let me know if you have any further questions.