What is the dual vector space of $\mathbb{C}\,$?

Question

Given that the complex numbers $\mathbb{C}$ are considered to be canonically isomorphic to $\mathbb{R^2}$, I am wondering if the dual space of $\mathbb{C}$ is the same es that of $\mathbb{R}$, i.e. the vector space of linear functionals. I never heard about dual space of $\mathbb{C}$ until today, so I couldnt find much information in the litterature. Also:

• How is the dual of $\mathbb{C}$ defined, what is a dual basis of $\mathbb{C}$ ?
• What is the field $K$ in which the linear functionals get their values ($\mathbb{C}$ or $\mathbb{R}$) ?

Many thanks.

Answer 1

The answer to your questions depends upon your answer to this question: do you see $\mathbb C$ as a real vector space or as a complex vector space?

Since you mention $\mathbb{R}^2$, I'll assume that you see it as a real vector space. In that case:

• A basis of $\mathbb{C}^*$ is $\{\alpha,\beta\}$, with $\alpha(z)=\operatorname{Re}z$ and $\beta(z)=\operatorname{Im}z$.
• The functionals get their values in $\mathbb R$.

On the other hand, if you see $\mathbb C$ as a complex vector space, then you can take any map $\alpha\colon\mathbb{C}\longrightarrow\mathbb{C}$ of the type $\alpha(z)=az$ (as long as $a\neq0$) and $\{\alpha\}$ will be a basis of $\mathbb{C}^*$. For instance, you can take $\alpha=\operatorname{Id}$ (which corresponds to choosing $a=1$).