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.