I wonder if my statements below are correct.
Let $V$ be an open domain in $\mathbb{R}^d$, and $U$ an open domain in $\mathbb{C}^d$ with $V=\operatorname{Re}U:=\{\operatorname{Re}z:z\in U\}$.
I want to define $\mathcal{H}(V,U)$ to be the space of complex analytic functions on $U$ which has a continuous extension to $\overline{U}$ and is real analytic when restricted to $V$. Let $\|f\|:=\sup_{z\in U}|f(z)|$. I think $\mathcal{H}(V, U)$ is a Banach space?
I am thinking, the space of real analytic functions on $V$ which has a complex analytic extension to $U$ and further continuous extension to $\overline{U}$ should be an equivalent definition for $\mathcal{H}(V,U)$? (Because I think if a real analytic function $f$ on $V$ can be analytically extended to $U$, this extension is unique.)
Any help is much appreciated! Thanks.