For $p \in [1, \infty)$, let $H^{p}(\mathbb{D},X)$ be the space of analytic functions from $\mathbb{D}$ into a complex Banach space $X$ such that \begin{equation} \label{him-p7-e-1.11} ||f||_{H^{p}(\mathbb{D},X)}= \sup \limits _{0<r<1} \left(\int_{0}^{2\pi} ||f(re^{it})||^p \frac{dt}{2\pi}\right)^{1/p} < \infty. \end{equation} For complex valued analytic functions, $H^{p}(\mathbb{D},\mathbb{C})$ is the usual Hardy spaces. For complex valued analytic function $f(z)=\sum_{n=0}^{\infty} a_{n} z^n$ in $\mathbb{D}$, it is known that \begin{equation} \label{e-5} ||f||^2_{H^{2}(\mathbb{D},\mathbb{C})}= \sum_{n=0}^{\infty} |a_{n}|^2 \end{equation} Let $X$ be an arbitrary complex Banach space. Let $f \in H^{2}(\mathbb{D},X)$ of the form $f(z)=\sum_{n=0}^{\infty} x_{n}z^n$ in $\mathbb{D}$, where $x_{n}\in X$. Then is the following \begin{equation} \label{e-3} ||f||^2_{H^{2}(\mathbb{D},X)}= \sum_{n=0}^{\infty} ||x_{n}||^2 \end{equation} holds always? Here $||.||$ is the norm in Banach space $X$. I don't know whether it is true for arbitrary Banach space $X$.
Kindly help! Thanx n regards