Let $X$ a Banach Space and $(\Omega, \Sigma, \mu)$ a measure space. A function $F:\Omega\rightarrow X$ is Dunford integrable is $x^\ast\circ F$ is $\mu$-integrable for every $x^\ast\in X^\ast$. The space of functions that are dunford integrable, denoted by $\mathbb{D}(\mu,X)$, is a normed space with $$ \|F\|:=\sup\left\{\int_\Omega|x^\ast\circ F|d\mu:x^\ast\in X^\ast,\,\|x^\ast\|\leq1\right\} $$ Does anybody knows if the space $\mathbb{D}(\mu,X)$ is complete?

Let $X$ a Banach Space and $(\Omega, \Sigma, \mu)$ a measure space. A function $F:\Omega\rightarrow X$ is Dunford integrable if $x^\ast\circ F$ is $\mu$-integrable for every $x^\ast\in X^\ast$. The space of functions that are Dunford integrable, denoted by $\mathbb{D}(\mu,X)$, is a normed space with $$ \|F\|:=\sup\left\{\int_\Omega|x^\ast\circ F|d\mu:x^\ast\in X^\ast,\,\|x^\ast\|\leq1\right\} $$ Does anybody knows if the space $\mathbb{D}(\mu,X)$ is complete?