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Let $(\Omega,\mathcal{F},\mu)$ be a measure space (say complete and $\sigma$-finite, for simplicity). Furthermore, let $(X,\Vert\cdot\Vert_{X})$ be an arbitrary Banach space. I denote by $(L^{p}(\...

Assume that $X$ is a Banach space such that the dual $X'$ has the Radon-Nikodym property. Moreover let $(\Omega,\Sigma,\mu)$ a say finite measure space. Then we know that for $1\leq p<\infty$ holds ...

Let's have a look to the unit interval $[0,1]$ and a Banach space $X$ and then to the space $$ E:=L^{\infty}([0,1],X), $$ i.e. all essentially bounded Banach-valued functions $f:[0,1]\rightarrow X$. ...