Let $1 \leq p<\infty$.
i)Find an isometry $j: l_{\infty} \rightarrow B\left(L_{p}[0,1]\right)$
ii). Let $1 \leq p<\infty$ and consider $U: l_{p} \rightarrow L_{p}[0, \infty) $
for case ii we can set : $U(x)=\sum_{n=1}^{\infty} x_{n} \chi_{[n-1, n)}$
$\forall x=\left(x_{1}, x_{2}, \ldots\right) \in l_{p}$
We have $|U(x)|^{p}=\sum_{n=1}^{\infty}\left|x_{n}\right|^{p} \chi_{[n-1, n)},$ and therefore
$$
\int_{0}^{\infty}|U(x)|^{p} d \mu=\sum_{n=1}^{\infty}\left|x_{n}\right|^{p} \mu([n-1, n))=\sum_{n=1}^{\infty}\left|x_{n}\right|^{p}
$$
i.e., $\|U(x)\|=\|x\| \forall x \in l_{p}$ so $U$ is an isometry.