Don't understand this $L^p$ space inequality (Bochner spaces, etc)

Don't understand this $L^p$ space inequality (Bochner spaces, etc)

For $p \geq 1,$ define $f \in L^p(0,T;X)$ by $$f = \sum_{i=1}^\infty x_i h
\chi_{E_i}$$ where $E_i$ are measurable disjoint partition of $[0,T]$. The
$x_i \in X$ with $|x_i|_X = 1$, and $h \geq 0$ is in $L^p(0,T)$ and is
such that $0 < |h|_{L^p(0,T)} \leq 1$.
Why is it true that: $$|f|_{L^p(0,T;X)} = |h|_{L^p(0,T)} \leq 1.$$
This is from page 98 of Diestel, Uhl: Vector measures.
Where to go from here: $$|f|_{L^p(0,T;X)}^p = \int_0^T |f(t)|_{X}^p =
\int_0^T |\sum_{i=1}^\infty x_i h \chi_{E_i}|_X^p$$ How to expand the
integrand??