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

Question

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: $$|f|_{L^p(0,T;X)}^p = \int_0^T |f(t)|_{X}^p = \int_0^T \left|\sum_{i=1}^\infty x_i h \chi_{E_i}\right|_X^p$$ How to expand the integrand??

Answer 1

The point is that $$ \left|\sum_{i=1}^\infty\,x_i\,h\,\chi_{e_i}\right|_X=\sum_{i=1}^\infty\,|x_i|_X\,|h|\,\chi_{e_i}=\sum_{i=1}^\infty\,\,|h|\,\chi_{e_i}=h $$ because you are calculating the norm at each point. And the same happens with taking a power, so $$ \left|\sum_{i=1}^\infty\,x_i\,h\,\chi_{e_i}\right|_X^p=\sum_{i=1}^\infty\,|x_i|_X^p\,|h|^p\,\chi_{e_i}=h^p $$

Answer 2

For every $t \in (0,T)$ we have $$ \Bigl| \sum_{i=1}^\infty x_i h \chi_{E_i} \Bigr|_X^p =\Bigl(\sum_{i=1}^\infty |x_i h|_X\chi_{E_i}\Bigr)^p =\Bigl(\sum_{i=1}^\infty |h|_X\chi_{E_i}\Bigr)^p =|h|^p_X. $$ edit: since the sets $E_i$ partition $(0,T)$.

edit2: fixed the equality signs