Let $E$ be a Banach lattice and $E_+$ denote its positive cone.
For $x_1, \ldots, x_n \in E_+$, is there any way to bound $\sum_1^n\|x_i\|$ with $\|\sum_1^n x_i\|$ without using n?
Similaryly, for a atomless finite measure space $X$, let $f:X \rightarrow E_+$ be an integrable function. Can I bound $\int_X\|f\|d\mu$ with $\|\int_Xfd\mu\|$?
I think this is not usually possible, but the range of the function is only restricted to a positive cone of $E$