What does $f \in L^p(I,V) + L^q(I,H)$ mean? What does the addition mean?? I see this in PDE theory.
It means that $f$ can be written as the sum of a function in $L^p(I,V)$ and a function in $L^q(I,H)$. Although deviations are possible, the notation suggests that $L^p(I,V)$ and $L^q(I,H)$ are probably Bochner spaces.