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What does $f \in L^p(I,V) + L^q(I,H)$ mean? What does the addition mean?? I see this in PDE theory.

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You should precise what are $I,V$ and $H$. –  FPE Jul 16 '13 at 12:45
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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. –  brom Jul 16 '13 at 12:51
    
@brom Your comment fully qualifies as an answer here. Please post as such. –  40 votes Jul 16 '13 at 17:53

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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.

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