# Banach dual space integral

Let $X$ be a Banach space and $f_t \in X^*$ for each $t \in [0,t_0]$. Suppose that $$\int_0^{t_0} f_t(x) = 0$$ for all $x \in X$.

1) Does it make sense to write $\int_0^{t_0}f_t = 0$?

2) If so, does this follow from the above property?

Thanks

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Are you familiar with the Bochner or Pettis integral? –  Nate Eldredge Jan 18 '13 at 18:32
@NateEldredge Yes, sort of. I wonder if the expression I write above in 1) makes sense though. –  george.s Jan 18 '13 at 18:33