Let $X$ be a (real) sequence Banach space, such as $c_{0},\ell_{p}$, etc. and $f:[0,1]\longrightarrow X$ continuous. Then, it seems "quite intuitive" that if $f(t):=(f_{1}(t),f_{2}(t),\ldots,f_{n}(t),\ldots)\in X$ for every $t\in I$, and certain continuous functions $f_{i}$ we have
$$ \int_{0}^{t}f(s)ds=\big(\int_{0}^{t}f_{1}(s)ds,\int_{0}^{t}f_{2}(s)ds,\ldots, \int_{0}^{t}f_{n}(s)ds,\ldots \big)\quad \textrm{for all }t\in I, $$
where the left-hand side integral means in the Bochner sense and the integrals in the right-hand side mean in the Lebesgue sense.
But, is true the above equality?
Many thanks in advance for your comments.
Fix $n$. Define the functional $g\in X^*$ by $g(f) = f_n$. Then $$ \left\langle g, \int_0^1f(s)ds \right\rangle =\int_0^1 \langle g,f(s)\rangle ds =\int_0^1 f_n(s)ds. $$