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Dec 6, 2016 at 21:02 comment added malwin Thank you for your answer. I edited some typos in my question. I wanted to ask for $\phi_j' := L^2(0,T,V) \to L^2(0,T,V^*)$. Your answer of course answers this question, too. Thanks. But could you give me a hint on how to prove this in general?
Dec 6, 2016 at 20:57 comment added gerw You need additionally a growth condition, e.g., $|\phi_j(u)| \le |u|$. Such things you can find in the book "Appell J., Zabrejko P.P.: Nonlinear Superposition Operators".
Dec 6, 2016 at 20:45 vote accept malwin
Dec 3, 2016 at 18:39 history answered gerw CC BY-SA 3.0