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Let $f \in H^{-1}(U)$ and $u \in H^1(U)$. I know that we write $f(u)$ as the pairing $$\langle f, u \rangle_{H^{-1}, H^1}$$.

Suppose that $v$ is the weak/distributional derivative of $u$. So $$\int_0^T u\phi' = -\int_0^T v\phi$$ holds for all $\phi \in C_c^\infty(0,T)$. What does it mean to write $\langle v, w \rangle_{H^{-1}, H^1}$ where $w \in H^1(U)$? It's more than just $v(w)$, right? What's the functional that it's associated to? I ask because I see expressions like $\langle v(t), \varphi(t) \rangle_{H^{-1}, H^1}$ for $\varphi$ a $H^1(U)$-valued smooth function with compact support in $(0,T)$ and I do not know what it means.

PS: I know this is an easy question but no one answers in math.SE..

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up vote 3 down vote accepted

Maybe Thm. 1.5.5. of these notes is the answer to your question.

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Thank you. I hope I am not being rude by kindly asking you to have a look, if you're inclined, at my other question from which this one springs (…). – Henry P Oct 28 '12 at 22:15
Have you checked Lions and Magenes book? They deal with this kind of question. – Liviu Nicolaescu Oct 28 '12 at 22:34
I had a look.. unfortunately no luck :| – Henry P Oct 29 '12 at 9:54
I am confused about some formulation in your question. . What do you mean by Let $S=S(s)$ be a set in the space, where $s\in (0,T)$ is fixed... What space are you talking about? What is the role of $s$. – Liviu Nicolaescu Oct 29 '12 at 12:26
Sorry. S(t) is supposed to represent a hypersurface in $\mathbb{R}^n$ for each $t$. So think of $\{S(t)\}_{t \in [0,T]}$ as a set of hypersurfaces. $s \in [0,T]$ is simply a fixed time, so $S(s)$ is a hypersurface. But I believe you can just ignore this; and think of $S$ as some domain. For that question, the specifics don't matter (I believe). – Henry P Oct 29 '12 at 13:07

It's more than just v(w), right?

No, it's not, as far as I know. It is just a (historically motivated) alternative notation for $v(w)$.

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