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What does the notation $$\langle u, v \rangle_{H^{-1}, H^1}$$ mean? Is it simply $u(v)$ or does it have something to do with inner products on $H^{-1}$ and $H^1$?

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Just a notation: it says that $u \in H^{-1}$, that $v \in H$, and that $u$ acts on $v$ as $u(v)$.

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If we are talking about Bochner spaces eg. $L^2(0,T,H^1)$, when do we use $\langle u, v \rangle_{something} = \int_0^T \langle u, v \rangle$? – hopo2 Oct 24 '12 at 20:36
Is $H^{-1}$ defined via the Fourier transform, the way it is for $H^s$ for any real $s$, or is $H^{-1}$ something else? – Stefan Smith Mar 14 at 0:17

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