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How can we define the Inner Product of multi-variable functions?

For example, what is the value of the inner product of $\nabla f$ and $\nabla g$?

$$\langle \nabla f, \nabla g\rangle = ?? $$

Here $\langle\cdot,\cdot\rangle$ is used for the inner product in $L^2$.

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$\langle \nabla f, \nabla g\rangle_{L^2(\Omega)} = \sum_i \int_\Omega \partial_i f\, \partial_i g\, dx$. – martini Apr 21 '12 at 4:53
@martini As I expected! Thanks a lot. – Misaj Apr 21 '12 at 4:57
@Ashuley Use '\langle \rangle' which gives $\langle \rangle$ instead of $<>$. – Davide Giraudo Apr 21 '12 at 9:39

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