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I'm studying distributions and just came along the formula $$\langle f\otimes g,\phi\otimes\psi\rangle=\langle f,\phi\rangle\langle g,\psi\rangle.$$

I understand what that means in the context of distributions, but I try to get some more intuition about the tensor product operator. Is there a more general setting (e.g. linear spaces) where we have $\otimes$, $\langle\cdot,\cdot\rangle$ and multiplication and can deduce above equality? Or does this only make sense in a very particular setting?

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    $\begingroup$ It helps to think of $\langle f,\cdot\rangle$ as a linear operator on the first inner product space. $\endgroup$ Jan 29, 2015 at 22:17

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Consider $k$-linear maps of $k$-modules ($k$ some ring, commutative say) $f:M\rightarrow N$, $f':M'\rightarrow N'$. Then one has (by using the definition of the tensor product of modules) an induced map $f\otimes f':M\otimes_k M'\rightarrow N\otimes_k N'$ which is such that $(f\otimes f')(m\otimes m')=f(m)\otimes f'(m')$. This latter equality is a generalization of the equation you mention: in your case $N=N'=k$ and one uses the canonical isomorphism $k\otimes_k k\rightarrow k$ which maps $x\otimes y\rightarrow xy$. (The bracket notation you use is not an inner product; it is the duality bracket, i.e. simply evaluation.)

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  • $\begingroup$ Sweet, many thanks! $\endgroup$
    – fweth
    Jan 30, 2015 at 0:27

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