Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this community
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?
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.)
