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When studying linear function spaces I encounter tensor product of the spaces and also tensor products of the vectors, for example: A standard exercise is to show that if $\left(\phi_n \right)_n$ is an orthonormal basis for $L^2([a,b])$ then $\left(\phi_n \otimes \phi _m \right)_{n,m}$ is an orthonormal basis for $L^2([a,b]^2)$, where we define the functions $\phi_n \otimes \phi_m (x,y) = \phi_n (x) \cdot \phi _m (y)$.

Is it meaningful, for any two functoins $f\colon X\rightarrow Z$ , $g\colon Y\rightarrow Z$ to defined a function $$ f\otimes g = h\colon X\times Y \rightarrow Z,\quad h(x,y)=f(x)\cdot g(y)$$ given that there is a multiplication operation in $Z$?

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    $f\otimes g$ is from $X\otimes Y\to Z\otimes Z$, I believe, with $h(x\otimes y)=f(x)\otimes g(y)$. The one you defined would be $f\cdot g$. – 2017-01-19
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    @AkivaWeinberger Ok so I get that $f\otimes g$ is not not to be understood as $f\cdot g$ (in the context of hilbert spaces)? I thought this was a pretty standard notation as it was used by two professors whose lectures I attended (in real analysis and functional analysis). – 2017-01-19
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    I might be wrong. – 2017-01-19

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