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Let $f$ and $g$ are two functions defined on $[0,1]$ and taking values in a Hilbert space $H$. Then how to define and compute the tensor product between $f$ and $g$, namely $f\otimes g$?

I know how to compute the tensor product between vectors and matrices (same as the kronecker product). But unable to do the same for functions.

Thanks.

Janak
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1 Answers1

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$f \otimes g$ is the function $[0,1]^2 \to \mathbb C$ given by $(f \otimes g)(x,y) = f(x) g(y) $.

Stephen Montgomery-Smith
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    Remember that $f$ and $g$ take values in a Hilbert space. What exactly is the product $f(x)g(y)$ you are referring to? The inner product? Or the tensor product? I'm pretty sure that the tensor product of functions is given by $(f \otimes g)(x,y) = f(x) \otimes g(y)$ – silvascientist May 24 '18 at 20:46
  • Yes, I missed that the functions took values in a Hilbert space. I agree with your answer. – Stephen Montgomery-Smith May 24 '18 at 21:05