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I'm trying to understand the concept of the tensor product of Hilbert spaces. Every source I'm reading has as part of the definition, an equation such as the following:

(v⊗w, v’⊗w’) = (v,v’)(w,w’)

I believe that I’m correctly understanding the left side of this equation as: the inner product of the tensor product of v and w, with the tensor product of v’ and w’.

However, I don’t know how to read the right side of the equation. I see the inner product of v and v’, next to the inner product of w and w’, though with no operator symbol between the two expressions.

Since I’m not familiar with the notation on the right side of the above equation, I don’t understand what it’s saying about how the inner products on that side are related to each other.

I may be missing something important regarding the definition of tensor products of Hilbert spaces, that would explain why the above equation is written the way it is. However, my first guess is that I just don’t understand the notation.

I’d appreciate any help, and I thank you very much for your thoughtful consideration.

From what I can gather, the expression

(v⊗w, v’⊗w’) = (v,v’)(w,w’)

represents the definition of an inner product for tensor products in a Hilbert space and the right hand side of the equation represents a simple multiplication of 2 scalars, i.e., the product of the scalars given by the inner (dot) product of Hilbert space vectors. See https://en.wikipedia.org/wiki/Tensor_product_of_Hilbert_spaces.

￼￼The key is that left hand side is a little more than just the seemingly straightforward "inner product of the tensor product of v and w, with the tensor product of v’ and w’" and is in fact a double dot product (news to me!), which also turns out to be a scalar; see https://en.wikipedia.org/wiki/Dyadics.

Thus, the expression you are puzzling over is apparently a definition that provides an inner product required to define a Hilbert space for tensor products comprised of vectors from Hilbert spaces.

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