Suppose we are given an infinite dimensional vector space V with a basis satisfying the following property: all the elements in V can be represented by a finite linear combinations of the elements of the basis. Now I should be able to talk about linear maps in this space, and their matrix representation, e.g.
where AB or BA are well defined too. This is what I understand from reading Lectures in Abstract Algebra, Vol. 2 by Nathan Jacobson (http://reslib.com/book/Lectures_in_A...ebra__Vol__2#1), chapter 9 on infinite dimensional vector spaces.
From here, I want to talk about the matrix that represents the kronecker product A⊗B, but I'm not sure if it is well defined.
The interesting thing is, that later in chapter 9 of this book (page 256 middle paragraph) the author says that we can talk about Kronecker product of arbitrary vector spaces. However, at that point I could not follow the details due to insufficient background, so I don't know if it indeed answers my question.
Finally, if such an infinite matrix is well defined, I'd like to know if it satisfies the Kronecker product properties that could be found in wikipedia on Kronecker product: bilinearity, associativity and the mixed-product property.
Thank you for your help.