In chapter 1 of Raymond A. Ryan's Introduction to Tensor Products of Banach Spaces, the alternating tensor product is defined in the exercise section (see below).
However, while I would like to solve exercise 1.9, I don't even understand the definition. First of all, there is a round bracket on the lhs and a angular bracket on the rhs in the definiton of $x\otimes_ay$. Is this a typo? I guess not, since there are other places in the book before where this kind of bracket type mismatch occurs. But I cannot find the definition. Moreover, shouldn't $x\otimes y-x\otimes y$ be $0$?
So, I'd really appreciate if someone could help me understanding the definition and indicate how we can solve this task. I'd also be interested in the question, if we are able to complete this space of alternating tensor products with respect to the Hilbert-Schmidt norm (assuming $X$ is a Hilbert space, of course).
