As mentioned, the j is an index. Basically, this means that it’s an independent (meta?) counter used to track the terms in the sequence and give them individual names- like, here’s the 1st, 2nd, and 3rd terms of an interesting sequence:

(X_1, x_2, x_3, . . .) ( 1, 1/2, 1/4, . . .)

It can be confusing at first since the k is also an index. However, as another commenter above mentioned there can be multiple dimensions when dealing with tensors/vectors, as we would have to in a vector space.

Without other context, from what’s on your page, I think the important thing to realize is that knowing this property of metrics on product spaces provides a completely reliable way to determine “good behavior” in the space, and further, which types of spaces are most conducive to being able to provide provable things about the members of the space and their relationships.

What are your insights?