1. In class I showed that the map is always injective. If V and W are finite-dimensional, then these two vector spaces have the same finite dimension, and are therefore isomorphic.

What happens when one (or both) of V and W is not finite-dimensional? Is this sometimes / never an isomorphism?

2. I also explained that if A is a finite-dimensional algebra, then the dual A* is a coalgebra, with its coalgebra structure from the algebra structure of A. The key is that the map above can be reversed, and everything works fine.

What happens when A is not finite-dimensional? Is A* sometimes/never a coalgebra? Is there some other natural coalgebra we can associate to A?

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I saw in a book that you can mod out A by an ideal to try to get it to be finite-dimensional, and then work with its finite-dual with respect to that ideal… I wonder if the coalgebra associated with A’s finite-dual could be extended via tensoring with the ideal basis to “undo” the effect of modding out?