Here is a very basic question that has come up in some work I’m doing with Diane Maclagan. There is lots of algebraic geometry in our intended application, but I think that what I really need is a better understanding of the underlying linear algebra. First, let me review some even more basic ideas. Let and be two finite dimensional vector spaces over a field and let be a bilinear pairing. For a subspace of , define the subspace to be the space of those vectors such that for all . We can also define for any subspace of .
Then the “Fundamental Theorem of Bilinear Pairings” is the following: for any subspace of , we have . In particular, if and only if
OK, that was pretty simple. My situation is that, instead of having a pairing down to the ground field, we have a bilinear pairing into another finite dimensional vector space.
What is the new fundamental theorem characterizing ? In particular, under what conditions do we have ?