A digression is that what multiplying A and A^T really means, in the sense that A represents a map of spaces, but A^T represents a map in the opposite direction

A digression is that what multiplying A and A^T really means, in the sense that A represents a map of spaces, but A^T represents a map in the opposite direction of the dual spaces. In finite dimension or in topological vector spaces, a space is isomorphic with its dual, but not naturalliy so. It is naturally isomorphic to the double dual! as in Mac Lane's CWM. So the multiplication is doable with an apparent excess good luck. On the other hand, in [1], linear adjoint operators bewteen two topological vector spaces are shown equivalent, up to an scalar, to adjoint functor pairs showing a Galois connection between the posetal categories being the lattice of subspaces of the original two spaces. This story should continue. [1] https://doi.org/10.1090/S0002-9939-1974-0346548-4 https://doi.org/10.1090/S0002-9939-1974-0346548-4