  In this sequence, we first prove an abstract Morse index theorem in a Hilbert
space modeling a variational problem with constraints. Then, our abstract
formulation is applied to study several optimization setups including closed
CMC hypersurfaces, capillary surfaces in a ball, and critical points of type-II
partitioning. In this paper, we study the index and nullity of a symmetric
bounded bilinear form in a Hilbert space. The main results determine precisely
how these notions change when restricting to a subspace of a finite
codimension.
