The authors propose quasi-Newton and limited memory quasi-Newton algorithms for functions defined on a Grassmannian Gr\((n, r)\) as well as a product of Grassmannians Gr\((n_1, r_1) \times \dots \times\) Gr\((n_k, r_k)\), with Broyden-Fletcher-Shanno-Goldfarb (BFGS) and limited memory BFGS updates. By focusing on the local coordinates approach they show that their BFGS update shares the same well-known optimality property of its Euclidean counterpart.