  We merge and extend recent results which prove the H1-stability of the
L2-orthogonal projection onto standard finite element spaces, provided that the
underlying simplicial triangulation is appropriately graded. For lowest-order
Courant finite elements S1(T) in Rd with d>=2, we prove that such a grading is
always ensured for adaptive meshes generated by newest vertex bisection. For
higher-order finite elements Sp(T) with p>=1, we extend existing bounds on the
polynomial degree with a computer-assisted proof. We also consider
L2-orthogonal projections onto certain subspaces of Sp(T) which incorporate
zero Dirichlet boundary conditions resp. an integral mean zero property.
