  We develop projection operators onto finite element differential forms over
simplicial meshes. Our projection is locally bounded in Lebesgue and
Sobolev-Slobodeckij norms, uniformly with respect to mesh parameters. Moreover,
it incorporates homogeneous boundary conditions and satisfies a local broken
Bramble-Hilbert estimate. The construction principle includes the Ern-Guermond
projection and a modified Cl\'ement-type interpolant with the projection
property. The latter seems to be a new result even for Lagrange elements. This
projection operator immediately enables an equivalence result on local- and
global-best approximations. We combine techniques for the Scott-Zhang and
Ern-Guermond projections and adopt the framework of finite element exterior
calculus. We instantiate the abstract projection for Brezzi-Douglas-Marini,
N\'ed\'elec, and Raviart-Thomas elements.
