Subspace decompositions of finite element spaces based on L2-1ike
orthogonal projections play an important role for the construction and analysis
of multigrid like iterative methods. Recently several authors have proved the
equivalence of the associated discrete norms with the Hi-norm. The present
paper gives an elementary, self-contained derivation of this result which is based
on the use of K-functionals known from the theory of interpolation spaces.