Mixed finite element methods employ numerical spaces that are, most of the time, something different than the full P^k spaces (polynomials with order up to k). For example, the BDM_k spaces for quadrilaterals employ numerical spaces that are more than P^k in each coordinate, while BDFM_{k-1} spaces for triangles employ numerical spaces that are less than P^k in each coordinate.
An essential part of the development of these numerical spaces is the construction of projection operators. The projection operators are directly related to the accuracy of the approximation in that space. For example, BDM_k spaces for quadrilateral provide an approximation order of k+1in L^2, while BDFM_{k-1} spaces for triangles provide an approximation order of only k in L^2.
As far as I know, it is a common understanding that the general requirement for having k+1 approximation order in L^2 is that the numerical space contains a full P^k space. In other words, that the projection operator gives a polynomial that lives at least in P^k.
That said, my question is: is there any reference that rigorously proves the affirmation above, showing the conditions necessary for projection operations provide optimal approximation orders in L^2?
I have found the answer myself. The conditions for the projection, if it is defined on reference shapes, is given in Ciarlet's "The Finite Element Method for Elliptic Problems", Theorem 3.1.4. For more general star-shaped elements, the conditions are given in Di Pietro's "The Hybrid High-Order Method for Polytopal Elements", Lemma 1.43.
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