I am interested in equivalence between Euclidean norm and finite element norm: $$ C_0\left\| \mathbf{v}^h \right\|_2 \leq \left\| P^h\mathbf{v}^h \right\|_{L^2(\Omega)} \leq C_1\left\| \mathbf{v}^h \right\|_2 $$
My question is related to previously posted question: The equivalence of Euclidean norm and finite element norm The equivalence of Euclidean norm and finite element norm
I am mostly interested in knowing what needs to be adjusted, such that proof holds also in 3D and for non-uniform finite elements(still P1, Lagrange).