Orthogonal polynomials are often preferred as basis functions. Recently I learned selecting orthonormal basis further simplifies the mass matrix from diagonal to simply the identity matrix when used as modal basis. But I am confused how the normalizing factors are obtained.
Say the orthonormal Legendre polynomials in each direction of a reference hexahedron are $P_1$, $P_2$, $P_3$. Then using tensor-product approach, is the orthonormal basis for hex simply $P_1\times P_2\times P_3$? What about for quads, simply $P_1\times P_2$, or do we need to multiply by some constant to keep the orthonormality of the basis? I learned from the text book (Hesthaven & Warburton) that orthonormal basis for triangles and tetrahedron have different normalizing factors, i.e. simply cancelling one of the directions of an orthonormal basis in tetrahedron does not result in a basis for the triangle.
Triangle:
$$ \sqrt{2} P_i^{(0,0)}(a) P_j^{(2i+1,0)}(b) (1-b)^i $$
Tetrahedron:
$$ 2\sqrt{2} P_i^{(0,0)}(a) P_j^{(2i+1,0)}(b) P_k^{(2i+2j+2,0)}(c) (1-b)^i (1-c)^{(i+j)} $$