  Constrained orthogonal polynomials have been recently introduced in the study
of the Hohenberg-Kohn functional to provide basis functions satisfying particle
number conservation for an expansion of the particle density. More generally,
we define block orthogonal (BO) polynomials which are orthogonal, with respect
to a first Euclidean scalar product, to a given $i$-dimensional subspace ${\cal
E}_i$ of polynomials associated with the constraints. In addition, they are
mutually orthogonal with respect to a second Euclidean scalar product. We
recast the determination of these polynomials into a general problem of finding
particular orthogonal bases in an Euclidean vector space endowed with distinct
scalar products. An explicit two step Gram-Schmidt orthogonalization (G-SO)
procedure to determine these bases is given. By definition, the standard block
orthogonal (SBO) polynomials are associated with a choice of ${\cal E}_i$ equal
to the subspace of polynomials of degree less than $i$. We investigate their
properties, emphasizing similarities to and differences from the standard
orthogonal polynomials. Applications to classical orthogonal polynomials will
be given in forthcoming papers.
