  Let $V$ be a set of isolated points in $\RR^d$. Define a linear functional
$\CL$ on the space of real polynomials restricted on $V$, $\CL f = \sum_{x \in
V} f(x)\rho(x)$, where $\rho$ is a nonzero function on $V$. Polynomial
subspaces that contain discrete orthogonal polynomials with respect to the
bilinear form $<f,g> = \CL(f g)$ are identified. One result shows that the
discrete orthogonal polynomials still satisfy a three-term relation and
Favard's theorem holds in this general setting.
