  We give necessary and sufficient conditions for interpolation inequalities of
the type considered by Marcinkiewicz and Zygmund to be true in the case of
Banach space-valued polynomials and Jacobi weights and nodes. We also study the
vector-valued expansion problem of $L_p$-functions in terms of Jacobi
polynomials and consider the question of unconditional convergence. The notion
of type $p$ with respect to orthonormal systems leads to some characterizations
of Hilbert spaces. It is also shown that various vector-valued Jacobi means are
equivalent.
