BASES IN THE SPACES OF HOMOGENEOUS POLYNOMIALS AND MULTILINEAR OPERATORS ON BANACH SPACES

For Banach spaces E-1, ..., E and F with their bases, we show that a particular monomial sequence forms a basis of P(E-m; F), the space of continuous m-homogeneous polynomials from E to F (resp. a basis of L(E-1, ..., E-m;F), the space of continuous m-linear operators from E-1 x ... x E-m to F) if and only if the basis of E (resp. the basis of E-1, ..., E-m) is a shrinking basis and every P is an element of P(E-m;F) (resp. every T is an element of L(E-1, ..., E-m;F)) is weakly continuous on bounded sets.