Bases in the spaces of homogeneous polynomials and multilinear operators on Banach spaces

Abstract

For Banach spaces $E_1, \dots ,E_m$, $E$ and $F$ with their bases, we show that a particular monomial sequence forms a basis of $\mathcal {P}(^mE; F)$, the space of continuous $m$-homogeneous polynomials from $E$ to $F$ (resp.\ a basis of $\mathcal {L}(E_1,\dots ,E_m;F)$, the space of continuous $m$-linear operators from $E_1\times \cdots \times E_m$ to $F$) if and only if the basis of $E$ (resp. the basis of $E_1,\dots ,E_m$) is a shrinking basis and every $P \in \mathcal {P}(^mE; F)$ (resp.\ every $T \in \mathcal {L}(E_1,\dots ,E_m;F)$) is weakly continuous on bounded sets.