$L_{\mu }^p$-Normed Spaces of Bounded Multilinear Operator

Abstract

We study the spaces of bounded multilinear operators ${\mathcal{L}}^i(X,Y)$ endowed with $L_{\mu }^p$-norms, where $X,Y$ are normed spaces and $\mu$ is a measure with $p$-moments. These are norms without taking a quotient, under suitable support conditions. The topology induced coincides with the strong operator and compact-open topologies on operator-norm-bounded subsets. Furthermore, we identify the completion of these spaces with spaces of Hilbert--Schmidt operators on the reproducing kernel Hilbert space (RKHS) for $p=2$; with iterated $\gamma$-radonifying multilinear operators on the RKHS in the Gaussian case, for general $p$.