Uniform approximation on ideals of multilinear mappings

Abstract

For each ideal of multilinear mappings $\mathcal{M}$ we explicitly construct a corresponding ideal ${}^{a}\mathcal{M}$ such that multilinear forms in ${}^{a}\mathcal{M}$ are exactly those which can be approximated, in the uniform norm, by multilinear forms in $\mathcal{M}$. This construction is then applied to finite type, compact, weakly compact and absolutely summing multilinear mappings. It is also proved that the correspondence $\mathcal{M} \mapsto {}^{a}\mathcal{M}$ is Aron-Berner stability preserving.