Uniform approximation on ideals of multilinear mappings

TL;DR

This paper constructs a new ideal of multilinear mappings that captures those forms approximable by a given ideal, and explores its properties and applications to various classes of multilinear mappings.

Contribution

It explicitly constructs an approximation ideal for multilinear mappings and studies its properties, including Aron-Berner stability, with applications to several classes.

Findings

The approximation ideal $^{a}{ m M}$ precisely characterizes approximable multilinear forms.

The construction preserves Aron-Berner stability.

Applications to finite type, compact, weakly compact, and absolutely summing multilinear mappings.

Abstract

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