Lipschitz tensor product

Abstract

Inspired by the ideas of R. Schatten in his celebrated monograph on a theory of cross-spaces, we introduce the notion of a Lipschitz tensor product X&E of a pointed metric space X and a Banach space E as a certain linear subspace of the algebraic dual of Lip0(X,E*). We prove that X&E is linearly isomorphic to the linear space of all finite-rank continuous linear operators from (X#, τp) into E, where X# denotes the space Lip0(X,K) and τp is the topology of pointwise convergence of X#. The concept of Lipschitz tensor product of elements of X# and E* yields the space X#&E* as a certain linear subspace of the algebraic dual of X&E. To ensure the good behavior of a norm on X&E with respect to the Lipschitz tensor product of Lipschitz functionals (mappings) and bounded linear functionals (operators), the concept of dualizable (respectively, uniform) Lipschitz cross-norm on X&E is defined. We show that the Lipschitz injective norm ε, the Lipschitz projective norm π and the Lipschitz p-nuclear norm dp (1≤ p≤∞) are uniform dualizable Lipschitz cross-norms on X&E. In fact, ε is the least dualizable Lipschitz cross-norm and π is the greatest Lipschitz cross-norm on X&E. Moreover, dualizable Lipschitz cross-norms α on X&E are characterized by satisfying the relation ε ≤ α ≤ π. In addition, the Lipschitz injective (projective) norm on X&E can be identified with the injective (respectively, projective) tensor norm on the Banach-space tensor product of the Lipschitz-free space over X and E, but this identification does not hold in general for the Lipschitz 2-nuclear norm and the corresponding Banach-space tensor norm. In terms of the space X#&E*, we describe the spaces of Lipschitz compact (finite-rank, approximable) operators from X to E*.