Lipschitz Tensor Product

Inspired by ideas of R. Schatten in his celebrated monograph [23] on a theory of cross-spaces, we introduce the notion of a Lipschitz tensor product $Xboxtimes E$ of a pointed metric space $X$ and a Banach space $E$ as a certain linear subspace of the algebraic dual of $text{Lip}0(X,E^*)$. We prove that $leftlangle text{Lip}0(X,E^*),Xboxtimes Erightrangle$ forms a dual pair.We prove that $Xboxtimes E$ is linearly isomorphic to the linear space of all finite-rank continuous linear operators from $(X^#,tau_p)$ into $E$, where $X^#$ denotes the space $text{Lip}0(X,mathbb{K})$ and $tau_p$ 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 $Xboxtimes E$. To ensure the good behavior of a norm on $Xboxtimes E$ 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 $Xboxtimes E$ is defined. We show that the Lipschitz injective norm $varepsilon$, the Lipschitz projective norm $pi$ and the Lipschitz $p$-nuclear norm $d_p$ $(1leq pleqinfty)$ are uniform dualizable Lipschitz cross-norms on $Xboxtimes E$. In fact, $varepsilon$ is the least dualizable Lipschitz cross-norm and $pi$ is the greatest Lipschitz cross-norm on $Xboxtimes E$. Moreover, dualizable Lipschitz cross-norms $alpha$ on $Xboxtimes E$ are characterized by satisfying the relation $varepsilonleqalphaleqpi$.In addition, the Lipschitz injective (projective) norm on $Xboxtimes E$ 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 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 $E^*$.