Abstract:
We present an original work laying the foundations for a general theory of tensor products of a pointed metric space X with a Banach space E. Earlier papers in this direction by authors as J. D. Farmer and W. B. Johnson in Lipschitz p-summing operators and D. Cheng and B. Zheng in Lipschitz p-integral operators and Lipschitz p-nuclear operators fully justifies such an investigation. Our purpose is to develop a theory of the Lipschitz tensor product of X and E by following the original ideas of R. Schatten to construct the algebraic tensor product of two Banach spaces in his famous book A theory of cross-spaces. We are also motivated by the problem of researching the spaces of Lipschitz compact (finite-rank, approximable) operators from X to E*. It is shown that, in the study of tensor norms in a Lipschitz setting, a surprisingly large amount of ground can be covered by the analogy with the linear case. We believe that results of this type have strong potential for further applications.
Acknowledgements. Other co-authors of this work are J. A. Chávez-Domínguez from the University of Texas at Austin (Texas, U.S.A), A. Jiménez-Vargas from the University of Almería (Spain) and Moisés Villegas-Vallecillos from the University of Cádiz (Spain). |