Multi-normed spaces, based on non-discrete measures, and their tensor products

It was A. Lambert who discovered a new type of structures, situated, in a sense, between normed spaces and (abstract) operator spaces. His definition was based on the notion of amplification a normed space by means of spaces $\ell_2^n$. Afterwards several mathematicians investigated more general structure, "p-multi-normed space", introduced with the help of spaces $\ell_p^n$; $1\le p\le\infty$. In the present paper we pass from $\ell_p$ to $L_p(X,\mu)$ with an arbitrary measure. This happened to be possible in the frame-work of the non-coordinate ("index-free") approach to the notion of amplification, equivalent in the case of a discrete counting measure to the approach in mentioned articles. Two categories arise. One consists of amplifications by means of an arbitrary normed space, and another one consists of p-convex amplifications by means of $L_p(X,\mu)$. Each of them has its own tensor product of its objects whose existence is proved by a respective explicit construction. As a final result, we show that the "p-convex" tensor product has especially transparent form for the so-called minimal $L_p$-amplifications of $L_q$-spaces, where q is the conjugate of p. Namely, tensoring $L_q(Y,\nu)$ and $L_q(Z,\lambda)$, we get $L_q(Y\times Z,\nu\times\lambda)$.