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

Lambert 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 amplifying a normed space by means of the spaces $\ell_2^n$. Later, several mathematicians studied more general structures (`$p$-multi-normed spaces') introduced by means of the spaces $\ell_p^n$, $1\le p\le\infty$. We pass from $\ell_p$ to $L_p(X,\mu)$ with an arbitrary measure. This becomes possible in the framework of the non-coordinate approach to the notion of amplification. In the case of a discrete counting measure, this approach is equivalent to the approach in the papers mentioned.
Two categories arise. One consists of amplifications by means of an arbitrary normed space, and the other consists of $p$-convex amplifications by means of $L_p(X,\mu)$. Each of them has its own tensor product of objects (the existence of each product is proved by a separate explicit construction). As a final result, we show that the `$p$-convex' tensor product has an especially transparent form for the minimal $L_p$-amplifications of $L_q$-spaces, where $q$ is conjugate to $p$. Namely, tensoring $L_q(Y,\nu)$ and $L_q(Z,\lambda)$, we obtain $L_q(Y\times Z,\,\nu\times\lambda)$.