0,$ $A$ and $B$ be subspaces of $L_p(\mu)$ and $L_p(\nu)$ respectively. Identify $A\otimes B$ with the set $$ \operatorname{span}_{L_p(\mu\times \nu)} \{h:\ h(s,t)=f(s)g(t),\, f\in A, g\in B\} $$ and put $A\otimes_p B= clos_{L_p(\mu\times\nu)} A\otimes B.$ A norm $\alpha$ on $X\otimes Y$ is said to be {\it $p$-regular}\ if for any $(S,\mu),$ $(T,\nu),$ $A$ and $B$ as above and for any operators $U:A\to X$ and $V: B\to Y$ the operator $U\otimes V$ can be extended to a continuous operator (still denoted by $U\otimes V)$ from $A\otimes_p B$ into $X\bar \otimes_\alpha Y$ (the completion with respect to $\alpha)$ and $||U\otimes V||\le ||U||\, ||V||.$ \vskip 0.1cm {\bf Theorem II.1.}\, {\it Let $C,D,X,Y$ be Banach spaces, $p>0,$ $S\in \Pi_p(C,X),$ $T\in \Pi_p(D,Y).$ If $\alpha$ is a $p$-regular norm on $X\otimes Y,$ then $S\otimes T\in \Pi_p(C\widehat{\widehat\otimes}D, X\bar \otimes_\alpha Y)$ and $\pi_p(S\otimes T)\le \pi_p(S)\, \pi_p(T).$ \rm \vskip 0.1cm {\it Proof}. Just apply Pietsch factorization theorem and the definition II.1. Thus, this theorem contains essentially only a modification of the definition; but it has some nice consequences which justify that the authors called it "the theorem". In follows immediately from the theorem, e. g. : \vskip 0.1cm {\bf Corollary II.1.}\, {\it If $T_i\in \operatorname{L}(l_1,l_2),$ $i=1,\dots, n,$ \, then $$T_1\otimes \dots \otimes T_n\in \Pi_1(l_1\widehat{\widehat\otimes}\dots \widehat{\widehat\otimes} l_1, l_2(\Bbb Z_{+}^n))\ \text{ and } \ \pi_1(T_1\otimes \dots \otimes T_n)\le K_G^n\, ||T_1||\dots ||T_n||.$$ \rm \vskip 0.1cm Therefore, the multi-dimensional generalization of Grothendieck's inequality is just a "right" definition (of a tensor norm) plus an application of the classical 1-dimensional inequality of A. Grothendieck. This "right" definition let us to get a lot of other applications. As an example, we obtain also another interesting consequence (with not very difficult proof). Bellow, it is denoted by $I_{\mu,p}$ the identity imbedding from $C(K)$ into $L_p(\mu)$ (where $\mu$ is a finite Radon measure on a compact $K).$ \vskip 0.1cm {\bf Corollary II.2.}\, {\it Let $X$ be a Banach spaces, $T$ is a linear continuous operator from $X$ to $C(K)$ and $p>0.$ Suppose that there is a sequence of finite dimensional projectors $\{P_n\}$ in $X$ with the following properties: $1)$\ $\sup_n ||P_n||<+\infty;$ $2)$\ $(id-P_n)X=(X_1^n\oplus\dots X_{k_n}^n)_p$ for some subspaces $X_1^n, \dots, X^n_{k_n}$ of the space $(id-P_n)X;$ $3)$\ for every $n$ there exists a family $I_1^n, \dots, I^n_{k_n}$ of pairwise disjoint Borel subsets of the compact $K$ such that all the functions from $T(X_j^n)$ vanish out of the set $I^n_{j}$ \, $(j=1, \dots, k_n).$ Let $\mu$ be a measure on $K$ with $\lim_n\, \sup_{1\le j\le k_n} \mu I^n_j=0.$ Then: \ $(a)$\, the operator $I_{\mu,p}T$ is compact for all $p, p>0;$ \ $(b)$\, if $1\le r<2$ and $1

0$ then every $p$-absolutely summing operator from $JT_r$ is compact.\ {\rm 2)\, } If $1\le r <2$ and $1

t \}= A \cup B, $ where each of the sets $ A $ and $ B $ has the lowest
element and any two elements $ a $ and $ b, $ $ a \in A, b \in B $ are
incomparable, and 3) no infinite chain in $ \mathcal T $ has an upper
%border
bound. Elements of $ \mathcal T $ are referred to as
vertexes. The root of the tree is the %pinnacle
vertex of the zero level,
two vertexes (directly following it) are the vertexes of level 1, the next incomparable four vertexes
are called the vertexes of level 2. In general, by induction, we can naturally define the vertexes of the
$ n $-th level (there are exactly $ 2^n $ %pieces
ones).
%%%% next item
If $ s \in \mathcal T $ then the set $ \{t: \, t \ge s \} $ is called a
subtree growing from $s.$ The branch growing
from a vertex $ s $ (of the $ n $-th level) is any totally ordered
set in which $ s $ is the smallest element
and that contains a vertex of the $ m $-th
level for every $ m, m \ge n. $ By subtrees (branches) of the $ n $-th level we understand any
subtrees (branches), growing from the vertexes of a $ n $-th level.
Branches of the zero level are in natural bijective
correspondence with the sequences of zeros and ones,
that is, with the points of the dyadic Cantor set $ \mathcal C. $ In this
correspondence (it is allowed some freedom of speech here), subtrees of the
$n $-th level correspond to $ 2^n$ dyadic intervals of the $ n $-th
rank (which form a partition of $ \mathcal C), $ which we
denote by $ I_1^n, \dots, I^n_{2^n}. $ In what follows, if $ I $ is any
dyadic interval then the corresponding subtree is denoted
by $ \mathcal T_I; $ \, $ F_s^n $ is the branch of $ n $-th level, corresponding to
$ s, s \in \mathcal C. $ Every branch $ F $ can be considered as a
sequence (if numbering its elements in ascending order)
and, therefore, the expression of the form $|| g|_F|| $ has a sense, where $ g $ is a
(finite) function on $ \mathcal T $ and $ || \cdot || $ is a norm in some %!!! finite ?
sequence space.
%Recall
The definition of the classical James's space $J$ can be found in \cite{Ja}. Recall it.
The space $J$ is the completion of the set of all finite sequences with respect to the norm $||\cdot||_J:$
$$
||x||_J:= \sup \bigg\{
\left(
\sum_{j=1}^m |\sum_{k=n_j}^{n_{j+1}-1} x_k|^2\right)^{1/2}:\ 1\le n_1< n_2<\dots