Embedding sums into products of Banach spaces

In this paper we study the problem of embedding sums $\oplus_1X$ of Banach spaces into large products $X^J$ of the same or different Banach spaces. The first result in this direction corresponds to Saxon[12], who solved it for $X$ finite-dimensional and $I$ countable. For $X$ a Hilbert space it was solved in [2].\newline In the first part we give solutions to this problem for general Banach spaces, completing in this way [12], [2] and [3]. Then we apply those results to subfactorizations of "diagonal" operators acting between vector valued sequence spaces. As a by-product, a creteria for a Banach space to contain non-separable $l_p$-spaces is given. In the second part we introduce tensor products in order to replace subfactorizations arguments by tensor product statements and show how the preceding tools can serve to explain some pathologies occurring in tensor products of locally convex spaces.\newline Finally, we give examples and counterexamples showing that most of the classical Banach spaces satisfy the countable embedding $(I=\mathbb{N})$