Tensor product bases and tensor diagonals

Let X and Y denote Banach spaces with bases $({x_i})$ and $({y_i})$, respectively, and let $X{ \otimes _\varepsilon }Y$ and $X{ \otimes _\pi }Y$ denote the completion in the $\varepsilon$ and $\pi$ crossnorms of the algebraic tensor product $X \otimes Y$. The purpose of this paper is to study the structure of the tensor product spaces $X{ \otimes _\varepsilon }Y$ and $X{ \otimes _\pi }Y$ through a consideration of the properties of the tensor product basis $({x_i} \otimes {y_j})$ for these spaces and the tensor diagonal $({x_i} \otimes {y_i})$ of such bases.