  In this paper we define Schwartz families in tempered distribution spaces and
prove many their properties. Schwartz families are the analogous of infinite
dimensional matrices of separable Hilbert spaces, but for the Schwartz test
function spaces, having elements (functions) realizable as vectors indexed by
real Euclidean spaces (ordered families of scalars indexed by real Euclidean
spaces). In the paper, indeed, one of the consequences of the principal result
(the characterization of summability for Schwartz families) is that the space
of linear continuous operators among Schwartz test function spaces is linearly
isomorphic with the space of Schwartz families. It should be noticed that this
theorem is possible because of the very good properties of Schwartz test
function spaces and because of the particular structures of the Schwartz
families viewed as generalized matrices; in fact, any family of tempered
distribution, regarded as generalized matrix, has one index belonging to a
Euclidean space and one belonging to a test function space, so that any
Schwartz family is a matrix in the sense of distributions. Another motivation
for the introduction and study of these families is that these are the families
which are summable with respect to every tempered system of coefficients, in
the sense of superpositions. The Schwartz families we present in this paper are
one possible rigorous and simply manageable mathematical model for the infinite
matrices used frequently in Quantum Mechanics.
