  In this note we define summable families in tempered distribution spaces and
we state some their properties and characterizations. Summable families are the
analogous of summable sequences in separable Hilbert spaces, but in tempered
distribution spaces, having elements (functional) realizable as generalized
vectors indexed by real Euclidean spaces (not pointwise defined ordered
families of scalars indexed by real Euclidean spaces in the sense of
distributions). Any family we introduce here is summable with respect to every
tempered system of coefficients belonging to a certain normal space of
distributions, in the sense of superpositions. The summable families we present
in this note are one possible rigorous and simply manageable mathematical model
for the infinite families of vector-states appearing in the formulation of the
continuous version of the celebrated Principle of Superpositions in Quantum
Mechanics.
