Motivations and origins of Schwartz Linear Algebra in Quantum Mechanics

We propose, by Schwartz Linear Algebra, a significant development of Laurent Schwartz Distribution Theory. The study is conducted by following the (natu- ral and straightforward) way of Weak Duality among topological vector spaces, aiming at the construction of a feasible, rigorous, quite elementary and manageable frame- work for Quantum Mechanics. It turns out that distribution spaces reveal themselves an environment more capable to help in Quantum Mechanics than previously thought. The goal of the research, introduced here, consists in showing that the most natural state-spaces of a quantum system, in the infinite dimensional case, are indeed dis- tribution spaces. Moreover, we show new, natural and straightforward mathematical structures that reproduce very closely several physical objects and many operational procedures required in Quantum Mechanics, systematizing the algorithms and nota- tions of Dirac Calculus, in such a way that it becomes a more versatile and more powerful tool, than we are used to think of.