Abstract
An account of the Schwartz space of rapidly decreasing functions as a topological vector space with additional special structures is presented in a manner that provides all the essential background ideas for some areas of quantum mechanics along with infinite-dimensional distribution theory.
Highlights
In 1950–1951, Laurent Schwartz published a two volumes work Théorie des Distributions [1,2], where he provided a convenient formalism for the theory of distributions
This operator arose in quantum mechanics as the Hamiltonian for a harmonic oscillator and, in that context as well as in white noise analysis, the operator N = T − 1 is called the number operator
The relevant notions concerning topological vector spaces are presented so that the reader need not wade through the many voluminous available works on this subject
Summary
In 1950–1951, Laurent Schwartz published a two volumes work Théorie des Distributions [1,2], where he provided a convenient formalism for the theory of distributions. The purpose of this paper is to present a self-contained account of the main ideas, results, techniques, and proofs that underlie the approach to distribution theory that is central to aspects of quantum mechanics and infinite dimensional analysis. This approach develops the structure of the space of Schwartz test functions by utilizing the operator d2. This operator arose in quantum mechanics as the Hamiltonian for a harmonic oscillator and, in that context as well as in white noise analysis, the operator N = T − 1 is called the number operator. The approach we take has a direct counterpart in the theory of distributions over infinite dimensional spaces [8,9]
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