Abstract
We tackle the problem of finding a suitable categorical framework for generalized functions used in mathematical physics for linear and non-linear PDEs. We are looking for a Cartesian closed category which contains both Schwartz distributions and Colombeau generalized functions as natural objects. We study Frolicher spaces, diffeological spaces and functionally generated spaces as frameworks for generalized functions. The latter are similar to Frolicher spaces, but starting from locally defined functionals. Functionally generated spaces strictly lie between Frolicher spaces and diffeological spaces, and they form a complete and cocomplete Cartesian closed category. We deeply study functionally generated spaces (and Frolicher spaces) as a framework for Schwartz distributions, and prove that in the category of diffeological spaces, both the special and the full Colombeau algebras are smooth differential algebras, with a smooth embedding of Schwartz distributions and smooth pointwise evaluations of Colombeau generalized functions.
Highlights
The problem of considering derivatives of locally integrable functions arises frequently in Physics, e.g., in idealized models like in shock mechanics, material points mechanics, charged particles in electrodynamics, gravitational waves in general relativity, etc
The foundation of a rigorous linear theory of generalized functions has been pioneered by Schwartz with a deep use of locally convex vector space theory [25,36], and heuristic multiplications of distributions early appeared, e.g., in quantum electrodynamics, elasticity, elastoplasticity, acoustics and other fields [12,34]
We present some results about linear functionals on the spaces DK ( ) and D( ) which are continuous with respect to the locally convex topology
Summary
The problem of considering (generalized) derivatives of locally integrable functions arises frequently in Physics, e.g., in idealized models like in shock mechanics, material points mechanics, charged particles in electrodynamics, gravitational waves in general relativity, etc. (see, e.g., [12,24,34]). As such, generalized functions find deep applications in solutions of singular differential equations [1,25,33,35] and are naturally framed in (several) theories of infinite-dimensional spaces, from locally convex vector spaces [27] and convenient vector spaces [29] up to diffeological [26,28] and Frölicher spaces [13]. The foundation of a rigorous linear theory of generalized functions has been pioneered by Schwartz with a deep use of locally convex vector space theory [25,36], and heuristic multiplications of distributions early appeared, e.g., in quantum electrodynamics, elasticity, elastoplasticity, acoustics and other fields [12,34]. The (special) Colombeau algebra on is defined as the quotient
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