Riesz-Schwartz distributions and vector-valued integration in closed categories

**Abstract:**

We study compactly-supported Radon measures, Schwartz distributions,
and their generalizations in an abstract setting through monoidal and
enriched category theory, developing a theory of vector-valued
integration with respect to these scalar functionals. Building on
ideas of Lawvere and Kock, we work in a given closed category, whose
objects we call spaces, and we study *R*-module objects therein (or
algebras of a commutative monad), which we call linear spaces. For
each space, we consider the space of scalar functionals on the
associated space of scalar-valued maps. For appropriate choices of
category, the latter functionals specialize to the classical notions
of compactly-supported Radon measure and Schwartz distribution. We
study three axiomatic approaches to vector integration, including an
abstract Pettis-type integral, showing that all are encompassed by an
axiomatization *via* monads and that all coincide in suitable
contexts.
We study the relation of this vector integration to novel relative
notions of completeness in linear spaces. One such notion of
completeness, defined *via* enriched orthogonality,
characterizes
exactly those separated linear spaces that support the vector
integral. We prove Fubini-type theorems for the vector integral.
Further, we develop aspects of several supporting topics in category
theory, including enriched factorization systems, enriched associated
idempotent monads and adjoint factorization, symmetric monoidal
adjunctions and commutative monads, and enriched commutative algebraic
theories