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We develop aspects of functional analysis in an abstract axiomatic setting,through monoidal and enriched category theory. We work in a given closedcategory, whose objects we call spaces, and we study R-module objects therein(or algebras of a commutative monad), which we call linear spaces. Building onideas of Lawvere and Kock, we study functionals on the space of scalar-valuedmaps, including compactly-supported Radon measures and Schwartz distributions.We develop an abstract theory of vector-valued integration with respect tothese scalar functionals and their relatives. We study three axiomaticapproaches to vector integration, including an abstract Pettis-type integral,showing that all are encompassed by an axiomatization via monads and that allcoincide in suitable contexts. We study the relation of this vector integrationto relative notions of completeness in linear spaces. One such notion ofcompleteness, defined via enriched orthogonality, determines a symmetricmonoidal closed reflective subcategory consisting of exactly those separatedlinear spaces that support the vector integral. We prove Fubini-type theoremsfor the vector integral. Further, we develop aspects of several supportingtopics in category theory, including enriched orthogonality and factorizationsystems, enriched associated idempotent monads and adjoint factorization,symmetric monoidal adjunctions and commutative monads, and enriched commutativealgebraic theories.

Riesz-Schwartz extensive quantities and vector-valued integration in closed categories