**Original manuscript:** 2021/03/19
**Manuscript last revised:** 2022/04/25

The theory of integration for functions with values in a topological vector
space is a bit of a messy subject. There are many different choices for the
integral, most distinct from one another in general, and with no compelling
reason to always adopt one definition or another as the "right" one. No
attempt is made here to resolve this question of which integral is the
"right" integral, but a couple of useful properties are proved for the
notion of integrability by seminorm. For this notion of integrability, the
completeness of the space of integrable functions with values in a complete
locally convex space is established. This space is then easily seen to be
isomorphic to the completion of the projective tensor product of the usual
L^{1} space of scalar functions with the vector space. Absolute
continuity for this notion of integrability is also considered. Here it is
shown that the familiar properties of differentiation for absolutely
continuous scalar functions holds in the vector-valued case.

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Last Updated: Fri Mar 15 08:16:31 2024