ABSTRACT
A charge is a distribution whose continuity properties resemble those of the distributional divergence of a continuous vector field. We define a locally convex topology https://www.w3.org/1998/Math/MathML"> T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429096679/ee3e4f1b-65eb-4f80-b900-911f329ff879/content/eq7425.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> in the space https://www.w3.org/1998/Math/MathML"> D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429096679/ee3e4f1b-65eb-4f80-b900-911f329ff879/content/eq7426.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> of all test functions so that charges are https://www.w3.org/1998/Math/MathML"> T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429096679/ee3e4f1b-65eb-4f80-b900-911f329ff879/content/eq7427.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> continuous, and that the space https://www.w3.org/1998/Math/MathML"> B V c https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429096679/ee3e4f1b-65eb-4f80-b900-911f329ff879/content/eq7428.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> of all BV functions with compact support is the sequential completion of https://www.w3.org/1998/Math/MathML"> ( D , T ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429096679/ee3e4f1b-65eb-4f80-b900-911f329ff879/content/eq7429.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . In the sense of Mackey-Arens theorem, the space of all charges is in duality with https://www.w3.org/1998/Math/MathML"> B V c https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429096679/ee3e4f1b-65eb-4f80-b900-911f329ff879/content/eq7430.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> — a fact we employ in Chapter 11. Some properties of locally convex spaces are stated without proofs. In such cases, we provide references to standard texts.
