Henstock Integration in a Hilbertian Countably Normed Space with Nuclearity

Abstract

Henstock integration of real-valued functions has been extended to functions with values in normed spaces. Cao, who considered Banach-valued functions, showed that Henstock’s lemma—which plays an important role in the real-valued case—does not always hold in infinite-dimensional Banach spaces. Nakanishi showed that Henstock’s lemma holds in a ranked space called a Hilbertian CN-space with nuclearity.
In this paper, we revisit this space, define r-differentiability of a function with values in an r-separated ranked space, and present results concerning the primitives of Henstock integrable functions with values in this space. Furthermore, we provide a descriptive definition of the Henstock integral as defined by Nakanishi.

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