Article
| Title: | The topology of the space of $\mathcal {HK}$ integrable functions in ${\mathbb R}^n$ (English) |
| Author: | Boonpogkrong, Varayu |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 1 |
| Year: | 2025 |
| Pages: | 85-102 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | It is known that there is no natural Banach norm on the space $\mathcal {HK}$ of $n$-dimensional Henstock-Kurzweil integrable functions on $[a,b]$. We show that the $\mathcal {HK}$ space is the uncountable union of Fréchet spaces $\mathcal {HK}(X)$. On each $\mathcal {HK}(X)$ space, an $F$-norm $\|{\cdot }\|^X$ is defined. A $\|{\cdot }\|^X$-convergent sequence is equivalent to a control-convergent sequence. Furthermore, an $F$-norm is also defined for a $\|{\cdot }\|^X$-continuous linear operator. Hence, many important results in functional analysis hold for the $\mathcal {HK}(X)$ space. It is well-known that every control-convergent sequence in the $\mathcal {HK}$ space always belongs to a $\mathcal {HK}(X)$ space. Hence, results in functional analysis can be applied to the $\mathcal {HK}$ space. Compact linear operators and the existence of solutions to integral equations are also given. The results for the one-dimensional case have been discussed in V. Boonpogkrong (2022). Proofs of many results for the $n$-dimensional and the one-dimensional cases are similar. (English) |
| Keyword: | compact operator |
| Keyword: | integral equation |
| Keyword: | controlled convergence |
| Keyword: | Henstock-Kurzweil integral in $\mathbb{R} ^n$ |
| MSC: | 26A39 |
| MSC: | 26A42 |
| DOI: | 10.21136/CMJ.2023.0313-22 |
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| Date available: | 2025-03-11T15:57:12Z |
| Last updated: | 2025-03-19 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152898 |
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