The topology of the space of $\mathrm{\mathscr{H}𝒦}$ integrable functions in ${ℝ}^n$

Boonpogkrong, Varayu. "The topology of the space of $\mathcal {HK}$ integrable functions in ${\mathbb {R}}^n$." Czechoslovak Mathematical Journal (2025): 85-102. <http://eudml.org/doc/299915>.

@article{Boonpogkrong2025,
abstract = {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 $\Vert \{\cdot \}\Vert ^X$ is defined. A $\Vert \{\cdot \}\Vert ^X$-convergent sequence is equivalent to a control-convergent sequence. Furthermore, an $F$-norm is also defined for a $\Vert \{\cdot \}\Vert ^X$-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.},
author = {Boonpogkrong, Varayu},
journal = {Czechoslovak Mathematical Journal},
keywords = {compact operator; integral equation; controlled convergence; Henstock-Kurzweil integral in $\mathbb \{R\} ^n$},
language = {eng},
number = {1},
pages = {85-102},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The topology of the space of $\mathcal \{HK\}$ integrable functions in $\{\mathbb \{R\}\}^n$},
url = {http://eudml.org/doc/299915},
year = {2025},
}

TY - JOUR
AU - Boonpogkrong, Varayu
TI - The topology of the space of $\mathcal {HK}$ integrable functions in ${\mathbb {R}}^n$
JO - Czechoslovak Mathematical Journal
PY - 2025
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 85
EP - 102
AB - 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 $\Vert {\cdot }\Vert ^X$ is defined. A $\Vert {\cdot }\Vert ^X$-convergent sequence is equivalent to a control-convergent sequence. Furthermore, an $F$-norm is also defined for a $\Vert {\cdot }\Vert ^X$-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.
LA - eng
KW - compact operator; integral equation; controlled convergence; Henstock-Kurzweil integral in $\mathbb {R} ^n$
UR - http://eudml.org/doc/299915
ER -