Compact operators and integral equations in the ℋ𝒦 space

Varayu Boonpogkrong

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 1, page 239-257
  • ISSN: 0011-4642

Abstract

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The space ℋ𝒦 of Henstock-Kurzweil integrable functions on [ a , b ] is the uncountable union of Fréchet spaces ℋ𝒦 ( X ) . In this paper, on each Fréchet space ℋ𝒦 ( X ) , an F -norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the ℋ𝒦 ( X ) space. It is known that every control-convergent sequence in the ℋ𝒦 space always belongs to a ℋ𝒦 ( X ) space for some X . We illustrate how to apply results for Fréchet spaces ℋ𝒦 ( X ) to control-convergent sequences in the ℋ𝒦 space. Examples of compact linear operators are given. Existence of solutions to linear and Hammerstein integral equations is proved.

How to cite

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Boonpogkrong, Varayu. "Compact operators and integral equations in the $\mathcal {HK}$ space." Czechoslovak Mathematical Journal 72.1 (2022): 239-257. <http://eudml.org/doc/297803>.

@article{Boonpogkrong2022,
abstract = {The space $\mathcal \{HK\}$ of Henstock-Kurzweil integrable functions on $[a,b]$ is the uncountable union of Fréchet spaces $\mathcal \{HK\}(X)$. In this paper, on each Fréchet space $\mathcal \{HK\}(X)$, an $F$-norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $\mathcal \{HK\}(X)$ space. It is known that every control-convergent sequence in the $\mathcal \{HK\}$ space always belongs to a $\mathcal \{HK\}(X)$ space for some $X$. We illustrate how to apply results for Fréchet spaces $\mathcal \{HK\}(X)$ to control-convergent sequences in the $\mathcal \{HK\}$ space. Examples of compact linear operators are given. Existence of solutions to linear and Hammerstein integral equations is proved.},
author = {Boonpogkrong, Varayu},
journal = {Czechoslovak Mathematical Journal},
keywords = {compact operator; integral equation; controlled convergence; Henstock-Kurzweil integral},
language = {eng},
number = {1},
pages = {239-257},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Compact operators and integral equations in the $\mathcal \{HK\}$ space},
url = {http://eudml.org/doc/297803},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Boonpogkrong, Varayu
TI - Compact operators and integral equations in the $\mathcal {HK}$ space
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 1
SP - 239
EP - 257
AB - The space $\mathcal {HK}$ of Henstock-Kurzweil integrable functions on $[a,b]$ is the uncountable union of Fréchet spaces $\mathcal {HK}(X)$. In this paper, on each Fréchet space $\mathcal {HK}(X)$, an $F$-norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $\mathcal {HK}(X)$ space. It is known that every control-convergent sequence in the $\mathcal {HK}$ space always belongs to a $\mathcal {HK}(X)$ space for some $X$. We illustrate how to apply results for Fréchet spaces $\mathcal {HK}(X)$ to control-convergent sequences in the $\mathcal {HK}$ space. Examples of compact linear operators are given. Existence of solutions to linear and Hammerstein integral equations is proved.
LA - eng
KW - compact operator; integral equation; controlled convergence; Henstock-Kurzweil integral
UR - http://eudml.org/doc/297803
ER -

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