Exploration of indispensable Banach-space valued functions

Yiheng Hu; Gang Lyu; Yuanfeng Jin; Qi Liu; Yiheng Hu; Gang Lyu; Yuanfeng Jin; Qi Liu

doi:10.3934/math.20231416

AIMS Mathematics

2023, Volume 8, Issue 11: 27670-27683. doi: 10.3934/math.20231416

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Research article

Exploration of indispensable Banach-space valued functions

Yiheng Hu ¹,
Gang Lyu ^{1
,
,},
Yuanfeng Jin ^{2
,
,},
Qi Liu ³

1.
School of General Education, Guangzhou College of Technology and Business, Guangzhou 510850, China
2.
Department of Mathematics, Yanbian University, Yanji 133001, China
3.
School of Mathematics and Physics, Anqing Normal University, Anqing 246001, China

Received: 03 June 2023 Revised: 03 September 2023 Accepted: 14 September 2023 Published: 07 October 2023
MSC : 39B52, 39B62, 46B25, 47H10

In the paper, we present a necessary and sufficient condition for the existence of a sequence of measurable functions with finite values, which converge to any given essential bounded function in the topology of essential supremum in a Banach space. A new convergence method is proposed, which allows for the discovery of an essential bounded function $ F $ that is valued in a Banach space. Generally speaking, there exists a Banach-valued essential bounded function $ F $ which $ F_n $ can't converge to $ F $ in the topology of essential supremum for any sequence of finite-valued measurable function.
- essential bounded Banach-valued function,
- sequential compactness,
- uniform convergence
Citation: Yiheng Hu, Gang Lyu, Yuanfeng Jin, Qi Liu. Exploration of indispensable Banach-space valued functions[J]. AIMS Mathematics, 2023, 8(11): 27670-27683. doi: 10.3934/math.20231416

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Abstract

In the paper, we present a necessary and sufficient condition for the existence of a sequence of measurable functions with finite values, which converge to any given essential bounded function in the topology of essential supremum in a Banach space. A new convergence method is proposed, which allows for the discovery of an essential bounded function $ F $ that is valued in a Banach space. Generally speaking, there exists a Banach-valued essential bounded function $ F $ which $ F_n $ can't converge to $ F $ in the topology of essential supremum for any sequence of finite-valued measurable function.

References

[1]	W. Rudin, Real and complex analysis, In: The mathematical gazette, New York: McGraw-Hill, 52 (1974), 412. https://doi.org/doi:10.2307/3611894" target="_blank">10.2307/3611894">https://doi.org/doi:10.2307/3611894
[2]	J. A. Yan, Lecture notes on measure theory (Chinese), 2 Eds., Beijing: Science Press, 2004.
[3]	M. Kreuter, Sobolev spaces of vector-valued functions, Ulam university, 2015.
[4]	G. Q. Zhang, Y. Q. Lin, The lecture of functional analysis, Peking University Press, 1990.
[5]	F. Zheng, C. Cui, A theorem on uniform convergence of operator series, J. Bohai Univ. (Nat. Sci. Ed.), 28 (2007), 338–339. https://doi.org/10.3969/j.issn.1673-0569.2007.04.010 doi: 10.3969/j.issn.1673-0569.2007.04.010
[6]	F. León-Saavedra, M. P. R. Rosa, S. Antonio, Orlicz-Pettis theorem through summability methods, Mathematics, 7 (2019), 895. https://doi.org/10.3390/math7100895 doi: 10.3390/math7100895
[7]	F. León-Saavedra, S. Moreno-Pulido, A. Sala, Orlicz-Pettis type theorems via strong $\rho$-Cesaro convergence, Numer. Funct. Anal. Optim., 40 (2019), 798–802. https://doi.org/10.1080/01630563.2018.1554587 doi: 10.1080/01630563.2018.1554587
[8]	F. León-Saavedra, F. J. Pérez-Fernández, F. P. R. Rosa, A. Sala, Ideal convergence and completeness of a normed space, Mathematics, 7 (2019), 897. https://doi.org/10.3390/math7100897 doi: 10.3390/math7100897
[9]	F. León-Saavedra, S. Moreno-Pulido, A. Sala-Pérez, Completeness of a normed space via strong $\rho$-Cesàro summability, Filomat, 33 (2019), 3013–3022. https://doi.org/10.2298/FIL1910013L doi: 10.2298/FIL1910013L
[10]	J. Diestel, J. J. Uhl, Jr., Vector measures, American mathematical society, 15 (1977). http://doi.org/10.1090/surv/015 doi: 10.1090/surv/015
[11]	D. X. Xia, S. Z. Yan, W. C. Shu, Y. S. Tong, Functional analysis (Second tutorial) (Chinese), Higher education press, 2008.

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