Compactness in Groups of Group-Valued Mappings
Abstract
:1. Introduction
2. Preliminaries
- (i)
- (regularity);
- (ii)
- (invariance under closure);
- (iii)
- (semi-additivity).
- (iv)
- implies (monotonicity);
- (v)
- for every one-element set S in L (non-singularity).
- (vi)
- (algebraic semi-additivity);
- (vii)
- for any (invariance under translations).
3. Compactness in Pseudonormed Subgroups of
4. Compactness in Seminormed Subgroups of
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Fréchet, M. Sur les ensembles compacts de fonctions mesurables. Fund. Math. 1927, 9, 25–32. [Google Scholar] [CrossRef] [Green Version]
- Šmulian, V.L. On compact sets in the space of measurable functions. Mat. Sbornik 1944, 15, 343–346. (In Russian) [Google Scholar]
- Appell, J.; De Pascale, E. Some parameters associated with the Hausdorff measure of noncompactness in spaces of measurable functions. Unione Mat. Ital. Boll. 1984, 3, 497–515. [Google Scholar]
- Avallone, A.; Trombetta, G. Measures of noncompactness in the space L0 and a generalization of the Arzelà-Ascoli theorem. Unione Mat. Ital. Boll. 1991, 5, 573–587. [Google Scholar]
- Caponetti, D.; Trombetta, A.; Trombetta, G. Monotonicity and total boundedness in spaces of “measurable” functions. Math. Slovaca 2017, 67, 1497–1508. [Google Scholar] [CrossRef] [Green Version]
- Nussbaum, R.D. A generalization of the Ascoli theorem and an application to functional differential equations. J. Math. Anal. Appl. 1971, 35, 600–610. [Google Scholar] [CrossRef]
- Trombetta, G.; Weber, H. The Hausdorff measure of noncompactness for balls of F-normed linear spaces and for subsets of L0. Boll. Della Unione Mat. Ital. 1986, 5, 213–232. [Google Scholar]
- Tavernise, M.; Trombetta, A.; Trombetta, G. On the lack of equi-measurability for certain sets of Lebesgue-measurable functions. Math. Slovaca 2017, 67, 1595–1601. [Google Scholar] [CrossRef]
- Dunford, N.; Schwartz, J.T. Linear Operators I. In General Theory; Wiley-Interscience Pub., Inc.: New York, NY, USA, 1964. [Google Scholar]
- Ambrosetti, A. Un teorema di esistenza per le equazioni differenziali sugli spazi di Banach. Rend. Sem. Mat. Univ. Padova 1967, 39, 349–361. [Google Scholar]
- Heinz, H.-P. Theorems of Ascoli type involving measures of noncompactness. Nonlinear Anal. 1981, 5, 277–286. [Google Scholar] [CrossRef]
- Bartle, R.G. On compactness in functional analysis. Trans. Am. Math. Soc. 1955, 79, 35–57. [Google Scholar] [CrossRef]
- Ayerbe Toledano, J.M.; Dominguez Benavides, T.; Lopez Acedo, G. Measures of Noncompactness in Metric Fixed Point Theory; Birkhäuser: Basel, Switzerland, 1997. [Google Scholar]
- Banaś, J.; Goebel, K. Measures of noncompactness in Banach Spaces. In Lecture Notes in Pure and Applied Mathematics, Vol. 60; Marcel Dekker, Inc.: New York, NY, USA, 1980. [Google Scholar]
- Sadovskij, B.N. Limit-compact and condensing operators. Russian Math. Surv. 1972, 27, 85–155. [Google Scholar] [CrossRef]
- Caponetti, D.; Trombetta, A.; Trombetta, G. Regular measures of noncompactness and Ascoli-Arzelà-type compactness criteria in spaces of vector-valued functions. arXiv 2022, arXiv:2210.13176. [Google Scholar]
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Caponetti, D.; Trombetta, A.; Trombetta, G. Compactness in Groups of Group-Valued Mappings. Mathematics 2022, 10, 3973. https://doi.org/10.3390/math10213973
Caponetti D, Trombetta A, Trombetta G. Compactness in Groups of Group-Valued Mappings. Mathematics. 2022; 10(21):3973. https://doi.org/10.3390/math10213973
Chicago/Turabian StyleCaponetti, Diana, Alessandro Trombetta, and Giulio Trombetta. 2022. "Compactness in Groups of Group-Valued Mappings" Mathematics 10, no. 21: 3973. https://doi.org/10.3390/math10213973
APA StyleCaponetti, D., Trombetta, A., & Trombetta, G. (2022). Compactness in Groups of Group-Valued Mappings. Mathematics, 10(21), 3973. https://doi.org/10.3390/math10213973