{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T13:42:48Z","timestamp":1762522968658,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,1,29]],"date-time":"2025-01-29T00:00:00Z","timestamp":1738108800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Education of Ukraine in the framework of the research project \u201cStudy of algebras generated by symmetric polynomial and rational mappings in Banach spaces\u201d","award":["0123U101791"],"award-info":[{"award-number":["0123U101791"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We study the space of continuous linear functionals approximated by weakly symmetric continuous linear functionals on the complex Banach space \u21131 of all absolutely summing complex sequences. We construct the sequence of groups of symmetries on \u21131, obtain the structure of corresponding weakly symmetric continuous linear functionals, find the completion of the space, and construct for it a Schauder basis.<\/jats:p>","DOI":"10.3390\/sym17020206","type":"journal-article","created":{"date-parts":[[2025,1,29]],"date-time":"2025-01-29T04:36:18Z","timestamp":1738125378000},"page":"206","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The Space of Continuous Linear Functionals on \u21131 Approximated by Weakly Symmetric Continuous Linear Functionals"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9055-6341","authenticated-orcid":false,"given":"Taras","family":"Vasylyshyn","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka Str., 76018 Ivano-Frankivsk, Ukraine"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5554-4342","authenticated-orcid":false,"given":"Andriy","family":"Zagorodnyuk","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka Str., 76018 Ivano-Frankivsk, Ukraine"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,29]]},"reference":[{"key":"ref_1","first-page":"51","article-title":"Spectra of algebras of analytic functions on a Banach space","volume":"415","author":"Aron","year":"1991","journal-title":"J. Reine Angew. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1090\/S0002-9947-96-01553-X","article-title":"Regularity and algebras of analytic functions in infinite dimensions","volume":"348","author":"Aron","year":"1996","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s00209-023-03279-6","article-title":"Homomorphisms on algebras of analytic functions on non-symmetrically regular spaces","volume":"304","author":"Carando","year":"2023","journal-title":"Math. Z."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"311","DOI":"10.15330\/cmp.11.2.311-320","article-title":"Spectra of Some Algebras of Entire Functions of Bounded Type, Generated by a Sequence of Polynomials","volume":"11","author":"Halushchak","year":"2019","journal-title":"Carpathian Math. Publ."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"104","DOI":"10.15330\/cmp.15.1.104-119","article-title":"Spectra of Algebras of Analytic Functions, Generated by Sequences of Polynomials on Banach Spaces, and Operations on Spectra","volume":"15","author":"Vasylyshyn","year":"2023","journal-title":"Carpathian Math. Publ."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Baziv, N., and Zagorodnyuk, A. (2023). Analytic Invariants of Semidirect Products of Symmetric Groups on Banach Spaces. Symmetry, 15.","DOI":"10.3390\/sym15122117"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"125977","DOI":"10.1016\/j.jmaa.2021.125977","article-title":"Symmetric analytic functions on the Cartesian power of the complex Banach space of Lebesgue measurable essentially bounded functions on [0,1]","volume":"509","author":"Vasylyshyn","year":"2022","journal-title":"J. Math. Anal. 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Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3835","DOI":"10.1002\/mana.202300415","article-title":"On the spectrum of the algebra of bounded-type symmetric analytic functions on l1","volume":"297","author":"Chernega","year":"2024","journal-title":"Math. Nachr."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"207","DOI":"10.5565\/PUBLMAT6612209","article-title":"Group-invariant separating polynomials on a Banach space","volume":"66","author":"Jung","year":"2022","journal-title":"Publ. Mat."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1674","DOI":"10.55730\/1300-0098.3225","article-title":"Symmetric polynomials in free associative algebras","volume":"46","author":"Boumova","year":"2022","journal-title":"Turk. J. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1006\/jcta.2001.3186","article-title":"MacMahon symmetric functions, the partition lattice, and Young subgroups","volume":"96","author":"Rosas","year":"2001","journal-title":"J. Comb. Theory Ser. A"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Kravtsiv, V. (2024). Block-supersymmetric polynomials on spaces of absolutely convergent series. Symmetry, 16.","DOI":"10.3390\/sym16020179"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"030010","DOI":"10.1063\/5.0115680","article-title":"Generating Elements of the Algebra of Block-Symmetric Polynomials on the Product of Banach Spaces Cs","volume":"2483","author":"Kravtsiv","year":"2022","journal-title":"AIP Conf. Proc."},{"key":"ref_17","unstructured":"Vasylyshyn, T. (2025). Isomorphisms of algebras of symmetric functions on spaces \u2113p. Mat. Stud., 63, to appear."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"066105\/1","DOI":"10.1103\/PhysRevE.64.066105","article-title":"Partition functions in statistical mechanics, symmetric functions, and group representation","volume":"64","author":"Balantekin","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"683","DOI":"10.3390\/quantum5040043","article-title":"Applications of Supersymmetric Polynomials in Statistical Quantum Physics","volume":"5","author":"Chernega","year":"2023","journal-title":"Quantum Rep."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"067701","DOI":"10.1103\/PhysRevE.83.067701","article-title":"Generating functions for canonical systems of fermions","volume":"83","author":"Pain","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1016\/S0378-4371(98)00288-X","article-title":"Investigations on finite ideal quantum gases","volume":"260","author":"Schmidt","year":"1998","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1119\/1.1412643","article-title":"Partition functions and symmetric polynomials","volume":"70","author":"Schmidt","year":"2002","journal-title":"Am. J. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Dineen, S. (1999). Complex Analysis on Infinite Dimensional Spaces, Springer.","DOI":"10.1007\/978-1-4471-0869-6"},{"key":"ref_24","unstructured":"Mujica, J. (1986). Complex Analysis in Banach Spaces, North-Holland."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Lindenstrauss, J., and Tzafriri, L. (1977). Classical Banach Spaces, Vol. I, Sequence Spaces, Springer.","DOI":"10.1007\/978-3-642-66557-8"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/2\/206\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:38:13Z","timestamp":1759919893000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/2\/206"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,29]]},"references-count":25,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["sym17020206"],"URL":"https:\/\/doi.org\/10.3390\/sym17020206","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,1,29]]}}}