{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:46:27Z","timestamp":1760150787710,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,12,31]],"date-time":"2023-12-31T00:00:00Z","timestamp":1703980800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we discuss some extensions of the Maligranda\u2013Orlicz lemma. It deals with the problem of constructing a norm in a subspace of the space of bounded functions, for which it becomes a normed algebra so that the norm introduced is equivalent to the initial norm of the subspace. This is done by satisfying some inequality between these norms. We show in this paper how this inequality is relevant to the study of operator equations in Banach algebras. In fact, we study how to equip a subspace of the space of bounded functions with a norm equivalent to a given one so that it is a normed algebra. We give a general condition for the construction of such norms, which allows us to easily check whether a space with a given norm is an algebra with a pointwise product and the consequences of such a choice for measures of noncompactness in such spaces. We also study quasi-normed spaces. We introduce a general property of measures of noncompactness that allows the study of quadratic operator equations, prove a fixed-point theorem suitable for such problems, and complete the whole with examples and applications.<\/jats:p>","DOI":"10.3390\/sym16010056","type":"journal-article","created":{"date-parts":[[2023,12,31]],"date-time":"2023-12-31T06:37:12Z","timestamp":1704004632000},"page":"56","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On Normed Algebras and the Generalized Maligranda\u2013Orlicz Lemma"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2255-9449","authenticated-orcid":false,"given":"Mieczys\u0142aw","family":"Cicho\u0144","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Adam Mickiewicz University, 61-614 Pozna\u0144, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9255-231X","authenticated-orcid":false,"given":"Kinga","family":"Cicho\u0144","sequence":"additional","affiliation":[{"name":"Faculty of Automatic Control, Robotics and Electrical Engineering, Poznan University of Technology, 60-965 Pozna\u0144, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/BF01540525","article-title":"On some properties of functions of generalized variation","volume":"104","author":"Maligranda","year":"1987","journal-title":"Monatshefte Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"475","DOI":"10.4171\/zaa\/1394","article-title":"On a class of measures of noncompactness in Banach algebras and their application to nonlinear integral equations","volume":"28","author":"Olszowy","year":"2009","journal-title":"Zeit. 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