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2018, vol. 66, iss. 1, pp. 1-8
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Some critical remarks on the paper “A note on the metrizability of tvs-cone metric spaces”
Neke kritičke napomene o radu “Beleška o metrizabilnosti tvp-konusnih metričkih prostora”
aUniversity of Kragujevac, Faculty of Science, Department of Mathematics and Informatics, Serbia bUniversity of Priština - Kosovska Mitrovica, Faculty of Teacher Education, Prizren-Leposavić, Serbia cUniversity of Belgrade, Faculty of Mechanical Engineering, Serbia dUniversity of Palermo, Department of Energy, Information Engineering and Mathematical Models (DEIM), Palermo, Italy
email: suzanasimic@kg.ac.rs, ljiljana.paunovic76@gmail.com, radens@beotel.net, francesca.vetro@unipa.it
Project: Methods of Functional and Harmonic Analysis and PDE with Singularities (MESTD - 174024)
Methods of Numerical and Nonlinear Analysis with Applications (MESTD - 174002)
Keywords: tvs-cone metric space; metrizable; solid; normal; non-normal
Ključne reči: tvp-konusni metrički prostor; metrizabilan; konus sa nepraznom unutrašnjošću; normalan; nije normalan
Abstract
This short and concise note provides a detailed exposition of the approach and results established by (Lin et al, 2015, pp.271-279). We show that the obtained results are not particularly surprising and new. Namely, using an old result due to K. Deimling it is indicated that tvs-cone metric spaces over solid cones are actually cone metric spaces over normal solid cones. Hence, there are only cone metric spaces over normal solid cones or over normal non-solid cones. One question still unanswered is whether an ordered topological vector space with a non-normal non-solid cone exists.
Sažetak
Ova kratka i pregledna beleška daje detaljan izveštaj o pristupu i rezultatima do kojih su došli Šou Lin i grupa autora (Lin et al, 2015, pp.271-279). U članku je pokazano da njihovi rezultati nisu naročito iznenađujući i novi. U stvari, korišćenjem jednog poznatog K. Demlingovog rezultata naznačeno je da su tvp-konusni metrički prostori sa konusima koji imaju nepraznu unutrašnjost zapravo konusni metrički prostori sa normalnim konusima i nepraznim unutrašnjostima. Stoga, postoje samo konusni metrički prostori sa normalnim konusima čija unutrašnjost nije prazna ili sa konusima koji su normalni, ali sa praznim unutrašnjostima. Još uvek se ne zna da li postoji uređen topološki vektorski prostor sa konusom koji nije normalan i čija unutrašnjost nije prazna.
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