Compactness in Banach space theory - selected problems

We list a number of problems in several topics related to compactness in nonseparable Banach spaces. Namely, about the Hilbertian ball in its weak topology, spaces of continuous functions on Eberlein compacta, WCG Banach spaces, Valdivia compacta and Radon-Nikod\'{y}m compacta

On realcompact topological vector spaces

[EN] This survey paper collects some of older and quite new concepts and results from descriptive set topology applied to study certain infinite-dimensional topological vector spaces appearing in Functional Analysis, including Frechet spaces, (L F)-spaces, and their duals, (D F)-spaces and spaces of continuous real-valued functions C(X) on a completely regular Hausdorff space X. Especially (L F)-spaces and their duals arise in many fields of Functional Analysis and its applications, for example in Distributions Theory, Differential Equations and Complex Analysis. The concept of a realcompact topological space, although originally introduced and studied in General Topology, has been also studied because of very concrete applications in Linear Functional Analysis.The research for the first named author was (partially) supported by Ministry of Science and Higher Education, Poland, Grant no. 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Validation of Heat and Mass Transfer Model in the Protective Clothing

W artykule przedstawiono zaawansowany model matematyczny i numeryczny transportu ciepła i masy w wielowarstwowych ubraniach ochronnych, które były poddane działaniu wysokiej temperatury otoczenia lub dużych radiacyjnych strumieni ciepła emitowanych przez gorące ciała. Zaproponowany model uwzględniał przewodzenie ciepła i promieniowanie cieplne w warstwach tkaniny o spektralnych właściwościach optycznych i w szczelinach powietrznych oraz transport energii związany z dyfuzją wilgoci przez ubranie ochronne oraz z procesami sorpcji i desorpcji wody w włóknach tkaniny. Dodatkowo do modelu włączono złożone warunki bilansu energii i masy oraz warunki optyczne na granicach warstw tkaniny. Do rozwiązania równań modelowych opracowano autorski iteracyjny algorytm numeryczny, który bazował na metodzie objętości kontrolnych. Następnie przeprowadzono analizę walidacyjną zaproponowanego modelu obliczeniowego poprzez porównanie otrzymanych wyników z wynikami pomiarów eksperymentalnych dla wybranego pakietu ubrania ochronnego ogrzewanego przez krótki czas radiacyjnym strumieniem ciepła emitowanym przez promiennik podczerwieni, a następnie chłodzonego w otoczeniu. Otrzymano dobrą zgodność czasowych przebiegów temperatury, co potwierdziło wiarygodność zaproponowanego modelu.An advanced mathematical and numerical model of heat and mass transfer in the multi-layer protective clothing, which was exposed to either high temperature environment or to high incident radiative heat flux emitted by hot objects was presented in this paper. The developed model accounted for heat conduction and thermal radiation in a non -grey layers of the protective garment. Additionally, heat transport associated with water vapour diffusion through the protective clothing and with sorption and desorption of liquid water in the fabric fibres were included. Complex energy and mass balances as well as optical conditions at the external and internal interfaces between clothing layers were formulated and incorporated into the model. A novel iterative numerical algorithm which was based on the Finite Volume Method was developed to solve the system of governing equations. Finally, the validation analysis of the model was carried out for selected multi-layer clothing which was exposed for a short time to radiative heat flux emitted by an infrared emitter and then cooled down in the surroundings. The simulated and experimentally measured time variations of temperatures were in good agreement, therefore the accuracy of the proposed model was validated