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In the book the author investigates spaces of sequences with components belonging to an vector space.These spaces represent a generalization of the known Musielak-Orlicz sequence spaces and are generated by so-called pregenfunctions and genfunctions M.A part of the book is devoted to a superposition operator in considered spaces.The author considers in the spaces in question two equivalent semimetrics (which under some conditions are metrics)and F-norms (if M is a genfunction).Some properties of the considered spaces and their subsets are established.Criteria of the action,boundedness and continuity of the superposition operator are found.The book represents an interest for specialists in functional analysis and is comprehensible to the students-mathematicians of older years.Besides,it is possible to find here themes for diploma theses.