Investigation of some problems of functional analysis in linear space with limit operation of a sequence

This work is based on the theory of spaces with limit operation of a sequence constructed by the author. In a vector space $X$ partially ordered set $L$ of all linear limit operations of a sequence is considered, each of them generates the same system of bounded subsets in $X$ as the given linear limit operation of a sequence. It is proved that $L$ contains the smallest element, each nonempty subset from $L$ has the greatest lower bound and the perfect ordered subset has tne least upper bound that is $L$ contains maximal elements. The characteristics of the smallest element and the maximal elements of $L$ are obtained. For linear spaces with limit operation of a sequence statements about a neighborhood of zero, convex sets and differentiable mappings as well as statements that generalize the classical Banach–Steinhaus theorem and the theorem on open mapping are proved. In particular we obtain results reinforcing some known versions of Banach–Steinhaus theorem for topological vector spaces.