Some topics of the theory of linear topological spaces (completeness, non-trivial expansions of zero and their generators)

The connection between the existence of non-trivial expansions of zero with respect to some system of elements in a separated linear topological space and the completeness of such systems is investigated. The notion of a generator of non-trivial expansions of zero with respect to the system $X$ is introduced and some sufficient conditions under which the element $x$ generates non-trivial expansions of zero with respect to certain systems of elements are found.