ANDRZEJ KAMIŃSKI ON AXIOMS OF CONVERGENCE IN LINEAR SPACES

A b s t r a c t. By a (general) convergence in a given lin ea r space X we m ean a m apping G: X N —> 2X, w here N denotes the set of a ll positive integers, and by a zero-convergence in X w e m ean a convergence G0 in X fo r w hich G0(x) ¥ = 0 im plies 0 6 G0(x) fo r each x 6 X N. In the paper, the two operations a re defined: 1° operation C, w hich to each zero-convergence G0 in X assigns some general con vergence G in X, an d 2 ° operation Co, w hich to each general convergence G in X assigns a zero-convergence G0 in X. V arious system s of axiom s fo r general conver gences and zero-convergences a re considered and th e ir connections w ith th e ope rations C and Co are studied. Also m u tua l independence of axiom s is studied. Very often convergences are defined by topology. However there exist im portant convergences which cannot be defined in this way, e.g., type I and type II convergences in the Mikusinski operational calculus (see [9], [2], [3]). These and other examples show the need of development of a gene ral theory, in which convergence in a given space is defined immediately by indicating convergent sequences and their limits and some general conditions (axioms) are supposed (see e.g. [1 0]). Of course, such a convergence can be treated as a function, which to every sequence assigns a set of limits (the empty set if a sequence is divergent; a one-element set if a convergent sequence has a un it limit). In particular, topological convergence (i.e. convergence defined by some topology) can be characterized in term s of conditions mentioned above. For Hausdorff convergences it is done in [7] and [8] and for m ul tivalued convergences (i.e. w ithout the assumption of uniqueness) in the paper [4] (see also [5] and [6]). One can consider convergences in spaces equipped w ith some alge braic structure, e.g., in groups or in linear spaces (see e.g. [11])- In linear R ece ived O ctober 10, 1981