The strictly singular operators and the strictly cosingular operators are characterized by the manner in which they can be approximated by continuous linear operators of finite-dimensional range. We make use of linear convergence structures to obtain each class as limit points of the operators with finite-dimensional range. The construction of a nonstandard model makes it possible to replace convergence structures by topologies. Our nonstandard models are called nonstandard locally convex

more »

... . A recent paper [4] showed that classes of "compact-like" operators, including compact, weakly compact, and completely continuous operators, can be characterized by "approximation topologies", i.e. topologies on the space of operators for which the given class is precisely the closure (or completion) of the continuous finite-dimensional range operators. In this paper, techniques providing a similar characterization for strictly singular and strictly cosingular operators are given. It is necessary, however, to generalize the concept of an approximation topology to include convergence structures ([5], [7]), after which it is possible to obtain these two classes of operators as limit points of the finite-dimensional range operators. Furthermore, these convergence structures have a characterization in terms of a nonstandard model ([14], [15]) of the operator space, and in this setting it is possible to obtain topological convergence. The paper is presented in three sections. §1 summarizes characterizations of strictly singular and strictly cosingular operators due to Kato [11], Goldberg [8], Pelczyriski [13], and Vladimirskiï [18], and some additional results are obtained utilizing nuclear operators. §2 contains the characterizations of the strictly singular and strictly cosingular operators as the limit points of the set of continuous finitedimensional range operators under certain linear convergence structures. To obtain