K-convergence

P. Antosik and J. Mikusinski have introduced a stronger form of sequential convergence (cf. also Sequential space), called -convergence, which has found applications in a number of areas of analysis. If is a sequence in a Hausdorff Abelian topological group , then is --convergent if every subsequence of has a further subsequence such that the subseries is -convergent in . Any --convergent sequence is obviously -null ( convergent to ), but the converse does not hold in general although it does hold in a complete metric linear space. A space in which null sequences are -convergent is called a -space; a complete metric linear space is a -space, but there are examples of normal -spaces that are not complete [a2].

One of the principal uses of the notion of -convergence is in formulating versions of some of the classical results of functional analysis without imposing completeness or barrelledness assumptions. A subset of a topological vector space is bounded if for every sequence and every null scalar sequence , the sequence is a null sequence in . A stronger form of boundedness is obtained by replacing the condition that be a null sequence by the stronger requirement that is -convergent; sets satisfying this stronger condition are called -bounded. In general, bounded sets are not -bounded; spaces for which the bounded sets are -bounded are called -spaces. Thus, -spaces are -spaces but there are examples of -spaces that are not -spaces. Using the notion of -boundedness, a version of the uniform boundedness principle (cf. Uniform boundedness) can be formulated which requires no completeness or barrelledness assumptions on the domain space of the operators. If and are topological vector spaces and is a family of continuous linear operators from into which is pointwise bounded on , then is uniformly bounded on -bounded subsets of . If is a complete metric linear space, this statement generalizes the classical uniform boundedness principle for -spaces since in this case is equicontinuous (cf. also Equicontinuity). Similar versions of the Banach–Steinhaus theorem and the Mazur–Orlicz theorem on the joint continuity of separately continuous bilinear operators are possible. See [a1] or [a3] for these and further results.

References

[a1]	P. Antosik, C. Swartz, "Matrix methods in analysis" , Lecture Notes Math. , 1113 , Springer (1985)
[a2]	C. Klis, "An example of a non-complete (K) space" Bull. Acad. Polon. Sci. , 26 (1978) pp. 415–420
[a3]	C. Swartz, "Infinite matrices and the gliding hump" , World Sci. (1996)