A General Theory of Limits

Abstract

I. As to Notions.-In this paper we investigate a simple general limit of which, as will appear in ? 5, the various classical limits of analysis are actually instances.* The general limit in question is an obvious generalization of the following two limits: 1. An infinite sequence {an} of real or (ordinary) complex numbers an (n = 1, 2, *..) converges to a number a as a limit, in notation: Lnoo an = a,-as clearly defined about a century ago,-in case for every positive number e there exists a positive integer ne of such a nature that for every integer n i ne it is true that in absolute value ana is. at most e. Here the numerical sequence {an } may be considered as a numerically valued function a (an In) of the positive integer n (or on the range [n] of positive integers n), viz., a(n) an for every n. 2. Relative to a general (i.e., any particular) class e[q] of general elements q and the class ( = [s] of all finite classes s of elements q, a numerically valued function a (a (s) Is) on the range S converges to a number a as limit, in notation: L,a(s) = a, in case for every positive number e there exists a class Se of such a nature that for every class s including Se it is true that in absolute value a(s) a is at most e. The limit (2) belongs to General Analysis, i.e., to that doctrine of analysis in which a general class, here {D, plays a fundamental role. The limit (2), introducedt in 1915 by the senior author, plays a central role in his second theory$ of Linear Integral Equations in General Analysis. This general theory has as notable instances: (a) Hilbert's theory of limited quadratic forms in a denumerable infinitude of variables; here the Hilbert space [a] of infinite sequences a (a (n) I n) of real numbers with convergent Ina (n) a (n) plays a central role; (b) an analogous theory in which the corresponding role