In this paper we prove the following result: an inductive limit is regular if and only if for each Mackey null sequence in there exists such that is contained and bounded in . From this we obtain a number of equivalent descriptions of regularity.
We introduce an algebraic notion-stability-for an element of a commutative ring. It is shown that the stable elements of Banach algebras, and of Fréchet algebras, may be simply described. Part of the theory of power-series embeddings, given in [1] and [4], is seen to be of a purely algebraic nature. This approach leads to other natural questions.
The elementary theory of stable inverse-limit sequences, introduced in stable inverse-limit sequences, is used to extend the 'stability lemma' of automatic continuity theory.
The notion of a stable inverse-limit sequence is introduced. It provides a sufficient (and, for sequences of abelian groups, necessary) condition for the preservation of exactness by the inverse-limit functor. Examples of stable sequences are provided through the abstract Mittag-Leffler theorem; the results are applied in the theory of Fréchet algebras.