Corrections to: “On sequential convergence”

Abstract

In my paper [2], Theorems 8.2 and 8.3 are false. I am indebted to Dennis Sentilles who sent me a counterexample to Theorem 8.2. The wrong statements concern LS-spaces. These are linear spaces S on which there is a metric p and a function f such that xn -x in S if and only if both p(xn, x) O-0 and f(xn) remains bounded. Without restating the further conditions imposed on p and f, we single out for the present two tractable subclasses of the class of LS-spaces: (I) LF-spaces, i.e. strict inductive limits of spaces ([1], [2, Theorem 7.7]). (II) If B is a separable Banach space, and S is its dual B* with weak-star convergence of sequences, then S is an LS-space [2, Theorem 7.1]. (Actually the same is true with Frechet in place of Banach: see [1, Theoreme 5, p. 84].) In case (I), Theorems 8.2 and 8.3 are true. The Banach-Steinhaus theorem for LF-spaces is well known [1, Theoreme 2, p. 73]. But in case (II), if B is infinitedimensional, the elements of its unit ball give pointwise bounded continuous linear forms on S which are not equicontinuous for the LS-topology. The proof of Theorem 8.2 in [2] errs in assuming that multiplication by a positive scalar is an open mapping in the relative topology of a (convex, symmetric, closed, metrizable) set. The proof of Theorem 8.3 is invalid since it rests on Theorem 8.2. Disproving the statement of 8.3 requires some further work. Here is one counterexample. Let ' be the space of continuous real functions on [0, 1] with supremum norm. Then its dual ' * is the space of finite signed measures on [0, 1]. W* with weak* sequential convergence is an LS-space. It has a countable dense set and is complete as an LS-space. The LS-topology on W* is the topology of uniform convergence on sequences {f,} in ' with f,11 ---> 0 [2, around Theorem 7.8]. By the MackeyArens theorem, the dual space of (V*, Y) is ' (see also [1, Theoreme 6, p. 85]). A sequence in W is weakly convergent if and only if it is uniformly bounded and converges pointwise. Thus the following fact contradicts Theorem 8.3: