On the Stability of Barrelled Topologies II

Abstract

1. Introduction. If £ is a Hausdorff barrelled space, which does not already have its finest locally convex topology, then the continuous dual E' may be enlarged within the algebraic dual E*. Robertson and Yeomans [10] have recently investigated whether E can retain the barrelled property under such enlargements. Whereas finite-dimensional enlargements of the dual preserve barrelledness, they have shown that this is not always so for countable-dimensional enlargements E' + M. In fact, if E contains an infinitedimensional bounded set, there always exists a countable-dimensional M for which the Mackey topology T(JB, E' + M) is not barrelled [10, Theorem 2].Here, in Theorem 1 we give a construction which shows that for a large class of barrelled spaces E, there exists a countable-dimensional M with ET\M = {0}, for which T(E, E' + M) does remain barrelled. Combining this with Theorem 2 of [10], we then see that if E contains a sufficiently large bounded set, there exist two countable-dimensional subspaces M 1( M 2 of E* such that E'nM 1 = {0} = B ' n M 2 and T(E, E' + M X ) is barrelled, while T(JE, E' + M 2 ) is not. Our Theorem 1 applies in several important special cases, for example Frechet spaces, barrelled normed spaces and spaces of continuous functions.The method of construction of M imposes a condition on the bounded sets of E' + M, namely that for each such set there is a finite-dimensional enlargement of E' which contains it. In the last section we show this condition is necessary whenever r{E, E' + M) is barrelled, and therefore fundamental to any approach to finding such an M.Generally we adopt the topological vector space notation of [9]. We use K to denote the scalars R or C, and c for the cardinality of the continuum. The term countable is reserved for the infinite case only.We are grateful to Dr. W. J. Robertson for suggesting this problem to us and for her constant encouragement.

Stability of barrelledness and related concepts in topological vector spaces

“…It therefore seems reasonable to consider related questions and procedures in topological vector spaces which are not necessarily locally convex or separated. We are able to extend the main stability results of [5] and [7] to barrelled topological vector spaces and we have some partial results on the preservation of ultrabarrelledness and hyperbarrelledness, which are rather more important concepts in the non-locally convex cases.…”

“…The duality arguments of [5] and [7] are reformulated in terms of the results of Section 2. ) of co to a linear functional on E by putting it equal to zero on L. The set M of all these extensions is a countable dimensional subspace of E* with £ ' n M = {0}.…”

“…It follows that C = BnG is a barrel in G(^[AT|) and so, G . As in [7] we consider an algebraic isomorphism t:G-*co. Since = sup{|<x,x'>|:xe/l}, we deduce in the same way that £' (</>) nG' taking the polar in G' + N, we have that C° is pointwise bounded on G. Using the same normalisation technique as in [7], we see that C°zG' + Q where Q is a finite dimensional subspace of N. Let P be the subspace of M obtained by extending the elements of Q in the prescribed manner.…”

Countable enlargements of norm topologies and the quasidistinguished property

“…There has been some interest in the question of when barrelledness is preserved under countable enlargements (see [4], [5], [6], [8], [9]). In this note we are concerned with the preservation of the quasidistinguished property for normed spaces under countable enlargements; this was posed as on open question by B. Tsirulnikov in [7].…”

Barrelled Countable Enlargements and the Dominating Cardinal

“…The question of whether every barrelled space E with w x E Ј/E * has a BCE remains open. Partial answers in 1,4,5,12 were w x extended in this paper's companion 10 ; related is the barrelledly fit w x question 8 . We give a fairly general positive answer in Section 3 via a Ž .…”