A Counterexample to a Question of Valdivia On Fréchet Spaces not Containing l₁

Abstract

In this note we construct a non-separable Frtchet space E with the following properties:(i) No subspace of E is isomorphic to el. (ii) There is a linear form on E' which is bounded on bounded sets but is not continuous (i.e., the Mackey topology p(E, E ) is not bornological). This is a negative answer to a question posed by VALDIVIA.VALDIVIA [ll] proved the following result: If E is a separable Frechet space that does not contain d,, then the topological dual of E, E , endowed with the Mackey topology p ( E ' , E ) is bornological (or equivalently, every locally bounded linear form on E' is continuous). In the same paper VALDIVIA asks whether the hypothesis that E is separable can be removed or not. In this note, under the Continuum Hypothesis, we answer in the negative. In fact, we introduce weighted Banach spaces of James type, defined on a suitable uncountable index set, to construct a counterexample to the question of VALDIVIA.We first give some definitions. The classical quasireflexive James space J is defined by where the sup is taken over all increasing sequences of integers 0 = Given a map h : IN + [l, + a), we define a weighted James space bywhere the sup is taken as above. To handle J(h) it is convenient to use also the increasing map 6: IN --* [l, + 00) defined by 6(n) := sup (h(i); i I n} .Note that l/ej/I2 = Gj, for every j E IN, where en is the n-th unit vector basis. (From now on, given any index set I and given i E I we denote by ei the generalized sequence that takes the value 1 on i and 0 elsewhere.) 5 Math. Nachr., Bd. 174