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Dense subspaces of quasinormable spaces. (English) Zbl 1109.46004
The class of quasinormable locally convex spaces was introduced by A. Grothendieck; it contains all Schwartz spaces, all normed spaces and many classical spaces of functions and distributions. A. Peris [Ann. Acad. Sci. Fenn., Ser A I, Math. 19, 167–203 (1994; Zbl 0789.46006)] investigated quasinormable spaces in connection with the “problème des topologies” of Grothendieck (on bounded sets in tensor products of Fréchet spaces). The first two authors of the present paper [On distinguished Fréchet spaces. Progress in Functional Analysis, North-Holland Math. Studies 170, 201–214 (1992; Zbl 0785.46003)] produced a reflexive quojection (hence, in particular, quasinormable) containing a dense linear subspace which is not even distinguished (and hence a fortiori not quasinormable).
The present article is devoted to a study which dense subspaces of quasinormable spaces are quasinormable. Here are some of the authors’ results on the positive side: A dense linear subspace of a quasinormable Fréchet space is quasinormable if and only if it is large; i.e., if for every bounded set \(B \subset E\) there is a bounded set \(C \subset F\) with \(B \subset \overline{C}\), where the closure is taken in \(E\). Every dense linear subspace of a separable quasinormable Fréchet space is again quasinormable. Every dense linear subspace of a (DFM)-space is quasinormable. Linear subspaces of finite codimension in a quasinormable space are quasinormable. Every locally dense linear subspace of a regular (LB)-space is quasinormable.
On the other hand, the authors construct two counterexamples: There is a sequence \((X_n)_n\) of Banach spaces such that the direct sum \(\bigoplus_n X_n\) contains a dense linear subspace which is not quasinormable. And there is a sequence \((Z_n)_n\) of normed spaces such that the direct sum \(\bigoplus_n Z_n\) contains a dense linear subspace of countable codimension which is not quasinormable. The corresponding constructions are of Moscatelli type.
MSC:
46A04 Locally convex Fréchet spaces and (DF)-spaces
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
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