On some dense subspaces of topological linear spaces. (English) Zbl 0552.46002

Lipecki, Z.

Stud. Math. 77, 413-421 (1984).

Let X be a complete metrizable topological linear space with dim X\(=2^{\aleph_ 0}\). The main results of the paper are concerned with products of linear subspaces of X with property (K) (as defind in the preceding abstract). Namely: (a) X contains dense subspaces \(X_ 1\) and \(X_ 2\), both with property (K), such that \(X_ 1\times X_ 2\) does not have property (K); (b) X contains dense subspaces \(X_ 1\) and \(X_ 2\) such that \(X=X_ 1\oplus X_ 2\) and \(X_ 1\times X_ 2\) has property (K).
Remarks: (1) On p. 420, condition (3) should read ”dim \(X_ i^{\alpha}\leq card \alpha\) for \(i=1,2''\). (2) It follows from a recent result of J. Burzyk that if X is separable, then, under CH, there exists a dense proper subspace Y of X such that \(Y\times Y\) has property (K). (3) Related results are contained in a paper by L. Drewnowski and the author (in preparation).

MSC:

46A04	Locally convex Fréchet spaces and (DF)-spaces
54E50	Complete metric spaces
54E52	Baire category, Baire spaces

Keywords:

series; \(\kappa \) -subspace; complete metrizable topological linear space; products of linear subspaces; property (K)

Citations:

Zbl 0552.46001