We introduce a new class of locally convex spaces E, under the name quasi-(DF)-spaces, containing strictly the class of (DF)-spaces. A locally convex space E is called a quasi-(DF)-space if (i) E admits a fundamental bounded resolution, i.e. an NN-increasing family of bounded sets in E which swallows all bounded set in E, and (ii) E belongs to the class G (in sense of Cascales--Orihuela). The class of quasi-(DF)-spaces is closed under taking subspaces, countable direct sums and countable products. Every regular (LM)-space (particularly, every metrizable locally convex space) and its strong dual are quasi-(DF)-spaces. We prove that Cp(X) has a fundamental bounded resolution iff Cp(X) is a quasi-(DF)-space iff the strong dual of Cp(X) is a quasi-(DF)-space iff X is countable. If X is a metrizable space, then Ck(X) is a quasi-(DF)-space iff X is a Polish σ-compact space. We provide numerous concrete examples which in particular clarify differences between (DF)-spaces and quasi-(DF)-spaces.
|State||Published - 2017|
- 46A03, 46A04, 46E10