The quotient/codimension problems. (English) Zbl 1427.46001
This article is concerned with the following two problems for classes of topological vector spaces:
The famous separable quotient problem of Banach is the first problem in the case where \(\mathfrak{C}\) is the class of all Banach spaces.
The authors investigate these problems for a number of interesting classes of topological vector spaces, including the class of all locally convex spaces, the class of Fréchet spaces, the class of \((LF)\)-spaces and the classes of \(L_p(X)\), \(C_p(X)\) and of \(C_c(X)\) spaces. Answers to these problems for many of the considered classes are provided.
At the end of the article, the available answers for the classes under consideration are collected into a table. In addition, the cases where the question remains open are pointed out.
- 1.
- The separable quotient problem: Given a class of topological vector spaces \(\mathfrak{C}\), does every space \(E\) in this class admit a separable quotient?
As a variant, the question of whether every space in \(\mathfrak{C}\) has a properly separable quotient is also considered. A properly separable space (in the sense of Robertson) is a space containing a proper dense \(\aleph_0\)-dimensional subspace. - 2.
- The finite and countable codimension problem: Given a class \(\mathfrak{C}\) of topological vector spaces and a property \(P_0\), this problem asks whether for every space \(E\) in \(\mathfrak{C}\), having \(P_0\) passes on to all finite/countable codimensional subspaces.
As a variant, the same question is asked for closed or dense hyperplanes instead of finite or countably codimensional subspaces.
The famous separable quotient problem of Banach is the first problem in the case where \(\mathfrak{C}\) is the class of all Banach spaces.
The authors investigate these problems for a number of interesting classes of topological vector spaces, including the class of all locally convex spaces, the class of Fréchet spaces, the class of \((LF)\)-spaces and the classes of \(L_p(X)\), \(C_p(X)\) and of \(C_c(X)\) spaces. Answers to these problems for many of the considered classes are provided.
At the end of the article, the available answers for the classes under consideration are collected into a table. In addition, the cases where the question remains open are pointed out.
Reviewer: Christian Bargetz (Innsbruck)
MSC:
| 46A03 | General theory of locally convex spaces |
| 46A08 | Barrelled spaces, bornological spaces |
| 54C35 | Function spaces in general topology |
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