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Solutions of D. A. Raikov’s problems in the theory of topological vector spaces. (English) Zbl 1190.46002
The Russian mathematician D. A. Raikov posed in the 1960’s several problems mainly concerned with the properties of the spaces \(\mathcal D\) and \(\mathcal D'\). They were closely related to classical problems of J. Dieudonné and L. Schwartz which were solved by A. Grothendieck. The problems of Raikov mostly dealt with various completeness properties of the mentioned spaces. Most of these problems were solved by the author of the present article and these solutions are reproduced here. The main method is the construction of nonclosed sequentially closed subsets of locally convex spaces. It is remarked that some of the mentioned problems of Dieudonné and Schwartz can also be solved by this method, i.e., in a different way from that of Grothendieck.
MSC:
46A03 General theory of locally convex spaces
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46A30 Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness)
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
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