Abstract
Recently, B. Mitiagin and N. Zobin constructed an example of nuclear Fréchet space without basis. The essential modification of their constructions gives the following results. There exists such a nuclear Fréchet space X that for any nuclear Fréchet space Y the space X × Y has no basis (Sections 1 and 2). This fact has a lot of corollaries (Sect. 3); e.g., the space X × C ∞( R 1) having the maximal diametral dimension among nuclear Fréchet spaces nevertheless has no basis. One can also construct (Sect. 4) a nuclear Fréchet space X ̃ without strongly finite-dimensional decomposition (see Definition 0.1). In Section 5 some comments and open questions are given.
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