Summary: Every countable-codimensional subspace of a non-normable Fréchet space has a quotient isomorphic to \(\omega\), the Fréchet space of all sequences, answering the question by Perez Carreras and Bonet who proved the finite-codimensional case. In [Stud. Math. 6, 139-148 (1936; Zbl 0015.35603)] M. Eidelheit proved the 0-codimensional case. In general,
(*) If \(E\) and \(F\) are locally convex spaces at least one of which is idempotent (isomorphic to its square) and some quotient of \(E\) is isomorphic to \(F\), then some quotient of \(E_ 0\) is also isomorphic to \(F\), where \(E_ 0\) is any finite-codimensional subspace of \(E\).
This, with the Eidelheit result, immediately gives the Perez Carreras/Bonet result, and also the fact that every separable Banach space is isomorphic to a quotient of any finite-codimensional subspace of \(\ell_ 1\) (here the 0-codimensional case is a well-known Banach-Mazur theorem): Now (*) is used to show that the familiar idempotent sequence spaces \(\omega\), \(c_ 0\), \(\ell_ p\) \((1\leq p\leq\infty)\) and \((s)\) are all regenerative. Applying [the author, this Bull. 39, No. 3/4, 161- 166 (1991; Zbl 0785.46004), Theorem 3] for the cases of \(\ell_ 1\) and \(\omega\) extends the Banach-Mazur result (our title theorem) and the Eidelheit/Perez Carreras/Bonet result, respectively.