Following Grothendieck, a Fréchet space E is said to be totally reflexive if every separated quotient of E is reflexive. The author proves the beautiful theorem that a Fréchet space E is totally reflexive if and only if E is the projective limit of a sequence of reflexive Banach spaces (or, equivalently, if E is isomorphic to a closed subspace of a countable product of reflexive Banach spaces). From this characterization it is easy to deduce that the product \(E_ 1\times E_ 2\) of two totally reflexive Fréchet spaces \(E_ 1\) and \(E_ 2\) must itself be totally reflexive, which solves a problem in Grothendieck’s article on (F)- and (DF)-spaces.
The proof of (the difficult direction of the) main theorem involves (the Davis-Figiel-Johnson-Pełczyński factorization theorem for weakly compact operators and) the construction of a Schauder basis of a certain type for a suitable quotient E/F of any separable Fréchet space E which admits a decreasing fundamental sequence \((V_ n)_ n\) of closed absolutely convex 0-neighborhoods such that \(V^ 0_ 1\) is not weakly compact in \(E'_{V^ 0_ m}\) for arbitrary \(m\in {\mathbb{N}}.\)
This construction also shows that for each separable Fréchet space E which is not Schwartz, there is a closed subspace F such that E/F has a bounded Schauder basis.