Bartle and Graves theorem for approximately surjective mappings with values in \(b\)-spaces. (English) Zbl 1095.46002

The authors develop the theory of bornological vector spaces, which differ from the usual topological vector spaces by the fact that instead of a topology (a system of open subsets), a system of bounded sets is postulated. In their previous papers [see, for example, B. Aqzzouz, C.R.Acad.Sci., Paris, Sér. I, Math.333, No. 10, 925–930 (2001; Zbl 1022.46005)], the authors gave several bornological analogs of the Bartle–Graves theorem, a generalization of the open mapping principle. Here they give another version, for the class of approximately surjective mappings. For Banach spaces they are just those with dense ranges. There is also such a description for Fréchet spaces. The class of approximately surjective mappings is found suitable also in the problem of describing the projective limit of quotient spaces.


46A17 Bornologies and related structures; Mackey convergence, etc.
46A08 Barrelled spaces, bornological spaces
46A30 Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness)


Zbl 1022.46005