Generalized prequojections and bounded maps
A locally convex space E is called a generalized prequojection if for each 0-neighborhood U there is another one V such that for every 0- neighborhood W one can find \(c>0\) with \[ W^ 0\cap E'_{U^ 0}\subset cV^ 0. \] The name derives from the fact that a Fréchet space E is a generalized prequojection if and only if it is a prequojection (or, equivalently, does not satisfy condition (*) of Bellenot-Dubinsky). - We recall that a Fréchet space E is a prequojection precisely if its strong bidual \(E''\) is a quojection; i.e., \(E''\) is the strict projective limit of a sequence of Banach spaces. The condition in the definition of generalized prequojection was introduced by \textit{T. Terzioǧlu} [Doǧa T. J. Math. 10, 338-344 (1986)] who also proved that \(L(E,F)=LB(E,F)\) holds if E is a generalized prequojection and F is a barrelled space with a continuous norm and a (Schauder) basis. The present authors first show how it is possible to construct non- trivial generalized prequojections with continuous norm by starting from an arbitrary non-quasi-reflexive Banach space X (using strongly non- norming subspaces of \(X')\). Then they generalize Terzioǧlu's result to the natural limits of its validity by proving that for a generalized prequojection E, the equality L(E,F)\(=LB(E,F)\) remains valid for all locally convex spaces F with bounded approximation property and a continuous norm. [This result was found independently by \textit{A. Galbis} in his dissertation, Valencia (1988).]
construct non-trivial generalized prequojections with continuous norm
scientific article; zbMATH DE number 4109345