Amaximal extension of Kothe\'s homomorphism theorem

Abstract

In 1958, Prof. T.Kato gave the following perturbation theorem: Let $E₀$, and $F₀$ be subspaces of Banach spaces E and F, respectively, and let $f : E₀ → F₀$ be a linear surjective map from $E₀$ onto $F₀$ with closed graph in $E ? F$.If $dim(F/F₀) < \aleph₀$, then f is open and $F₀$ is closed in F [4]. Ten years later, Prof Dr. G. Köthe gave two generalizations [5] which enhanced and were enhanced by considerations of codimension [7], Baire-like (BL) spaces [11a], and quasi-Baire (QB) spaces [11a, 9], and thus, together with a Robertson-Robertson Closed Graph Theorem (cf. [14]), provided significant external impetus for the early study of strong barelledness conditions. Viewed as yet another version of the Kato result, Köthe's Homomorphism Theorem replaces «Banach spaces» with the more general «(LF)-spaces» (cf. 8.4.13 of [6]). Here, again, strong barelledness [12] kindly repays Köthe and allows us to replace «< \aleph_0» with «< c ». This is, easily, the best possible extension as regards codimension of $F₀$.