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At the meeting of the University of Wales Pure Mathematics Colloquium held in the delightful setting of Gregynog Hall, the University conference centre in the old county of Montgomeryshire (now part of Powys), in 1972, I gave a short talk on ‘Generalized Fredholm theory in Banach spaces’. In thanking me, Jeffrey Weston expressed the hope that I should some day collect my contributions to Fredholm theory in Banach spaces into a single book. Other friends have made similar suggestions from time to time. This tract is the outcome.
In this tract we present analogues, for operators on a Banach space, of Fredholm's solution of integral equations (of the second kind). In the first place we imitate Fredholm's construction for operators of finite rank. Then, ignoring questions of their construction, we consider formulae which have a structure similar to that of Fredholm's formulae. In studying these, we use the Riesz theory. We study further the relations between these formulae and the Riesz theory, in particular obtaining bases for the finite-dimensional subspaces figuring in the Riesz theory. Then we return to the study of specific constructions for various classes of operators. A fuller summary of the tract appears in §1.2.
The theorems, lemmas and (formal) definitions are separately numbered in each section, so that for instance Definition 1.3.6 is the sixth (formal) definition in the third section of the first chapter.
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