Abstract
We start with summarizing some necessary auxiliary material from functional analysis and operator theory with special emphasis on local invertibility theories in Banach algebras. Then to give the reader a first idea of the usefulness of abstract local principles, we examine the so-called finite section method for Toeplitz operators. From this application we are finally going to extract a general scheme of using Banach algebra techniques which allows to tackle a lot of different invertibility problems (including the Fredholmness of Toeplitz and Wiener-Hopf operators and the stability of spline approximation methods for singular integral and Mellin operators) from a common point of view.
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