This monograph has grown out of a course which I have been giving at the Indian Institute of Technology Bombay (India) since 1981. A rough draft of the monograph was written while I was visiting Institute IMAG, Grenoble (France in 1984-1985. It was revised and finalized during my visits to the Centre for Mathematical Analysis at the Australian National University, Canberra (Australia) in 1986 and 1987.

The purpose of this project is to introduce the reader who has already taken a course in Functional Analysis to the beautiful as well as useful area of spectral approximation. Instead of aiming at full generality, I have tried to deal with simpler situations in order to highlight the main ideas. Thus, only bounded linear operators on a Banach space are considered here, although much of the theory and practice can be extended to densely defined closed operators.