Algebraic Properties. Operators of Finite Rank

Abstract

Exercise 1. For finite-dimensional vector spaces the notion of Fredholm operator is empty, since then every linear map is a Fredholm operator. Moreover, the index no longer depends on the explicit form of the map, but only on the dimensions of the vector spaces between which it operates. More precisely, show that every linear map T: H → H’ where H and H’ are finite-dimensional vector spaces has index given by index T = dim H — dim H’.