ABSTRACT
Many problems in science lead to the equation Tx = y in infinite dimensional spaces rather than to the finite dimensional case in Chapter 1. In particular, ordinary and partial differential equations, and integral equations can be formulated as abstract equations on infinite dimensional spaces of functions. For the equation Tx = y, we again are interested in the questions raised at the beginning of Chapter 1. In 1934, Leray and Schauder [185] generalized Brouwer degree theory to an
infinite Banach space and established the so-called the Leray Schauder degree. It turns out that the Leray Schauder degree is a very powerful tool in proving various existence results for nonlinear partial differential equations (see [135], [185], [203], [228], etc.). In this chapter, we will introduce the Leray Schauder degree. This chapter
consists of five sections.
