Banach Algebras

This chapter is an introduction to Banach algebras as an axiomatic approach to the spaces of continuous linear operators on Banach spaces. Their morphisms, subalgebras and ideals are defined, but the main emphasis is on operator-valued analytic functions defined by power series and Laurent series. The Cauchy- Hadamard formula for the radius of convergence is proved, and applied to the definition of the exponential and logarithm functions. The group of invertible elements is then shown to be an open set, with a boundary consisting of ‘topological divisors of zero’.