Topics on Linear Operators
Fabio Botelho · 2020
This chapter discusses the topologies for bounded operators and recalls that the set of all bounded linear operators is a Banach space with a specific norm. The topology related to the metric induced by this norm is called the uniform operator topology. The chapter defines the adjoint operator and proves the Hahn-Banach theorem. It describes compact operators and shows that a compact operator maps weakly convergent sequences into norm convergent sequences. The chapter also discusses the square root of a positive operator in a Hilbert space and shows that under a sequence of self-adjoint commuting operators there exists a self-adjoint, bounded, linear operator.