1: Hilbert Spaces. 2: Operators on Hilbert Space. 3: Banach Spaces. 4: Locally Convex Spaces. 5: Weak Topologies. 6: Linear Operators on a Banach Space. 7: Banach Algebras and Spectral Theory for Operators on a Banach Space. 8: C^* Algebras. 9: Normal Operators on Hilbert Space. 10: Unbounded Operators. 11: Fredholm Theory.

Functional analysis has become a sufficiently large area of mathematics that it is possible to find two research mathematicians, both of whom call themselves functional analysts, who have great difficulty understanding the work of the other. The common thread is the existence of a linear space with a topology or two (or more). Here the paths diverge in the choice of how that topology is defined and in whether to study the geometry of the linear space, or the linear operators on the space, or both. In this book I have tried to follow the common thread rather than any special topic. I have included some topics that a few years ago might have been thought of as specialized but which impress me as interesting and basic. Near the end of this work I gave into my natural temptation and included some operator theory that, though basic for operator theory, might be considered specialized by some functional analysts.