Preface Preface
This text is inspired by and adapts the general shape of the excellent Introduction to Hilbert Space by Nicholas Young. The reader is encouraged to track down a copy - Professor Young’s exposition and in particular his presentation of the context and historical development of Hilbert spaces is an outstanding introduction to the origins and applications of the field. I do not attempt to recreate his deep historical knowledge nor his wide perspective on mathematics here. The other classical textbook on Hilbert spaces for undergraduates that I draw on is Kreyszig’s Introductory Functional Analysis with Applications.
Hilbert space theory is a natural bridge between undergraduate and graduate mathematical perspectives. It extends the ideas of finite dimensional linear algebra, blends in ideas from real and complex analysis, and points to a very active area of research in algebra. To get the most out of this text, the reader will need to be familiar with (or be willing to learn) ideas from real and complex analysis (e.g. uniform convergence of sequences of functions, analyticity in the complex plane), linear algebra (e.g. the spectral theorem for symmetric matrices), and basic point-set topology (e.g. metrics).
Of course the study and use of Hilbert spaces of functions goes well beyond what can be addressed in this text. A reader in need of an advanced course in linear algebra flavored with functional analysis should work through Axler’s Linear Algebra Done Right. Axler has also recently published Measure, Integration, and Real Analysis, which covers the basics of functional analysis up to the spectral theorem for compact operators. The reader is also encouraged to work through Halmos’s Introduction to Hilbert Space and the Theory of Spectral Multiplicity and especially the Hilbert space problem book, which treat more advanced topics than we consider here. My own path into the study of functional analysis ran through J. B. Conway’s A Course in Functional Analysis and P. Lax’s Functional Analysis. A reader interested in pursuing research in a modern flavor of Hilbert space theory might look to Pick interpolation and Hilbert function spaces and Operator Analysis by J. Agler, J. E. McCarthy, and N. J. Young (the same N. Young we pay homage to here). For the interaction of finite dimensional Hilbert spaces (i.e. matrices) and function theory, the reader should seek out Bhatia’s excellent Matrix Analysis.
Another area of research arises not from studying the spaces but instead the maps that act on them. In this way, Hilbert space theory provides the foundation for the study of linear operators on infinte dimensional spaces. There is a major fork in the road here; one branch, concerned with single or classes of related operators and functions becomes operator theory. The study of the algebraic properties of the class of all operators on some space is the beginning of operator algebra. Both fields are vibrant and active.
