## Description

In a nutshell

The purpose of this book is to serve as the accompanying text for a first course in functional analysis, taken typically by second- and third-year undergraduate students majoring in mathematics.

As I prepared for my first time teaching such a course, I found nothing among the countless excellent textbooks in functional analysis available that perfectly suited my needs. I ended up writing my own lecture notes, which evolved into this book (an earlier version appeared on my blog [31]).

The main goals of the course this book is designed to serve are to introduce the student to key notions in functional analysis (complete normed spaces, bounded operators, compact operators), alongside significant applications, with a special emphasis on the Hilbert space setting. The emphasis on Hilbert spaces allows for a rapid development of several topics: Fourier series and the Fourier transform, as well as the spectral theorem for compact normal operators on a Hilbert space. I did not try to give a comprehensive treatment of the subject, the opposite is true. I did my best to arrange the material in a coherent and effective way, leaving large portions of the theory for a later course. The students who finish this course will be ready (and hopefully, eager) for further study in functional analysis and operator theory, and will have at their disposal a set of tools and a state of mind that may come in handy in any mathematical endeavor they embark on.

The text is written for a reader who is either an undergraduate student, or the instructor in a particular kind of undergraduate course on functional analysis. The background required from the undergraduate student taking this course is minimal: basic linear algebra, calculus up to Riemann integration, and some acquaintance with topological and metric spaces (in fact, the basics of metric spaces will suffice; and all the required material in topology/metric spaces is collected in the appendix).

Some “mathematical maturity” is also assumed. This means that the readers are expected to be able to fill in some details here and there, not freak out when bumping into a slight abuse of notation, and so forth.