Elements of the Theory of Functions and Functional Analysis - Dover Mathematics Book | Advanced Math Textbook for Students & Researchers | Perfect for University Studies & Mathematical Research

I've gotten through most of the first volume (First volume is on linear spaces, second one on measure theory, both volumes are included in this book) and this has been an excellent reference and supplement to the text we are using. The terminology is somewhat outdated: for example there is mention of contact points (which would just be called a limit point today) and the codimension of a subspace is called its deficiency (which some people still use actually), but again as a reference and supplement to a more recent text it is superb. In particular I found its treatment of the contraction principle to solving nonlinear differential equations very useful in understanding that found in Royden. Eventhough Royden is the text used in my graduate class, its obvious the professor is using this book (probably a first edition of it lol) because the order in which he presents the material is pretty much the same as presented in this. All in all it is far too dense to be wielded properly by a beginner in functional analysis and operator theory but for the price it is a must to have even if just to get a different point of view on something you might have trouble understanding somewhere else. I cannot comment on the second volume as I haven't gone through it at all.