# Looking for an easy lightning introduction to Hilbert spaces and Banach spaces

I'm co-organizing a reading seminar on Higson and Roe's Analytic K-homology. Most participants are graduate students and faculty, but there are a number of undergraduates who might like to participate, and who have never taken a course in functional analysis. They are strong students though, and they do have decent analysis, linear algebra, point-set topology, algebraic topology...

Question: Could anyone here recommend a very soft, easy, hand-wavy reference I could recommend to these undergraduates, which covers and motivates basic definitions and results of Hilbert spaces, Banach spaces, Banach algebras, Gelfand transform, and functional calculus?

It doesn't need to be rigourous at all- it just needs to introduce and to motivate the main definitions and results so that they can "black box" the prerequisites and get something out of the reading seminar. They can do back and do things properly when they take a functional analysis course next year or so.

• I've had good experience with "An Introduction to Hilbert Space" by N. Young. It's a small and very readable book in my opinion. – Tunococ Jun 23 '13 at 4:12
• I liked Kreyszig's Introductory Functional Analysis text for a basic introduction. It has proofs but is more bird's eye view and doesn't delve into a lot of the headier mathematics. It's by no means a great text but it's a good introduction, easily accessible to undergraduates (at least at the level that you're interested). – Cameron Williams Jun 23 '13 at 4:14
• Richard V. Kadison & John R. Ringrose, Fundamentals of the theory of operator algebras, Elementary theory Volume 1. A very good book, having almost all the topics you say. It requires a certain knowledge of measure theory. – Felipe Pérez Jun 23 '13 at 4:15
• IIRC, for Higson and Roe, the functional analysis one needs to know is the spectral theorem and basic $C^*$-algebra theory. For the spectral theorem, you might want to consider Reed and Simon, vol 1. For $C^*$-algebras, Arveson's An Invitation to $C^*$-algebras might be useful. – Michael Jun 23 '13 at 15:56
• Another suggestion: A User's Guide to Operator Algebras by Fillmore. Really light and probably more in line with Higson and Roe's NCG point of view. For example, it sketches a development of operator $K$-theory. Arveson is more old-school and representation theory-centric. – Michael Jun 23 '13 at 20:21