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.
Best Answer
I don't know how useful this will be, but I have some lecture notes that motivate the last three things on your list by first reinterpreting the finite dimensional spectral theorem in terms of the functional calculus. (There is also a section on the spectral theorem for compact operators, but this is just pulled from Zimmer's Essential Results of Functional Analysis.) I gave these lectures at the end of an undergraduate course on functional analysis, though, so they assume familiarity with Banach and Hilbert spaces.