I wish to study the book 'Morse Theory' by J. Milnor, but I am not sure whether I have the necessary prerequisites. I know basic point set topology, real analysis (limits, continuity, differentiation, Riemann integration, sequences and series of functions, functions of several variables), differential geometry (metrics, connections, curvature, etc.) and algebra (one year graduate level course), but do not know any algebraic topology beyond the definition of fundamental group, and not much of functional analysis. Should I study some algebraic topology and functional analysis in order to be able to study this book? Are there any more prerequisites? Unfortunately the preface does not mention anything regarding this.
You should just dive in! Sounds like you will be ok. Maybe for some of the later chapters it is useful to have some more background in differential and Riemannian geometry, but one could also argue that this is the place to learn it!