Do there exist some relations between Functional Analysis and Algebraic Topology? As the title: does there exist some relations between Functional Analysis and Algebraic Topology.
As we have known, the tools developed in Algebraic Topology are used to classify spaces, especially the geometrical structures in finite dimensional Euclidean space. But when we come across some infinite dimensional spaces, such as Banach spaces, do the tools in Algebraic Topology also take effect?
Moreover, are there some books discussing such relation? My learning background is listed following:basic algebra(group, ring, field, polynomial); Rudin's real & complex analysis and functional analysis; general topology(Munkres level).
Any viewpoint will be appreciated.
 A: Yes, there is a whole field of study called Non-Commutative Topology/Geometry that is centred around this idea. K-theory for C* algebras, Brown-Douglas-Filmore's study of essentially normal operators, Kasparov's work on the Novikov conjecture, etc. are all very beautiful.
Edit: Given your background, there are two possible ways I can think of to begin :
a) Start learning some Operator Algebras' theory (Gerard Murphy's book is good for this). Then graduate to K-theory for C* algebras (from Rordam/Laustsen/Larsen's book)
b) Start with Vector bundles and K-theory for topological spaces (Allen Hatcher has some notes on this). Then read Atiyah's manuscript on K-theory.
This is just my 2c.
A: See Atiyah–Singer index theorem:
http://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem.
A: An interesting fact is that the space of Fredholm operators on a Hilbert space classifies stable equivalence classes of vector bundles over any space: This is the topological $K$ theory of the space. This is the Atiyah-Jänich Theorem. 
A: Weak * Topologies are also very interesting, these topologies concern bounded operators on Hilbert Spaces, also von Neumann algebras too 
http://en.wikipedia.org/wiki/Von_Neumann_algebra
