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There are analogues of certain basic notions in algebraic topology in the theory of Banach spaces. For example, the Brouwer fixed point theorem generalizes to the Schauder fixed point theorem, and the idea of the degree of a map generalizes to the Leray-Schauder degree. In both cases, you must restrict yourself to considering (nonlinear) compact operators: operators that take bounded sets to relatively compact sets.

How far can this analogy be extended? How much of algebraic topology can be made to work in this setting? Suppose we consider arbitrary bounded subsets (not necessarily convex) of a Banach space as our class of spaces, and then compact operators between those. (If that's the wrong choice, feel free to correct it.) Can you define some sort of infinite-dimensional homology group so that the degree is an element of it? is there an analogue of the Lefschetz fixed point theorem? Or does the analogy break down if you try to go beyond defining a degree?

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I think it is a mistake to view the Schauder fixed point theorem as in any way a generalization of the Banach fixed point theorem. First of all, the latter is really a purely metric space (not Banach space) theorem and is connected to the ideas of distance decreasing and completeness; it also does not depend in any way with the global topology of the space. The Schauder theorem on the other hand is a convexity related result, and the proof is in no way related to that of the Banach fixed point theorem. – Dick Palais Oct 27 '10 at 17:19
He possibly means the Brouwer's fixed point theorem, rather than Banach's contraction principle. – Pietro Majer Oct 27 '10 at 17:24
I did mean Brouwer, not Banach. Sorry about that. – arsmath Oct 28 '10 at 7:16

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