By Robert F. Brown

This 3rd version is addressed to the mathematician or graduate scholar of arithmetic - or maybe the well-prepared undergraduate - who would favor, with at least history and practise, to appreciate a few of the attractive effects on the center of nonlinear research. in response to carefully-expounded principles from a number of branches of topology, and illustrated by way of a wealth of figures that attest to the geometric nature of the exposition, the e-book can be of mammoth assist in delivering its readers with an figuring out of the math of the nonlinear phenomena that represent our genuine global. integrated during this re-creation are a number of new chapters that current the mounted aspect index and its purposes. The exposition and mathematical content material is more suitable all through. This booklet is perfect for self-study for mathematicians and scholars drawn to such components of geometric and algebraic topology, practical research, differential equations, and utilized arithmetic. it's a sharply centred and hugely readable view of nonlinear research via a working towards topologist who has noticeable a transparent route to figuring out. "For the topology-minded reader, the publication certainly has much to supply: written in a really own, eloquent and instructive variety it makes one of the highlights of nonlinear research obtainable to a large audience."-Monatshefte fur Mathematik (2006)

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**Sample text**

Furthermore, it preserves integer products, that is, az x bz' = ab(z x z') for integers a and b . In fact, when the homology groups are all infinite cyclic , the cross product may be identified with integer multiplication. The third property concerns maps f : (X, A) -+ (W, C) and g : (Y, B) -+ (Z, D) which have a well-defined cartesian product f x g : (X, A) x (Y, B) -+ (W, C) x (Z, D). The naturality property of the cross product states that, at the level of homology, the cartesian product of maps preserves cross products, as follows.

The generalized Brouwer theorem tells us that [« has fixed points ; choose one of them and call it Yn. Since K is compact, the sequence {f (Yn)} has a convergent subsequence, which we will still write as (f(Yn)} to avoid messy subscripts. Call the limit of the subsequence Y and note that it is in the closed set C . We claim that Y is a fixed point of f. The argument depends on that approximation property of the Schauder projection, which in this case states that d(Pn (x), x)) < *. When we let x = f (Yn) we find that dUn (Yn), f (Yn)) < * so the sequence {fn (Yn)} = {Yn} must converge to the same point that (f(Yn)} does, namely y.

3. (Kakutani's Example) There is a closed, bounded convex subset C ofa normed linear space X and a map f : C -+ C without fixed points. 4. Schauder Fixed Point Theory 25 x Figure 4. Proof. The space X is the Hilbert space £2 which consists of all infinite sequences of reals x = {Xl, x2, . } for which the series L~I x; converges. This forms a nonned vector space under term-by -term addition of sequences and the obvious scalar product, and with norm IIxli ~ Jta x; The unit ball C in X , that is, the set of points X such that IIx II ::: I, is certainly closed, bounded and convex.