Two fixed point theorems in Banach space

Here is a famous fixed point theorem in finite dimension by Brouwer:

Theorem 1 (Brouwer fixed point theorem) Let {M \subset {\mathbb R}^n} be a convex compact set, for any continuous function {f : M \rightarrow M}, there exists a point {x_0 \in M} such that {f (x_0) = x_0}.

There are couple of ways to extend this theorem to Banach spaces. First recall that a mapping between two Banach spaces is called compact, if the mapping is continuous (not necessarily linear) and the images of bounded sets are pre-compact.

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Geometry Hahn Banach theorem for weak* topology

We have geometric Hahn Banach theorem in standard functional analysis course, saying that there exists a separating hyperplane separates two special sets in usual topology. However this theorem can be used for some special sets in weak* topology. This is the problem 9 in the functional analysis book by Prof. Brezis.

Theorem 1 Let {E} be a Banach space, {A,B \subset E^*} be two nonempty disjoint convex subsets. Assume {A} is open in weak* topology. Then there exist some {x \in E, x \neq 0} and a constant {\alpha} such that the hyperplane {\{ f \in E^* : \langle f, x \rangle_{E^* \times E} = \alpha \}} separates {A} and {B}.

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Composition of sobolev functions

The following materials are exercise 8.10 and 8.11 from the functional analysis book by Prof. Brezis. The questions arise from whether we can well define a composition of two W^{1,p} functions or not. From Corollary 8.11 we know if G \in C^1, u \in W^{1,p}, G(u) \in W^{1,p} is well defined. However bad things might happen if G \in W^{1,p}. For instance, G(x) = |x|, then G(u) might be bad on the set \{ u = 0\}. I thought we could still define things in the sense of distribution, but Prof. Brezis said in this special case we could have something more than that, because of the following propositions. The proof is not hard by following the hints, but the idea is really clever (IMO)!

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