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Gn ). 2) Proof. By the chain rule, n ∇(h ◦ g) = ∇h ◦ ∇g = i=1 ∂h (g)∇gi . 2) can be written as a linear com∂h (g)∇gi from the bination of the remaining n − 1 columns. For 2 ≤ i ≤ n, subtract ∂xi ∂h (g). We are left with ﬁrst column of the determinant, and then factor out the scalar ∂x1 ∂h (g) det (∇g1 , ∇g2 , . . ∇gn ). From the deﬁnition of h, we see this is exactly (f ◦g) det ∇g. ∂x1 We state the following classical identity without proof. CHAPTER 2. 7 Let {Mi }ni=1 be the set of cofactors obtained by expanding the Jacobian den ∂ Mi ≡ 0.

Note that if x ∈ X, then r(x) ∈ ∂X, and so G(x) ⊆ X. Thus any ﬁxed point of G must be in X. It follows that x0 is a ﬁxed point of F . Now we turn to our ﬁrst result by Browder. The previous multifunction results have required F to be usc. In the following, we ask that the inverse image of open sets under F be open. In this case, F is said to be lower semicontinuous. 2 Let X be a nonempty compact convex subset of a topological vector space E, and F : X → 2X such that F (x) is nonempty and convex for each x ∈ X.

The following was proven by Hukuhara in 1950. 4 Let X be a subset of a space E, and f : X → E. 1. X : convex. 2. E : locally convex topological vector space. 3. f : compact, continuous self–map. Then f has a ﬁxed point in X. 2 Fixed Point Properties for Closed Bounded Convex Sets As noted, compactness is important in establishing the topological ﬁxed point theorem in inﬁnite dimensions. In inﬁnite dimensions, the property fails for closed bounded sets. In this section we pause from our consideration of continuous functions to consider some classes of functions for which closed bounded convex sets have ﬁxed points.

### Brouwer's Fixed Point Theorem. Methods of proof and generalizations by Stuckless T.

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