Topics In Analysis

14.4 Exercises

1. Suppose you have a Banach space X and a set A ⊆ X. Suppose A is a retract of B where B has the fixed point property. By this is meant that A ⊆ B and there is a continuous function f : B → A such that f equals the identity on A. Show that it follows that then A also has the fixed point property.
2. Show that the fixed point property is a topological property. That is, if you have A,B two topological spaces and there is a continuous one to one onto mapping f : A → B which has continuous inverse, then the two topological spaces either both have the fixed point property or neither one does.
3. The Brouwer fixed point theorem says that every closed ball in ℝⁿ centered at 0 has the fixed point property. Show that it follows that every bounded convex closed set in ℝⁿ has the fixed point property. Hint: You know that the closed convex set is a retract of ℝⁿ. Now if it is also a bounded set, then you could enclose it in B
   (0,r)
   for some large enough r.
4. Convex closed sets in ℝⁿ are retracts. Are there other examples of retracts not considered by Theorem 14.2.3?
5. In ℝ², consider an annulus,
   {x : 1 ≤ |x| ≤ 2}
   . Show that this set does not have the fixed point property. Could it be a retract of ℝ²?
6. Does
   {x ∈ ℝn : |x| = 1}
   have the fixed point property?
7. Suppose you have a closed subset H of X a metric space and suppose also that C is an open cover of H. Show there is another open cover Ĉ such that the closure of each open set in Ĉ is contained in some set of C. Hint: You might want to use the fact that metric space is normal.
8. If H is a closed nonempty subset of ℝⁿ and C is an open cover of H, show that there is a refined open cover such that each of the new open sets are bounded. In the partition of unity result obtained above, applied to H show that the functions in the partition of unity can be assumed to be infinitely differentiable with compact support.
9. Check that the conclusion of Theorem 14.2.3 applies for X just a metric space. Then apply it to give another proof of the Tietze extension theorem.
10. Suppose you have that h_k : B → B for B a compact set and each h_k has a fixed point. Suppose also that h_k converges to h uniformly on B. Then h also has a fixed point. Verify this.
11. The Brouwer fixed point theorem is a finite dimensional creature. Consider a separable Hilbert space H with a complete orthonormal basis
    {ek}
    _k=1^∞. Then define the following map. For x = ∑ _i=1^∞x_ie_i, define L
    ∑ ∞ ( i=1xiei)
    ≡ ∑ _i=1^∞x_ie_i+1. Now let f
    (x)
    ≡
    1 2
    (1 − ∥x∥H )
    e₁+Lx. Verify that f : B
    (0,1)
    → B
    (0,1)
    is continuous and yet it has no fixed point. This example is in [?].