- 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.
- 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.
- 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 for some large enough
- Convex closed sets in ℝn are retracts. Are there other examples of retracts
not considered by Theorem 14.2.3?
- In ℝ2, consider an annulus, . Show that this set does not
have the fixed point property. Could it be a retract of
- Does have the fixed point property?
- 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.
- 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.
- 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.
- Suppose you have that hk : B → B for B a compact set and each hk has a
fixed point. Suppose also that hk converges to h uniformly on B. Then h also
has a fixed point. Verify this.
- The Brouwer fixed point theorem is a finite dimensional creature. Consider
a separable Hilbert space H with a complete orthonormal basis
Then define the following map. For x = ∑
i=1∞xiei, define L
i=1∞xiei+1. Now let f
e1+Lx. Verify that f : B
is continuous and yet it has no fixed point. This example is in [?].