0
$\begingroup$

This question is from Foundations of mathematical analysis by Richard Johnsonbaugh enter image description here

The thing with this question is that there is a question that seems to prove the opposite claim Prove the map has a fixed point - someone look into this

How should one go about dealing with this question?

$\endgroup$
  • 7
    $\begingroup$ The magic word is "onto": the contraction mapping in (c) can't be surjective. $\endgroup$ – Rob Arthan Feb 29 '16 at 22:10
  • $\begingroup$ @RobArthan Are you saying that no contraction mapping can be surjective? $\endgroup$ – Carlos - the Mongoose - Danger Feb 29 '16 at 22:10
  • $\begingroup$ Not if the space is compact. $\endgroup$ – Rob Arthan Feb 29 '16 at 22:11
  • 1
    $\begingroup$ And actually any 1-Lipschitz surjective function from a compact to itself is an isometry. $\endgroup$ – Captain Lama Feb 29 '16 at 22:15
  • $\begingroup$ @CaptainLama What would be an example of such function and what is 1 lipschitz $\endgroup$ – Carlos - the Mongoose - Danger Feb 29 '16 at 23:27
8
$\begingroup$

Suppose that $f: M \to M $ was onto. Then for every $x,y \in M$ $x \not=y$, there exists an $x',y' \in M$ s.t. $f(x')=x$ and $f(y')=y$. Then $$d(x,y)=d(f(x'),f(y'))\leq c d(x',y')<d(x',y').$$

Let $B=\max_{x,y \in M^2} d(x,y)$, this exists since $M$ is compact and $d:M^2 \to \mathbb{R}$ is continuous. But, by the above fact, for any $x,y \in M$ there exist an $x',y'$ s.t. $$d(x,y)<d(x',y'),$$ which contradicts the existence of a maximizer $B$.

$\endgroup$
  • $\begingroup$ What is the purpose of making the function "on-to" specifically. How does it fail if it were not onto? $\endgroup$ – Carlos - the Mongoose - Danger Feb 29 '16 at 22:22
  • $\begingroup$ The function needs to be onto for the existence of $x'$ and $y'$. Without that step, we wouldn't be able to guarantee that $d(x,y)$ wasn't the maximizer for any $x$ and $y$. $\endgroup$ – Nick Feb 29 '16 at 22:53
4
$\begingroup$

This probably works. Define a distance function $r:M\rightarrow M$ such that $$ r(x,y) = d(x,y). $$ Note that $r(\cdot)$ is a continuous function, whose proof can be seen here: Is the distance function in a metric space (uniformly) continuous?

Thus, since $r$ is continuous on compact $M$, it attains its supremum, say at $(x^\ast, y^\ast)$. Note that $f(x^\ast),f(y^\ast)\in M$, which means that $$ r(f(x^\ast),f(y^\ast)) = d(f(x^\ast),f(y^\ast)) \leq d(x^\ast, y^\ast) $$ by definition of $(x^\ast, y^\ast)$ resulting in a contradiction.

$\endgroup$
3
$\begingroup$

HINT: Let $p$ be the fixed point guaranteed by the earlier question. Show that there is an $x\in X$ that maximizes $d(p,x)$. Then show that $x\notin f[X]$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.