# Tag Info

Accepted

### Does iterating the derivative infinitely many times give a smooth function whenever it converges?

I was able to adapt the accepted answer to this MathOverflow post to positively answer the question. The point is that one can squeeze more out of Petrov's Baire category argument if one applies it ...
• 107k
Accepted

• 6,414

### Fixed point theorem for the uncountable power of an interval

Let $f:[0,1]^\kappa\to[0,1]^\kappa$ be given. For a finite subset $F$ of $\kappa$ let $A_F=\{x\in[0,1]^\kappa: \pi_F(x)=\pi_F(f(x))\}$, where $\pi_F$ is the projection onto $[0,1]^F$. The set $A_F$ is ...
• 9,665
Accepted

### leray schauder fixed point and schauder fixed point

Note that Leray-Schauder is usually proven by using the hypotheses to construct a mapping that satisfies the conditions of the Schauder fixed point theorem, and then appealing to the Schauder fixed ...
• 36.3k

### Does the Brouwer fixed point theorem admit a constructive proof?

One constructive version of the Brouwer fixed-point theorem which I find very elegant and illustrating runs like this: Theorem (constructive Brouwer fixed-point theorem) Let $B$ be the closed unit ...
• 1,616

### Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem?

Edit: Will Sawin has pointed out some difficulties with this answer. I'm going to leave this up for at least a while, in case anyone has any ideas about repairing it. Or perhaps this could be a ...
• 52.2k

### A variation of the Ryll-Nardzewski fixed point theorem

The claim does not hold, as the nice example by Naratuka Ozawa shows. The purpose of this answer (or rather, extended comment) is to share a related fixed point theorem. Theorem: Let $X$ be a non-...
• 11.4k
Accepted

### Connected vertex-transitive graph with the fixed-point property

Let me convert my comments into an answer. There is no such [EDIT:] finite graph $G$. Indeed, something stronger can be said. Suppose that a group $\Gamma$ acts transitively by permutations on a ...
• 22.4k