62 votes
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 ...
Terry Tao's user avatar
  • 107k
19 votes
Accepted

How is this fixed point theorem related to the axiom of choice?

I'll deduce Zorn's Lemma from your fixed-point theorem. Suppose $P$ is a poset violating Zorn's Lemma; so all chains in $P$ have upper bounds, but there's no maximal element. Consider the poset $Q=P\...
Andreas Blass's user avatar
17 votes

What are the major differences between real and complex Banach space?

There are some differences. For example Bishop-Phelps theorem, which holds only in real Banach spaces. In my opinion, this qualifies as a "major theorem". MR1749671 Lomonosov, Victor A counterexample ...
Alexandre Eremenko's user avatar
17 votes

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

(Edit) A simplified shorter version of this answer is in Fedor Petrov's comments (as Iosif Pinellis pointed out) : Assume the sequence $f^{(n)}$ is dominated, that is, there exists $h$ locally ...
username's user avatar
  • 2,464
16 votes

Is it possible to find a $\mathbb{C}$-algebra endomorphism of $\mathbb{C}[x,y]$ that fixes a given $w \in \mathbb{C}[x,y]-\mathbb{C}$?

If we view $w$ as a map $\mathbb A^2 \to \mathbb A^1$ and the geometric generic fiber has a trivial automorphism group, then there will be no nontrivial automorphisms of $\mathbb C[x,y]$ fixing $w$. ...
Will Sawin's user avatar
  • 133k
12 votes
Accepted

A variation of the Ryll-Nardzewski fixed point theorem

The claim does not hold. Let $F$ be the free group on $\{a,b\}$ and $X⊂F$ be the words whose last letter is $a$ or $b$. Let $\xi=\delta_a+\delta_b−\delta_{a^{-1}}−\delta_{b^{-1}}\in\ell_1(F)$. Then ...
Narutaka OZAWA's user avatar
12 votes
Accepted

Has Goedel's Second Incompleteness Theorem been proven using Lawvere's Fixed Point Theorem?

As suggested by the OP, I'm turning my comment into an affirmative answer. WARNING: The presentation of Joyal's proof in the paper I cited contains an incorrect conclusion about Joyal's sentence (that ...
godelian's user avatar
  • 5,347
11 votes

Closed manifolds with the fixed point property

According to this paper, there are various such manifolds constructed as cartesian products and connected sums of real, complex and quaternion projective spaces. For example, it is shown that $\...
Alon Amit's user avatar
  • 6,414
11 votes

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 ...
KP Hart's user avatar
  • 9,665
10 votes
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 ...
Willie Wong's user avatar
  • 36.3k
10 votes

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 ...
Franka Waaldijk's user avatar
10 votes

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 ...
Todd Trimble's user avatar
  • 52.2k
9 votes

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-...
Uri Bader's user avatar
  • 11.4k
9 votes
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 ...
Sam Hopkins's user avatar
  • 22.4k
9 votes

Fixed point theorem for the uncountable power of an interval

The basic form of Brouwer's fixed point theorem does still hold. Fix an uncountable $\kappa$ and a continuous function $f:[0,1]^\kappa \to [0,1]^\kappa$. For any $X \subseteq \kappa$, let $\pi_X : [0,...
James Hanson's user avatar
  • 10.3k
8 votes

What are the major differences between real and complex Banach space?

In optimization, you typically have a function $f \colon X \to \mathbb{R}$ which you are going to minimize. In order to apply first-order optimality conditions or first-order methods, you would like ...
gerw's user avatar
  • 1,474
8 votes

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

Here is a sort of partial solution. I doubt it will be very helpful, but anyone who wants to read it is free to do so. Let $I$ be any space with the fixed point property. We will construct a space $X$...
Will Sawin's user avatar
  • 133k
8 votes
Accepted

Fixed point of a group action

I don't think so. Let $\Gamma$ be the free group with 2 generators. It has a natural action on the boundary $\partial\Gamma$ and therefore on the space of probability measures on the boundary as well. ...
R W's user avatar
  • 16.6k
8 votes
Accepted

Does the Banach space $( \ell ^2 \oplus \ell ^2 )$ have F.P.P?

I think that the answer is yes, and that it should follow from the following facts: every Hilbert space is uniformly convex, hence it has normal structure; the direct sum of two Banach spaces with ...
Francesco Polizzi's user avatar
8 votes

Fixed points under a finite group action on projective variety

Setup Let $G$ be a finite group acting on a smooth projective variety $X$, and let $$ \rho: G \times X \to X $$ be the action morphism. For any $g \in G$ let $\rho_g$ denote the composition $$ X \...
cgodfrey's user avatar
  • 778
7 votes

Closed manifolds with the fixed point property

Products do not always have the property, but products with "disjoint" rational homology (in other words $X\times Y$ where $H_i(X, \mathbb{Q}) \neq 0$ implies that $H_i(Y, \mathbb{Q}) = 0)$ do. For ...
Igor Rivin's user avatar
  • 95.5k
7 votes
Accepted

Continuity of mapping sending a function to its (brouwer) fixed point

The answer is yes. Fixed points of $f$ are zeros of the continuous $g(x)=f(x)-x$. So one has to prove the following: If $g:[0,1]^n\times[0,1]\to[0,1]^n$ is continuos, and for each $r\in[0,1]$, $g(.,r)$...
Alexandre Eremenko's user avatar
7 votes

Fixed points under a finite group action on projective variety

The answer to the edited question is no. Take a double covering $\pi :X\rightarrow \mathbb{P}^2$ branched along a smooth quartic curve $C$ (so $X$ is a Del Pezzo surface), and $G=\langle \sigma \...
abx's user avatar
  • 37k
7 votes

Fixed point theorem for the uncountable power of an interval

By collapsing cardinals with forcing, one can derive Brouwer's fixed point theorem for $[0,1]^\kappa$ where $\kappa$ is uncountable from Brouwer's fixed point theorem for $[0,1]^{\aleph_0}$. Suppose ...
Joseph Van Name's user avatar
7 votes
Accepted

Invariant subspaces for matrices via fixed points on Grassmannians

I think the Lefschetz fixed point theorem still applies. If a self-map M→M of a compact orientable manifold M has no fixed points than the Euler characteristic of M is zero. But if M is a complex ...
Noam D. Elkies's user avatar
6 votes

A property stronger than the fixed point property

Here is an example of two surjective continuous self-maps $f$ and $g$ on the closed unit disk whose graphs are disjoint. Consider the surjective continuous maps $F(x,y):=x e^{2i\pi y }$ and $G(x,y)...
Pietro Majer's user avatar
  • 56.1k
6 votes

Is Schauder's conjecture resolved?

There is R. Cauty paper from 2012 titled 'Un theoreme de Lefschetz-Hopf pour les fonctions a iterees compactes' which from what I heard was reviewed to establish a correct proof of Schauder's ...
jaco's user avatar
  • 161
6 votes

Does a certain contractive mapping have a fixed point?

It seems that the answer is yes. We may replace $\alpha(d)$ by $\beta(d)=\sup_{t\geq d}\alpha(t)$ (surely, $\beta(q)$ is the limit of some sequence $\alpha(t_n)$ with $t_n\geq d$, so $\beta(d)<1$)....
Ilya Bogdanov's user avatar
6 votes

What are the major differences between real and complex Banach space?

Consider a bounded operator $T$ from a Hilbert space $H$ into itself, i.e. $T \in B(H)$. You can then define the numerical radius $r(T)$ as the radius of the numerical range $W(T)$, i.e. $$ W(T) = \{ \...
anonymous's user avatar
  • 436

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