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Let $X$ be a infinite dimensional Banach space and $B=\{x\in X:\ \|x\|\leq 1\}$.

How to define a continuous $f:B\rightarrow B$ without fixed points?

Edit 1: I have changed the question a litlle, because if $f$ is linear then $f(0)=0$.

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A linear functional has scalar values. A linear operator fixes $0$. –  GEdgar Jan 11 '13 at 16:35
    
Yes, you are right. Let me delete the second item. –  Tomás Jan 11 '13 at 16:37
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For some $X$ the unit ball is weakly compact, for others not. –  GEdgar Jan 11 '13 at 16:52
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A particular example for the space $c_0$ is the mapping $(x_1,x_2,\ldots)\mapsto(1-\Vert x\Vert, x_1,x_2,\ldots)$. –  David Mitra Jan 11 '13 at 17:05
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This paper gives examples for an arbitrary normed, infinite dimensional space. According to the authors, that every infinite dimensional normed space admits a fixed-point free, continuous map on its unit ball was first proved by J. Dugundji in An extension of Tietze's theorem, Pacific Journal of math., I (1951), 353-367 using the Axiom of Choice, and later by V. L. Klee, Jr. in Some topological properties of convex sets, Trans. Amer. Math. Soc., 78, (1955), 30-45, using "a more constructive proof". –  David Mitra Jan 11 '13 at 17:52

2 Answers 2

up vote 6 down vote accepted

This paper shows that in an arbitrary normed, infinite dimensional space, there is a continuous map on its unit ball (into its unit ball) that is fixed-point free with even stronger properties.

According to the authors, the fact that every infinite dimensional normed space admits a fixed-point free, continuous map on its unit ball was first proved, using the Axiom of Choice, in Theorem 6.3 of [1], below, and was proven later using "a more constructive proof", in [2], below.

Of course, for specific spaces, it is not too hard to find fixed-free maps. For example, if $X=c_0$, then the map from $B_X$ to $B_X$ that sends the element $x=(x_1, x_2,\ldots)$ to $(1-\Vert x\Vert, x_1,x_2,\ldots)$ is continuous (in fact, non-expansive) and fixed-point free. An example on the space $\ell_2$ is given in Patrick Da Silva's answer; and this can be generalized to examples for $\ell_p$ spaces as suggested by Tomás in the comments to your post.


[1] J. Dugundji, An extension of Tietze's theorem, Pacific Journal of math., I (1951), 353-367

[2] V. L. Klee, Jr., Some topological properties of convex sets, Trans. Amer. Math. Soc., 78, (1955), 30-45.

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A link to [1] has been provided by Tomás in the comments to the OP. –  David Mitra Jan 11 '13 at 18:17

I think a good place to start would be if $X$ is an Hilbert space with a countable basis (yeah I know... lots of extras). This is because these spaces have a basis, hence we can write $$ \mathcal E = \{ e_1, e_2, \dots, e_n, \dots \} $$ as a Hilbert basis of the space. For $x \in B$, we have $$ x = \sum_{n=1}^{\infty} (x,e_n) e_n = \sum_{n=1}^{\infty} \underset{(x,e_n)}{\underbrace{x_n}} e_n. $$ Define $$ f(x) \overset{def}{=} \frac 12 e_1 + \sum_{n=1}^{\infty} \frac{x_n}2 e_{n+1} = \frac 12 e_1 + \sum_{n=2}^{\infty} \frac{ x_{n-1} }2 e_n $$ It is continuous as the series part of the function is a linear bounded function and we simply translated it. (The coefficients $\frac 12$ are there to preserve $\| x \| \le 1$. I just realized I wrote all this answer assuming that this would be sufficient for $f(x) \in B$, but I now see that it is not. I guess this answer restricts to the case where $f(x)$, as defined above, is in $B$.) It has no fixed point, for if $f(x) = x$, we must have $x_{n-1} = x_n = 1/2$ by induction, which means $x \notin H$ because the norm of this candidate fixed point $x$ is not finite. \prg Now that the idea is there, we can go to Hilbert spaces with arbitrary (infinite) basis : choose a non-surjective injection $$ \omega : \mathcal E \to \omega(\mathcal E) $$ which has no finite cycles (and in particular, no fixed points). I believe this requires the axiom of choice at some point. Afterwards, letting $$ x = \sum_{e \in \mathcal E} (x,e) e = \sum_{e \in \mathcal E} x_e e, $$ let $e_0 \in \mathcal E \backslash \omega(\mathcal E)$ and define $$ f(x) = \frac 12 e_0 + \sum_{e \in \mathcal E} x_e \omega(e). $$ If $f(x) = x$, we have $x_e = x_{\omega(e)}$. If there exists an $e$ with $x_e \neq 0$, by assumption that $\omega$ has no finite cycles, this means $$ x_e = x_{\omega(e)} = x_{\omega(\omega(e))} = \dots = x_{\omega^n(e)} $$ for every integer $n$, which leads to a contradiction because $x$ must have "bounded norm". Therefore the only possible fixed point left would be $x = 0$, but it is not fixed. Note that the same comment about $f(x) \in B$ applies here...

I believe one way to solve the problem of $f(x) \in B$ would be to replace my series on the right of my definition of $f(x)$ by a linear functional with no fixed point other than $0$ and with operator norm smaller than $\frac 12$ which is not surjective (so that you can find $e_0$). As for the general $X$ Banach space, I have no idea, but this is as close as I could get. My answer was meant to be a big comment.

Hope that helps,

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