2
$\begingroup$

Let $M = \{1, \dots, n\}$ be a metric space with the metric $d$ and, in $\Omega = M^{\mathbb{N}}$, define $\tilde{d}(x, y) = \sum_{k=1}^{+\infty} \frac{d(x_k, y_k)}{2^k}$.

We say that $f\colon \Omega \rightarrow \mathbb{R}$ depens only on finite coordinates if there exist $m \in \mathbb{N}$ such that $f(x_1, x_2, \dots) = f(x_1, \dots, x_m)$.

I'm trying to show that f $f: \Omega \rightarrow \mathbb{R}$ depends only on finite coordinates, then $f$ is $\alpha$-Holder

$\endgroup$
4
  • $\begingroup$ What if $\tilde d(x,y)<2^{-m}$? This question isn't really suitable for MathOverflow - probably it would be more appropriate for math.stackexchange.com $\endgroup$ Feb 7, 2018 at 22:18
  • $\begingroup$ what is $A(x_1,\dots)$? $\endgroup$ Feb 8, 2018 at 0:04
  • $\begingroup$ Sorry! It’s f... $\endgroup$ Feb 8, 2018 at 0:08
  • $\begingroup$ But even if $f$ only depends on the first coordinate, $f(x_1)$, in general it will not be Hölder $\endgroup$ Feb 8, 2018 at 16:34

0

Browse other questions tagged or ask your own question.