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Put a relation on $C[0,1]$ by $f\sim g$ if $f(k/10)=g(k/10)$ for $k$ with $0\le k \le10$

I have shown that this is an equivalence relation, but I have no idea what are the equivalence classes. It requires me to prove together with addition of eq class and scalar multiplication, it makes $C[0,1]/\sim$ into a vector space of dimension 11.

So I guess I have something to do with $k/10$. Any hints?

Thank you!

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  • $\begingroup$ $k\in \mathbb Z$? $\endgroup$ – Alessandro Blasetti Nov 7 '16 at 8:01
  • $\begingroup$ @AlessandroBlasetti It doesn't say anything on $k$, is there any difference if $k \in \Bbb Z$ and $k \notin \Bbb Z$ ? $\endgroup$ – mathshungry Nov 7 '16 at 10:47
  • $\begingroup$ Is it $\forall\ k\in \mathbb Z\cap[0,10]$ or $\forall\ k\in \mathbb R\cap[0,10]$? $\endgroup$ – Alessandro Blasetti Nov 7 '16 at 11:22
  • $\begingroup$ Maybe I am missing something $\endgroup$ – Alessandro Blasetti Nov 7 '16 at 12:19
  • $\begingroup$ You didn't miss anything, the first line in this article is all the question have written down, it doesn't specify what is $k$. I find this question in the book called Real Analysis and Applications by Kenneth R. Davidson and Allan P. Donsig. $\endgroup$ – mathshungry Nov 7 '16 at 12:21

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