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Suppose we have the space $c_0=\{f:\mathbb{N}\rightarrow\mathbb{C}: \lim_{n\rightarrow\infty}{f(n)=0}\}$. I can prove that $c_0$ is a Rieszspace. But now i want to prove more: All positiv linear functions $\varphi$ on $c_0$ have the following property: $$f_1\geq f_2\geq\ldots\geq0\Longrightarrow \varphi(f_n)\rightarrow 0$$ We assume that for all $i\in\mathbb{N}$ holds $f_i:\mathbb{N}\rightarrow[0,\infty)$ and that the sequence of functions converges pointwise to $0$. Thus we have to prove: $$\forall\epsilon>0\ \exists N\in\mathbb{N}\ \forall n\geq N: \varphi(f_n)<\epsilon$$

Now my problem: We know nothing about $\varphi$ instead of linearity etc. How can we make an estimation of $\varphi(f_n)$ to prove the result we are searching for?

Can someone help me??

Thanks

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  • $\begingroup$ A hint: under these conditions you can prove that the $f_n$ go to 0 uniformly. You can also show that a positive linear functional must be continuous (hint: $\varphi(1)$ is a finite number). $\endgroup$ – Nate Eldredge Sep 5 '13 at 13:33

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