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I would like to determine the dualspace of some normed vectorspace.

Namley, $$c_0:=\{x=(x_n)_{n\mathbb N}\subset\mathbb R:\lim_{n\rightarrow\infty}x_n=0\; \text{ and } ||x||=\sup_n|x_n|\}$$

I hoped someone could give me a general intuition or a hint how to start finding the dual space of this vectorspace.

EDIT I am pretty sure that I can not give the explicit dual-space but I have to find some "known" vectorspace which is isomorphic to the dual-space of $c_0$.

Thanks

Thorben
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  • No, this isn't possible without more details. Some are very simple, some are very complicated (e.g. http://math.stackexchange.com/questions/47395/the-duals-of-l-infty-and-l-infty). If you make it clear in you question that you just want a small hint, and not a complete solution, then people will be sure to pay attention to this request. – Frank Mar 28 '14 at 15:34
  • @Frank Thanks for the comment. I am sorry, I hoped that there is a general way. I just edited the post. – Thorben Mar 28 '14 at 15:50
  • Here's a hint: what is the dual space for $\ell^p$ if $p>1$? This can help guide you to an answer. – Cameron Williams Mar 28 '14 at 15:57
  • @CameronWilliams From $L^p$ spaces I know that the dualspace is $L^q$ where $q$ is conugate with p i.e $1/p+1/q=1$. Does this also hold for spaces of sequences? Then, since $c_0\subset l^{\infty}$ it must have something to do with the dualspace of $l^{\infty}$? – Thorben Mar 28 '14 at 16:17
  • @Thorben It also holds for sequences as well. $L^p$ spaces are very nice in that respect. The dual of $\ell^{\infty}$ is very, very bizarre which is why you are looking at $c_0$. However, with the fact that $(\ell^p)^{*} \cong \ell^p$ for $p>1$, this suggests maybe that the dual of $c_0$ is somehow related to $\ell^1$ since $(\ell^1)^ = \ell^{\infty}$ and $c_0$ lies inside of $\ell^{\infty}$. – Cameron Williams Mar 28 '14 at 16:58

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