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What is the dual space of $X = (C_\infty(\mathbb{R}^n),\|\cdot\|_\infty)$, the continous functions that approach zero at infinity? Is it reflexive?

Additionally, is there a Banach space $Y$ such that $X=Y'$?

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The dual is the space of finite signed measures on $\mathbb R^n$. No, it is not reflexive. No, it is not a dual of a Banach space. It looks like you need to read a textbook on Banach spaces... –  GEdgar Feb 18 '13 at 14:36
    
Probably I do need precisely that. Any suggestions? –  Nemis L. Feb 18 '13 at 14:52
    
Think yourself about the "discrete analogue" $c_0$. –  Jochen Feb 18 '13 at 16:23

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