I am trying to determine some properties of Lipschitz distributions. To do so, I need to know the dual space for $l^\infty$. The sequences tending to zero are certainly in the dual space to $l^\infty$, but are there other elements?
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$\begingroup$ You need to tell us more if you want an answer. What is $l^\infty$? On what vector space? for what topology? How do you define the dual? $\endgroup$ – Joël Oct 25 '13 at 21:25
Assuming you mean $l^\infty = l^\infty(\mathbb N)$ as a Banach space over a complete nonarchimedean field, then the dual space is $c_0(\mathbb{N})$.
See Grusonvan der Put, Theorem (4.3), in Banach spaces. Mémoires de la Société Mathématique de France, 3940 (1974), p. 55100, page 75.
In the answer by user41871, one should add the assumption that the field $K$, over which the spaces are considered, is NOT spherically complete, for example $K=\mathbb C_p$ (this condition is present in the paper by Gruson and van der Put). Otherwise (for example, for $K=\mathbb Q_p$) the dual is not of countable type. For all the details on this and related subjects see
C. PerezGarcia and W.H. Schikhof. Locally convex spaces over nonArchimedean valued fields. Cambridge University Press, 2010.