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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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closed as unclear what you're asking by BS., j.c., Carlo Beenakker, Daniel Moskovich, Willie Wong Oct 28 '13 at 8:41

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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
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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 Gruson-van der Put, Theorem (4.3), in Banach spaces. Mémoires de la Société Mathématique de France, 39-40 (1974), p. 55-100, page 75.

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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. Perez-Garcia and W.H. Schikhof. Locally convex spaces over non-Archimedean valued fields. Cambridge University Press, 2010.

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