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?
2 Answers
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.
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.