See

   1.  G.L. Litvinov, V.P. Maslov and G.B. Shpiz. Idempotent functional analysis.
An algebraic approach // Mathematical Notes, v. 69, # 5, 2001, p. 696-729. 
E-print math.FA/0009128 (http://ArXiv.org).

   2.  G.L. Litvinov and G.B. Shpiz. Kernel  theorems  and nuclearity in 
idempotent  mathematics.  An  algebraic  approach,  Journal of 
Mathematical Sciences, v. 141, #4, 2007, p. 1417-1428. See also 
E-print arXiv: math.FA/0609033 (http://arXiv.org), 2006.

   3.  G.L. Litvinov. Tropical mathematics, idempotent analysis, classical
mechanics and geometry. - in: Spectral Theory and Geometric Analysis M.Braverman et al., 
Eds.,  AMS Contemporary Mathematics, vol. 535, 2011, p. 159-186. See also E-print 
arXiv: 1005.1247 (http://arXiv.org).