See
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/0009128math.FA/0009128 (http://ArXiv.org).
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 math.FA/0609033math.FA/0609033 (http://arXiv.org), 2006.
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: arXiv: 1005.1247arXiv: 1005.1247 (http://arXiv.org).