In classical functional analysis, one can construct a reproducing kernel Hilbert space by starting with a positive definite kernel, say
$K: [0,1]\times [0,1] \rightarrow \mathbb{R}$.
One then creates linear combinations of the form
$f(x) = \sum^n a\_i k(x\_i,x)$, together with an inner product


$\langle f,g \rangle = \sum \sum a\_i b\_j k(x\_i,x\_j),$


and completes the space as usual.


As far as I can see, the 'essential' properties of a reproducing kernel Hilbert space are
the ability to represent linear operators as integral kernels, and the Riesz representation theorem.


I'm just wondering if there is a similar construction to the one outlined above in the framework of the [max-plus](https://en.wikipedia.org/wiki/Max-plus_algebra) algebra. Both of the properties I mentioned above have max-plus analogues (see [this](https://arxiv.org/pdf/math/0507014v1.pdf) introduction). If such a construction exists, how far is it possibe to take it? Is there an idempotent version of Mercer's theorem, for example?



Is there a tropical analogue of a reproducing kernel Hilbert space?
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](https://arxiv.org/abs/math/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 [math.FA/0609033](https://arxiv.org/abs/math/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: [arXiv: 1005.1247](https://arxiv.org/abs/1005.1247/) (<http://arXiv.org>).


