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Adaptive dynamics so far has been put on a rigorous footing only for clonalinheritance. We extend this to sexually reproducing diploids, althoughadmittedly still under the restriction of an unstructured population withLotka-Volterra-like dynamics and single locus genetics (as in Kimura's 1965infinite allele model). We prove under the usual smoothness assumptions,starting from a stochastic birth and death process model, that, whenadvantageous mutations are rare and mutational steps are not too large, thepopulation behaves on the mutational time scale (the 'long' time scale of theliterature on the genetical foundations of ESS theory) as a jump process movingbetween homozygous states (the trait substitution sequence of the adaptivedynamics literature). Essential technical ingredients are a rigorous estimatefor the probability of invasion in a dynamic diploid population, a rigorous,geometric singular perturbation theory based, invasion implies substitutiontheorem, and the use of the Skorohod $M_1$ topology to arrive at a functionalconvergence result. In the small mutational steps limit this process in turngives rise to a differential equation in allele or in phenotype space of a typereferred to in the adaptive dynamics literature as 'canonical equation'.

A rigorous model study of the adaptative dynamics of Mendelian diploids