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We present accurate upper and lower bounds for the covering numbers, withexplicit constants, of the unit ball for two general classes of ReproducingKernel Hilbert Space (RKHS) on the unit sphere of $\mathbb{R}^{d+1}$. In bothclasses, the RKHS is generated by an isotropic continuous positive definitekernel. The upper and lower bounds we present carry precise information aboutthe asymptotic constants, depending on the dimension of the sphere and themonotonic behavior of the Schoenberg/Fourier coefficients of the isotropickernel.

A close look at the entropy numbers of the unit ball of the Reproducing Hilbert Space of isotropic positive definite kernels