Mercer theorem for a limit of Reproducing Kernel Hilbert Spaces

Let (X n ) n0 be an increasing sequence of compact metric spaces, let K n be a Mercer kernel on X n for any n N and let H n be the reproducing kernel Hilbert space associated with K n for any n N. In this paper, we construct a reproducing kernel Hilbert space H , on X = + n=0 X n from the sequence (H n ) n0 and determine its reproducing kernel.We end by establishing a version of Mercer theorem in this non-compact context.