Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures

We present new classes of positive definite kernels on non-standard spaces that are integrally strictly positive definite or characteristic. In particular, we discuss radial kernels on separable Hilbert spaces, and introduce broad classes of kernels on Banach spaces and on metric spaces of strong negative type. The general results are used to give explicit classes of kernels on separable $L^p$ spaces and on sets of measures.