  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.
