Closure property of positive-semidefinite kernels

Nonnegative scalar multiples, sums, pointwise products, pointwise limits, and pullbacks of positive-semidefinite kernels are positive semidefinite whenever the displayed expressions are defined.

The pointwise product of two positive-semidefinite kernels is positive semidefinite because its finite kernel matrix is the entrywise product of two positive-semidefinite matrices, to which the Schur product theorem applies.