In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving integral operator equations. Since then, positive-definite functions and their various analogues and generalizations have arisen in diverse parts of mathematics. They occur naturally in Fourier analysis, probability theory, operator theory, complex function-theory, moment problems, integral equations, boundary-value problems for partial differential equations, machine learning, embedding problem, information theory, and other areas. Let be a nonempty set, sometimes referred to as the index set. A symmetric function is called a positive-definite (p.d.) kernel on if holds for any , given . In probability theory, a distinction is sometimes made between positive-definite kernels, for which equality in (1.1) implies , and positive semi-definite (p.s.d.) kernels, which do not impose this condition. Note that this is equivalent to requiring that any finite matrix constructed by pairwise evaluation, , has either entirely positive (p.d.) or nonnegative (p.s.d.) eigenvalues. In mathematical literature, kernels are usually complex valued functions, but in this article we assume real-valued functions, which is the common practice in applications of p.d. kernels. For a family of p.d. kernels The conical sum is p.d., given The product is p.d., given The limit is p.d. if the limit exists. If is a sequence of sets, and a sequence of p.d. kernels, then both and are p.d. kernels on . Let . Then the restriction of to is also a p.d. kernel. Common examples of p.d. kernels defined on Euclidean space include: Linear kernel: . Polynomial kernel: . Gaussian kernel (RBF kernel): . Laplacian kernel: . Abel kernel: . Kernel generating Sobolev spaces : , where is the Bessel function of the third kind. Kernel generating Paley–Wiener space: . If is a Hilbert space, then its corresponding inner product is a p.