Mercer's theorem

Mercer's theorem is one of the most popular results of the work of James Mercer. It is used for the kernel trick, an important method in machine learning.

Mercer's theorem states that any positive definite kernel K(x, y) can be expressed as a dot product in a high-dimensional space.

More specifically, if a kernel is positive semi-definite, i.e.,

then there exists a function whose image is in an inner product space of possibly high dimension, such that