Reproducing kernel Hilbert space

Summary

A reproducing kernel Hilbert space is a Hilbert space of functions on a set $X$. In it, evaluation at each point is a continuous linear functional. By the Riesz representation theorem, evaluation at $x$ is the inner product with a function $k_x$, and $K(x, y) = k_y(x)$ is the reproducing kernel. The kernel is a positive-definite kernel, and by the Moore–Aronszajn theorem each positive-definite kernel comes from exactly one such space. These spaces appear in complex analysis, as Hardy and Bergman spaces, and in statistics and machine learning, through kernel methods.