A real <positive-semidefinite kernel> is a symmetric function $k:X\times X\to\mathbb R$ such that every finite Gram matrix $(k(x_i,x_j))$ is positive semidefinite. The <Moore-Aronszajn theorem> says that there are a Hilbert space $H$ and a feature map $\phi:X\to H$ such that
$$
k(x,y)=\langle\phi(x),\phi(y)\rangle_H.
$$
Equivalently, $H$ can be chosen as the unique <Reproducing-kernel Hilbert space> with reproducing kernel $k$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 218 / 4 / a / Solution

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{wiki=Reproducing_kernel_Hilbert_space}

Every positive-semidefinite kernel $k$ has a Hilbert-space feature map $\phi$ satisfying $k(x,y)=\langle\phi(x),\phi(y)\rangle$, and determines a unique reproducing-kernel Hilbert space.


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 Incoming links: Moore-Aronszajn theorem