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  "body": "\n## Summary\n\nA reproducing kernel Hilbert space is a [[Hilbert space|HM2HHE3J51]] of functions on a set $X$.\nIn it, evaluation at each point is a continuous linear functional.\nBy the [[Riesz representation theorem|TTC40N49M6]], evaluation at $x$ is the inner product with a function $k_x$, and $K(x, y) = k_y(x)$ is the reproducing kernel.\nThe kernel is a positive-definite kernel, and by the Moore–Aronszajn theorem each positive-definite kernel comes from exactly one such space.\nThese spaces appear in complex analysis, as Hardy and Bergman spaces, and in statistics and machine learning, through kernel methods.\n\n## Explanations\n\n### L1 human\n\nIn such a space, functions that are close in norm are also close at every single point.\nThe value of a function at a point can be read off by taking the inner product with one special function for that point.\nA simple example: the vectors of $\\C^n$, seen as functions on $\\set{1, \\dots, n}$, form such a space.\nThe special function for the point $i$ is the unit vector $e_i$.\n",
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