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  "body": "\n## Summary\n\nA sequence of operators $K_n$ is a collectively compact approximation of an operator $K$ if $K_n \\to K$ pointwise.\nIn addition, the images of the unit ball under all $K_n$ together must form a relatively compact set.\nThis weaker notion replaces norm convergence, which often fails for numerical methods such as the Nyström method.\nIt still gives the main conclusion when $I - K$ is invertible.\nThen $I - K_n$ is invertible for large $n$, with uniformly bounded inverses, and the approximate solutions converge.\nThe theory was developed by P. M. Anselone for the numerical solution of integral equations.\n\n## Explanations\n\n### L1 human\n\nIn the Nyström method one replaces an integral by a weighted sum over finitely many points.\nThe resulting operators do not get close to the integral operator in the strongest sense, because a function can be large between the quadrature points.\nBut all of them smooth functions in the same uniform way, and that is enough for the approximate equations to have solutions close to the true one.\nA simple example: with the rectangle rule, $(K_n\\varphi)(x) = \\frac{1}{n} \\sum_{j=1}^n k(x, \\frac{j}{n}) \\varphi(\\frac{j}{n})$ approximates $\\int_0^1 k(x, t) \\varphi(t) \\, dt$.\n",
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