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  "body": "\n## Summary\n\nAn integral operator is a [[linear map|0NRYHJPQG2]] between spaces of functions that acts by integration against a kernel: $(Tf)(x) = \\int K(x, y) f(y)\\, dy$.\nIntegral operators are the infinite-dimensional analogue of matrices.\nThey appear as [[integral transforms|YCK709QB06]], as solution operators of differential equations, and in [[Volterra integral equations|DND8CBEVXT]] and Fredholm integral equations.\nWith a square-integrable kernel they are [[compact operators|M2NEP1K5AE]], which gives them a spectral theory close to that of matrices.\n\n## Explanations\n\n### L1 human\n\nA matrix turns a vector $y$ into the vector with entries $\\sum_j K_{ij} y_j$.\nAn integral operator does the same with functions, and the sum becomes an integral.\nThe simplest example is $(Vf)(x) = \\int_0^x f(y)\\, dy$ on $[0, 1]$, which takes a function to its running total.\nIts kernel is $1$ when $y \u003c x$ and $0$ otherwise, a \"triangular matrix\" with infinitely many entries.\n",
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