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  "body": "\n## Summary\n\nAn inner function is a bounded [[holomorphic function|0DG0Z72CQD]] on the [[unit disc|8G5ABXRK85]] whose boundary values have modulus $1$ almost everywhere on the circle.\nInner functions are one half of the [[inner-outer factorization|X6ZQPAQF76]] of functions in [[Hardy spaces|PNNNFJVF6W]].\nThey describe the closed subspaces of $H^2$ that are invariant under the shift, by [[Beurling's theorem|2V1DXXM9FA]].\n\n## Explanations\n\n### L1 human\n\nThink of an inner function as a map of the disc into itself that pushes almost all of the boundary circle onto the boundary circle.\nThe simplest example is $f(z) = z$.\nEvery [[Blaschke product|9N6R5ADJXT]] is inner, and its zeros can be placed almost anywhere in the disc.\nThere are also inner functions without zeros, such as $\\exp\\left(-\\frac{1 + z}{1 - z}\\right)$.\nAn inner function carries the \"zeros and singular part\" of a function, while its modulus on the circle carries no information.\n",
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