---
tag: NN8XH5RJYB
kind: concept
name: Inner function
msc: [30J05, 30H10]
sources:
  - url: https://en.wikipedia.org/wiki/Hardy_space
---

## Summary

An inner function is a bounded [[holomorphic function|0DG0Z72CQD]] on the [[unit disc|8G5ABXRK85]] whose boundary values have modulus $1$ almost everywhere on the circle.
Inner functions are one half of the [[inner-outer factorization|X6ZQPAQF76]] of functions in [[Hardy spaces|PNNNFJVF6W]].
They describe the closed subspaces of $H^2$ that are invariant under the shift, by [[Beurling's theorem|2V1DXXM9FA]].

## Explanations

### L1 human

Think of an inner function as a map of the disc into itself that pushes almost all of the boundary circle onto the boundary circle.
The simplest example is $f(z) = z$.
Every [[Blaschke product|9N6R5ADJXT]] is inner, and its zeros can be placed almost anywhere in the disc.
There are also inner functions without zeros, such as $\exp\left(-\frac{1 + z}{1 - z}\right)$.
An inner function carries the "zeros and singular part" of a function, while its modulus on the circle carries no information.