factorization theorem for $H^{\mathrm{\infty }}$ functions

Let $H^{\mathrm{\infty }}$ denote the bounded analytic functions on the unit disc.

Theorem.

Every $f\mathrm{\in }{H}^{\mathrm{\infty }}$ can be written as

$$f(z)=\alpha I(z)F(z)$$

where $\mathrm{|}\alpha \mathrm{|}\mathrm{=}\mathrm{1}$, $I$ is an inner function and $F$ is a bounded outer function. Conversely any function which can be so written is bounded.