Embedding $H^{\mathrm{\infty }}(\text{D})$ into $L^{\mathrm{\infty }}(\text{T})$: a proof without non-tangential limits

The purpose of this note is to show in an accessible and self-contained way the existence of an isometric algebra embedding from $H^{\mathrm{\infty }}(\text{D})$ into $L^{\mathrm{\infty }}(\text{T})$, without appealing to Fatou’s classical theorem on non-tangential limits of analytic functions, and relying only on results from complex and functional analysis that are typically covered in a standard undergraduate course.