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

TL;DR

This paper provides an accessible proof of an isometric algebra embedding from $H^(\u0019)$ into $L^()$ without using Fatou's theorem, relying solely on basic complex and functional analysis.

Contribution

It offers a self-contained, elementary proof of the embedding, avoiding the classical non-tangential limits approach.

Findings

Established the existence of the embedding without Fatou's theorem

Provided a proof accessible to undergraduates

Clarified the relationship between $H^()$ and $L^()$

Abstract

The purpose of this note is to show in an accessible and self-contained way the existence of an isometric algebra embedding from $H^\infty(\D)$ into $L^\infty(\T)$, 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.