Non-normable spaces of analytic functions

TL;DR

This paper constructs explicit examples of functions in Hardy and Bergman spaces for 0<p<1 that violate the triangle inequality, simplifying proofs for Hardy spaces and providing new insights for Bergman spaces.

Contribution

It offers the first known explicit examples of non-normable functions in Bergman spaces and simplifies existing proofs for Hardy spaces.

Findings

Explicit non-normable functions in Hardy spaces for 0<p<1

First known examples in Bergman spaces for 0<p<1

Simplified proof of non-normability in Hardy spaces

Abstract

For each value of $p$ such that $0<p<1$, we give a specific example of two functions in the Hardy space $H^p$ and in the Bergman space $A^p$ that do not satisfy the triangle inequality. For Hardy spaces, this provides a much simpler proof than the one due to Livingston that involves abstract functional analysis arguments and an approximation theorem. For Bergman spaces, we have not been able to locate any examples or proofs in the existing literature.