Non-normable spaces of analytic functions

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.