Function Theory and $\ell ^p$ Spaces (University Lecture Series)

The classical $\ell^{p}$ sequence spaces have been a mainstay in Banach spaces. This book reviews some of the foundational results in this area (the basic inequalities, duality, convexity, geometry) as well as connects them to the function theory (boundary growth conditions, zero sets, extremal functions, multipliers, operator theory) of the associated spaces $\ell^{p}_{A}$ of analytic functions whose Taylor coefficients belong to $\ell^p$. Relations between the Banach space $\ell^p$ and its associated function space are uncovered using tools from Banach space geometry, including Birkhoff-James orthogonality and the resulting Pythagorean inequalities. The authors survey the literature on all of this material, including a discussion of the multipliers of $\ell^{p}_{A}$ and a discussion of the Wiener algebra $\ell^{1}_{A}$. Except for some basic measure theory, functional analysis, and complex analysis, which the reader is expected to know, the material in this book is self-contained and detailed proofs of nearly all the results are given. Each chapter concludes with some end notes that give proper references, historical background, and avenues for further exploration.