This book surveys results concerning bases and various approximation properties in the classical spaces of analytical functions. It contains extensive bibliographical comments.

Readership

Reviews

"An almost complete exposition of the results concerning linear topological properties of Banach spaces of analytic functions (mainly of the disc algebras \(A\) and Hardy spaces \(H^p\)) obtained up to 1975 ... Written by one of the pioneers of the theory discussed, who has contributed very much to it.

"The book is worth reading for anyone who enjoys the interplay between function theory and functional analysis."

-- S. V. Kisljakov, Mathematical Reviews

Table of Contents

• Preliminaries
• The F. and M. Riesz theorem and duals of the disc algebra
• Absolutely summing operators from the disc algebra
• Absolutely summing operators from the disc algebra into Hilbert space
• The nonexistence of local unconditional structure for the disc algebra and for its duals
• Application to uniform algebras
• Uniformly peaking families of functions in \(A\) and \(H^\infty\). The Havin lemma
• Characterizations of weakly compact sets in \(L^1/H^1_0\) and in \(A^*\)
• Weakly compact operators from \(A\), \(L^1/H^1_0\) and \(A^*\) and complemented subspaces of these spaces
• Complementation of finite dimensional subspaces in \(A\), \(L^1/H^1_0\) and <\(H^\infty\)
• Bases and the approximation property in some spaces of analytic functions
• The polydisc algebra and the \(n\)-ball algebra, and their duals
• References