Introduction to the Theory of Bases (Springer Tracts in Natural Philosophy, 18)

Since the publication of Banach's treatise on the theory of linear operators, the literature on the theory of bases in topological vector spaces has grown enormously. Much of this literature has for its origin a question raised in Banach's book, the question whether every sepa­ rable Banach space possesses a basis or not. The notion of a basis employed here is a generalization of that of a Hamel basis for a finite dimensional vector space. For a vector space X of infinite dimension, the concept of a basis is closely related to the convergence of the series which uniquely correspond to each point of X. Thus there are different types of bases for X, according to the topology imposed on X and the chosen type of convergence for the series. Although almost four decades have elapsed since Banach's query, the conjectured existence of a basis for every separable Banach space is not yet proved. On the other hand, no counter examples have been found to show the existence of a special Banach space having no basis. However, as a result of the apparent overconfidence of a group of mathematicians, who it is assumed tried to solve the problem, we have many elegant works which show the tight connection between the theory of bases and structure of linear spaces. Front Matter....Pages I-XII Linear Transformations....Pages 1-17 Convergence of Series in Banach Spaces....Pages 18-27 Bases for Banach Spaces....Pages 28-54 Orthogonality, Projections and Equivalent Bases....Pages 55-68 Bases and Structure of the Space....Pages 69-78 Bases for Hilbert Spaces....Pages 79-85 Decompositions....Pages 86-97 Applications to the Theory of Banach Algebras....Pages 98-112 Some Results on Generalized Bases for Linear Topological Spaces....Pages 113-129 Back Matter....Pages 130-151