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.
Table of ContentsI. Linear Transformations.- 1. Linear Topological Spaces.- 2. Linear Transformations.- 3. Conjugate Spaces and Weak Topologies.- 4. Special Banach Spaces.- II Convergence of Series in Banach Spaces.- 1. Relations among Different Types of Convergence.- 2. Unconditional and Absolute Convergence.- III. Bases for Banach Spaces.- 1. Bases Corresponding to Different Topologies.- 2. Biorthogonal Systems.- 3. Shrinking and Boundedly Complete Bases.- 4. Unconditional Bases.- 5. Absolutely Convergent Bases and Uniform Bases.- 6. T-Bases.- 7. Bases for Special Spaces.- IV. Orthogonality, Projections and Equivalent Bases.- 1. Bases and Projections.- 2. Orthogonality, simple N1-Spaces and Monotone Bases.- 3. Equivalent Bases.- V. Bases and Structure of the Space.- 1. Bases, Completeness and Separability.- 2. Bases and Reflexivity.- 3. Criteria for Finite Dimension.- VI. Bases for Hilbert Spaces.- 1. Monotone and Orthonormal Bases.- 2. Unconditional Bases for Hilbert Spaces.- VII. Decompositions.- 1. Decompositions of F-Spaces.- 2. Decompositions of Banach Spaces.- VIII. Applications to the Theory of Banach Algebras.- 1. Two-Sided Ideals of Operators of Finite Rank.- 2. ?-Rings.- 3. Proper ?-Rings of Schauder Decompositions.- 4. Minimal Schauder Decompositions.- 5. Banach Algebras and Unconditional Bases.- IX. Some Results on Generalized Bases for Linear Topological Spaces.- 1. Definition and Fundamental Properties of Generalized Bases.- 2. Dual Generalized Bases.- 3. Examples.- 4. Similar Bases.- 5. Continuity of the Coefficient Functionals.- Author and Subject Index.