Classical Banach Spaces I: Sequence SpacesThe appearance of Banach's book [8] in 1932 signified the beginning of a syste matic study of normed linear spaces, which have been the subject of continuous research ever since. In the sixties, and especially in the last decade, the research activity in this area grew considerably. As a result, Ban:ach space theory gained very much in depth as well as in scope: Most of its well known classical problems were solved, many interesting new directions were developed, and deep connections between Banach space theory and other areas of mathematics were established. The purpose of this book is to present the main results and current research directions in the geometry of Banach spaces, with an emphasis on the study of the structure of the classical Banach spaces, that is C(K) and Lip.) and related spaces. We did not attempt to write a comprehensive survey of Banach space theory, or even only of the theory of classical Banach spaces, since the amount of interesting results on the subject makes such a survey practically impossible. |
Contents
Schauder Bases | 1 |
The Spaces co and | 53 |
b Absolutely Summing Operators and Uniqueness of Unconditional | 63 |
Fredholm Operators Strictly Singular Operators and Complemented | 75 |
Subspaces of co and I and the Approximation Property Complement | 84 |
e Banach Spaces Containing 1 or | 95 |
f Extension and Lifting Properties Automorphisms of l co and l₁ | 105 |
Symmetric Bases | 111 |
Orlicz Sequence Spaces | 124 |
| 180 | |
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Common terms and phrases
42-condition at zero anxn approximation property assume automorphism bases basic sequence basis constant biorthogonal system bounded linear operator boundedly complete choice of scalars compact operator conclude the proof constant of x}=1 construct decomposition defined definition denote dual easily checked elements example exists fact finite rank operator Fredholm operator Hence implies increasing sequence inequality infinite dimensional subspace infinite-dimensional Banach space integers isometric Lemma linear span Lorentz sequence space M₁ matrix N₁ non-equivalent normalized block basis normalized unconditional basis Observe Orlicz sequence spaces permutation Proposition prove quotient map quotient space reflexive reflexive space satisfies the 42-condition scalars a}=1 Schauder basis separable Banach space separable space sequence of integers sequence of scalars sequence x}=1 shrinking subsequence subset subsymmetric symmetric basis T₁ Theorem topology U₁ unconditional constant unique unit ball unit vector basis X₁ Y₁