Geometry of Müntz Spaces and Related Questions (Lecture Notes in Mathematics (1870))

starting Point And Motivation For This Volume Is The Classical Muentz Theorem Which States That The Space Of All Polynomials On The Unit Interval, Whose Exponents Have Too Many Gaps, Is No Longer Dense In The Space Of All Continuous Functions. The Resulting Spaces Of Muentz Polynomials Are Largely Unexplored As Far As The Banach Space Geometry Is Concerned And Deserve The Attention That The Authors Arouse. They Present The Known Theorems And Prove New Results Concerning, For Example, The Isomorphic And Isometric Classification And The Existence Of Bases In These Spaces. Moreover They State Many Open Problems. Although The Viewpoint Is That Of The Geometry Of Banach Spaces They Only Assume That The Reader Is Familiar With Basic Functional Analysis. In The First Part Of The Book The Banach Spaces Notions Are Systematically Introduced And Are Later On Applied For Muentz Spaces. They Include The Opening And Inclination Of Subspaces, Bases And Bounded Approximation Properties And Versions Of Universality. Part I Subspaces and Sequences in Banach Spaces 1 Disposition of Subspaces 1.1 Different Definitions of the Opening of Subspaces 1.2 Inclination 1.3 Connection between the Opening and Inclination 1.4 Conditions for the Closure of Sums of Subspaces 1.5 Projection Constants 1.6 The Projection Function of a Banach Space 2 Sequences in Normed Spaces 2.1 Complete, Supercomplete and {ck }-Complete Sequences 2.2 Minimal and Uniformly Minimal Sequences 2.3 Bases 2.4 Uniformly Convex and Uniformly Smooth Spaces 2.5 Bases in Uniformly Convex Spaces 2.6 Bases of Subspaces 2.7 Stability of Sequences 3 Isomorphisms, Isometries and Embeddings 3.1 Classical Isomorphisms 3.2 The Banach-Mazur Distance 3.3 Minkowski Representationof n-Dimensional Banach Spaces and Limiting Spaces 4 Spaces of Universal Disposition 4.1 Coincidence of Embeddings 4.2 Existence of Spaces of Almost Universal Disposition 4.3 Uniqueness and Universality of Spacesof Almost Universal Disposition 4.4 Synthesis of Sequences 5 Bounded Approximation Properties 5.1 Basic Definitions 5.2 The Shrinking CBAP Part II On the Geometry of Müntz Sequences 6 Coefficient Estimates and the Müntz Theorem 6.1 The Müntz Theorem 6.2 The Clarkson-Erdös Theorem 6.3 Lacunary and Quasilacunary Müntz Sequences 6.4 On "7016c-Completeness of Müntz Sequences 6.5 Coefficient Estimates on [a,b] where a>0 7 Classification and Elementary Propertiesof Müntz Sequences 7.1 Different Classes of 7.2 Iterated Differences 7.3 Elementary Properties of Müntz Sequencesand Polynomials 7.4 Differences of Müntz Sequences 7.5 The Inclination of Positive Octants of Müntz Sequences 8 More on the Geometry of Müntz Sequencesand Müntz Polynomials 8.1 Lorentz-Saff-Varga-Type Theorems 8.2 The Newman Inequality 8.3 A Bernstein-Type Inequality in C 8.4 Some Applications of the Clarkson-Erdös Theorem 9 Operators of Finite Rank and Bases in Müntz Spaces 9.1 Special Finite Rank Operators 9.2 Lacunary Müntz Spaces 9.3 Quasilacunary Müntz Spaces 9.4 Non-Existence of Monotone Basesin Non-dense Müntz Subspaces of C 9.5 Balanced Sequences in C 10 Projection Types and the Isomorphism Problemfor Müntz Spaces 10.1 The Projection Function of Müntz Spaces 10.2 Some Müntz Spaces of L-Projection Type 10.3 Bases in Some Müntz Spaces of L-Projection Type 10.4 Non-Isometric Müntz Spaces in C 11 The Classes [M], A, P and P 11.1 The Classes [M] and A 11.2 The Class P 11.3 Interpolation and Approximationin the Spaces of Class P 11.4 Embeddings of the Spaces X P into c0 11.5 Universality for Finite Müntz Sequences 11.6 Sequences of Universal Disposition 12 Finite Dimensional Müntz Limiting Spaces in C 12.1 Angle Convergence 12.2 Special Limiting Müntz Spaces of Finite Dimension Starting point and motivation for this volume is the Müntz theorem. In the first part of the book the Banach spaces notions are introduced and are later on applied for Müntz spaces. They include the opening and inclination of subspaces, bases and boundedapproximation properties and versions of universality