The bidual of C(X) I (North-Holland mathematics studies)

The most commonly occurring, and probably the most important, non-reflexive real Banach spaces' (all our spaces are over the reals) are those of the form [italic capital]C([italic capital]X), [italic capital]X compact, and those of the form L¹ ([lowercase Greek]Mu), [lowercase Greek]Mu a given measure. [italic capital]C([italic capital]X) and its bidual can be studied through their ring structure or, equivalently, their vector-space-cum-lattice structure. We follow the latter route , examining them as Banach lattices, that is, norm complete normed Riesz spaces. Part I is an introduction to Riesz spaces, with the concepts and notations we will use. Part II does the same for Ml-spaces (spaces of the form [italic capital]C([italic capital]X)) and L-spaces (those of the form L¹ ([lowercase Greek]Mu)). Part III consists of classical material on [italic capital]C([italic capital]X), its dual, and bidual. The present work is concerned with the embedding of a space [italic capital]C([italic capital]X) in its bidual [italic capital]Cʹʹ([italic capital]X) and the study of what properties of the latter can be found starting from this embedding The most commonly occurring, and probably the most important, non-reflexive real Banach spaces' (all our spaces are over the reals) are those of the form [italic capital]C([italic capital]X), [italic capital]X compact, and those of the form L1 ([lowercase Greek]Mu), [lowercase Greek]Mu a given measure. [italic capital]C([italic capital]X) and its bidual can be studied through their ring structure or, equivalently, their vector-space-cum-lattice structure. We follow the latter route , examining them as Banach lattices, that is, norm complete normed Riesz spaces. Part I is an introduction to Riesz spaces, with the concepts and notations we will use. Part II does the same for Ml-spaces (spaces of the form [italic capital]C([italic capital]X)) and L-spaces (those of the form L1 ([lowercase Greek]Mu)). Part III consists of classical material on [italic capital]C([italic capital]X), its dual, and bidual. The present work is concerned with the embedding of a space [italic capital]C([italic capital]X) in its bidual [italic capital]Cʹʹ([italic capital]X) and the study of what properties of the latter can be found starting from this embedding Content: Edited by Page iii Copyright page Page iv Preface Pages v-viii Chapter 1 Riesz Spaces Pages 2-49 Chapter 2 Riesz Space Duality Pages 50-91 Part II L-Spaces and MII-Spaces Page 92 Chapter 3 MII-Spaces and L-Spaces Pages 93-119 Chapter 4 Dual L-Spaces and MII-Spaces Pages 120-143 Chapter 5 Duality Relations Between an L-Space and its Dual Pages 144-163 Chapter 6 (C(X),X)-Duality Pages 165-185 Chapter 7 (C(X),C′(X))-Duality Pages 186-205 Chapter 8 C′′(X) Pages 206-211 Chapter 9 The Fundamental Subspaces of C′′(X) Pages 213-247 Chapter 10 The Operators u and l Pages 248-274 Chapter 11 U(X) Pages 275-300 Part V Riemann Integration Page 301 Chapter 12 The Riemann Subspace of a Band Pages 302-314 Chapter 13 Riemann Integrability Pages 315-334 Part VI The Rare Elements Page 335 Chapter 14 The Lean Elements Pages 336-347 Chapter 15 The Rare Elements Pages 348-362 Chapter 16 The Decomposition C′(X) = Ra(X) c ⊕ C(X) c Pages 363-376 Chapter 17 The Dedekind Completion of C(X) Pages 377-388 Chapter 18 The Meager Elements Pages 389-409 Bibliography Pages 410-414 Index of Symbols Pages 415-417 Index of Terminology Pages 418-423