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