Order Structures

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The present chapter is devoted to a systematic study of order structures within the framework of topological vector spaces. No attempt has been made to give an account of the extensive literature on Banach lattices, for a survey of which we refer the reader to Day [2], nor is any special emphasis placed on ordered normed spaces. Our efforts are directed towards developing a theory that is in conformity with the modern theory of topological vector spaces, that is to say, a theory in which duality plays the central role. This approach to ordered topological vector spaces is of fairly recent origin, and thus cannot be presented in a form as definite as a mature theory; it is nonetheless hoped that the reader who has encountered parts of it in the literature (e.g., Gordon [1], [2], Kist [1], Namioka [1], Schaefer [1]–[5]) will obtain a certain survey of the methods available and of the results to which they lead. The fact that ordered topological vector spaces abound in analysis is perhaps motivation enough for a systematic study; beyond this, the present chapter is followed by an appendix intended to illustrate some applications to spectral theory. As in the preceding chapters, further information can be found in the exercises.KeywordsVector LatticeBanach LatticeTopological Vector SpacePositive ConeOrder IntervalThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.