A linear mapping T from the Archimedean Riesz space E into itself is called an orthomorphism of E if it is order bounded and band preserving (that is T(B) £ B for every band B in E). The band preserving property is easily seen to be equivalent to the statement that x J_ y in E implies that Tx ± y (we recall that x _L y means |x| A \y\ = 0). Orthomorphisms have been studied by many authors and it is not possible to quote all of them in our references.The starting point of the theory consists of proving that the collection Orth (E) of all orthomorphisms of £ is a Riesz space under pointwise operations. Several proofs of this fact are known, but some make use of non-trivial representation theorems (for example [8,9]), others involve the use of the Dedekind completion (for example [1,4]), and another one is elementary but rather technical [2]. In this short paper we give a new proof, probably the most simple and the most elegant (but not completely elementary because of its use of Zorn's Lemma, via a result on "Riesz homomorphisms").However, instead of proving the result in question directly, we first establish a more general result which is interesting on its own and from which it will immediately follow that Orth (£), and even Orth 00 (E) (the extended orthomorphisms in the sense of [4,5]), is a Riesz space. This more general result has already been obtained by Meyer [7,8] by means of a representation theorem, but the proof given here avoids this representation theorem.
The maximum and anti-maximum principles are extended to the case of eigenvalue Sturm–Liouville problemswith boundary conditions of Dirichlet type (if possible) on a bounded interval [a, b]. The function r is assumed to be continuous and > 0 on ]a, b[, but the function 1/r is not necessarily integrable on [a, b]. The conditions on the functions p, m and h depend on the integrability or nonintegrability of 1/r on [a, c] and/or [c, b], for some c ∈ ]a, b[. The weight function m is not necessarily of constant sign.
The concept of a-directedness in a partially ordered Banach space X was first introduced by Asimow [1], Some studies of this property have since been made by Rogalski [7] and Wickstead [8]. In [8], it is shown that a-directedness is equivalent to the order-boundedness of compact subsets of X and is relevant to the positively generating property of the spaces of compact operators into X. In this paper, we generalize the result of Rogalski and Wickstead to the setting of metrizable locally convex spaces.From now and onward, E will denote a real locally convex space partially ordered by a wedge C (i.e. C # 0, C + C £ C and R+. C £ C; if C n -C = {0}, C is called a cone). E' is the topological dual of E and is partially ordered by the dual wedge C. H(C) denotes the space of all positively homogeneous real-valued functions on C \ each of which is >v*-continuous on every equicontinuous subset of C'\ H(C) is equipped with the topology of uniform convergence on equicontinuous subsets of C . The evaluation map is obviously continuous. °U will denote the system of all neighbourhoods of 0 in E.E is said to be quasi-directed if for each U e °ll there exists V e °ll such that V is directed by U in the sense that each finite subset of V is majorized by some element of U. E is said to be additive if each U e / contains some convex V e °U such that the Minkowski functional p y of V is additive on C.If F is any locally convex space, P(F, E) will denote the space of all precompact operators from F into E and K(F, E) will denote the space of compact operators. THEOREM 1. Suppose that E is pseudo-metrizable and that C is closed. Then the following statements are equivalent.(1°) Every convergent sequence in E is majorized.(2°) E is quasi-directed and every increasing Cauchy sequence in E is majorized.(4°) Every precompact subset of E is majorized.(5°) Every compact (not necessarily Hausdorff) subset of E is majorized.(6°) P(F, E) is positively generated whenever F is an additive partially ordered locally convex space.(7°) P(F, E) is positively generated whenever F is a base semi-normed space with a closed cone. (If E is Hausdorjf, it suffices to consider normed spaces F).Proof. We first show that (1°) implies that E is quasi-directed. Suppose E is not quasi-directed and take a decreasing countable neighbourhood-base (U n ) at 0 in E.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.