Finitely Extendable Functionals on Vector Lattices

If the vector space \({\mathcal{L}}^r(E, F)\) of all regular operators between the vector lattices E and F is ordered by the collection of its positive operators, then the Dedekind completeness of F is a sufficient condition for \({\mathcal{L}}^r(E, F)\) to be a vector lattice. \({\mathcal{L}}^r(E, F)\) and some of its subspaces might be vector lattices also in a more general situation. In the paper we deal with ordered vector spaces \({\mathcal{V}}(E, F)\) of linear operators and ask under which conditions are they vector lattices, lattice-subspaces of the ordered vector space \({\mathcal{L}}^r(E, F)\) or, in the case that \({\mathcal{L}}^r(E, F)\) is a vector lattice, sublattices or even Banach lattices when equipped with the regular norm. The answer is affirmative for many classes of operators such as compact, weakly compact, regular AM-compact, regular Dunford-Pettis operators and others if acting between appropriate Banach lattices. Then it is possible to study the finite elements in such vector lattices \({\mathcal{V}}(E, F)\), where F is not necessary Dedekind complete. In the last part of the paper there will be considered the question how the order structures of E, F and \({\mathcal{V}}(E, F)\) are mutually related. It is also shown that those rank one and finite rank operators, which are constructed by means of finite elements from E′ and F, are finite elements in \({\mathcal{V}}(E,F)\). The paper contains also some generalization of results obtained for the case \({\mathcal{V}}(E, F) = {\mathcal{L}}^r(E, F)\) in [10].

Riesz homomorphisms between vector lattices are characterized by interval preserving order adjoints. Buskes, van Rooij, and van Haandel generalized the notion of a Riesz homomorphism to linear maps between general ordered vector spaces. In this paper, we discuss how far the characterization between vector lattices extends to directed ordered vector spaces and pre-Riesz spaces.

Let A" be a set, A a family of real-valued functions on X which contains the constants, pA the weak uniformity generated by A, and U{nAX) the collection of uniformly continuous functions to the real line R.The problem is how to construct U(\iAX) from A. The main result here is: For A a vector lattice, the collection of suprema of countable, finitely Aequiuniform, order-one subsets of A + is uniformly dense in U(pAX).Two less technical corollaries: If A is a vector lattice (resp., vector space), then the collection of functions which are finitely A-uniform and uniformly localfy-A (resp., uniformly locally piecewise-A) is uniformly dense in U(pAX).Further, for any A, a finitely A-uniform function is fust a composition F (a,,... ,af) for some a{.^ e A and F uniformly continuous on the range of (fli.ap) in Rf.Thus, such compositions are dense in U(iiAX).For BU(iiAX), the compositions with F e BU(Rr) are dense (B denoting bounded functions).So, in a sense, to know U(pAX) it suffices to know A and subspaces of the spaces Rp, and to know BU(pAX), A and the spaces Rr suffice.In case A is a vector lattice and A ** BA (i.e., A consists of bounded functions), the problem of describing U(pAX) in terms of A has an elegant solution: A is uniformly dense in U(pAX) (i.e., if / G U(pAX) and t > 0 there is a E A with \f(x) -a(x)\ < e for all x E A").This can be seen to be essentially equivalent to the usual Stone-Weierstrass Theorem.The result appears to have been first published in 1955 by Maak [M] and Nbeling and Bauer [NB]; we give a short proof in the course of proving our main theorem.If A contains unbounded functions, the situation is more complicated: A = U(R)\N is a "uniformly closed" vector lattice, while pA is (uniformly) discrete.With X = R, let A be the piecewise linear functions (finitely many pieces); pA is the usual uniformity on R and the closure of A consists only of continuous eventually linear functions.

We present some of the classical inequalities in analysis in the context of Archimedean (real or complex) vector lattices and f-algebras. In particular, we prove an identity for sesquilinear maps from the Cartesian square of a vector space to a geometric mean closed Archimedean vector lattice, from which a Cauchy–Schwarz inequality follows. A reformulation of this result for sesquilinear maps with a geometric mean closed semiprime Archimedean f-algebra as codomain is also given. In addition, a sufficient and necessary condition for equality is presented. We also prove a Hölder inequality for weighted geometric mean closed Archimedean Φ-algebras, substantially improving results by K. Boulabiar and M. A. Toumi. As a consequence, a Minkowski inequality for weighted geometric mean closed Archimedean Φ-algebras is obtained.

A.G. Vulih has shown how an essentially unique intrinsic multiplication can be defined in certain types of Riesz spaces (vector lattices) L. In general, the multiplication is not universally defined in L, but L can always be imbedded in a large space L# in which multiplication is universally defined. If ф is a normal integral in L, then ф can be extended to a normal integral on a large space L1(ф) in L#, and L1(ф) may be regarded as an abstract integral space. A very general form of the Radon-Nikodym theorem can be proved in L1(ф), and this can be used to give a relatively simple proof of a theorem of Segal giving a necessary and sufficient condition that the Radon-Nikodym theorem hold in a measure space. In another application, the multiplication is used to give a representation of certain Riesz spaces as rings of operators on a Hilbert space.

§1. Introduction.The motivation of this work comes from two different directions: infinite abelian groups, and ordered algebraic structures. A challenging problem in both cases is that of classification. In the first case, it is known for example (cf. [KA]) that the classification of abelian torsion groups amounts to that of reducedp-groups by numerical invariants called theUlm invariants(given by Ulm in [U]). Ulm's theorem was later generalized by P. Hill to the class of totally projective groups. As to the second case, let us consider for instance the class of divisible ordered abelian groups. These may be viewed as ordered ℚ-vector spaces. Their theory being unstable, we cannot hope to classify them by numerical invariants. On the other hand, being o-minimal, the theory enjoys several good model theoretic properties (cf. [P-S]), so the search for some reasonable invariants is well motivated. The common denominator of the two cases, as well as of many others, is valuation theory. Indeed given an ordered vector space, one can consider it as a valued vector space, endowed with the natural valuation. Also, the socleG[p] of a reduced abelianp-groupG, endowed with the height functionhG, is a valued vector space over(the prime field of characteristicp) with values in the ordinals (cf. [F]).

Fuzzy topological ordered vector spaces I

Bartle–Dunford–Schwartz integration

We study completions of Archimedean vector lattices relative to any nonempty set of positively homogeneous functions on finite-dimensional real vector spaces. Examples of such completions include square mean closed and geometric mean closed vector lattices, amongst others. These functional completions also lead to a universal definition of the complexification of any Archimedean vector lattice and a theory of tensor products and powers of complex vector lattices in a companion paper.

In an Archimedean directed partially ordered vector space X, one can define the concept of a band in terms of disjointness. Bands can be studied by using a vector lattice cover Y of X. If X has an order unit, Y can be represented as a subspace of $$C(\Omega )$$ , where $$\Omega $$ is a compact Hausdorff space. We characterize bands in X, and their disjoint complements, in terms of subsets of $$\Omega $$ . We also analyze two methods to extend bands in X to $$C(\Omega )$$ and show how the carriers of a band and its extensions are related. We use the results to show that in each n-dimensional partially ordered vector space with a closed generating cone, the number of bands is bounded by $$\frac{1}{4}2^{2^n}$$ for $$n\ge 2$$ . We also construct examples of $$(n+1)$$ -dimensional partially ordered vector spaces with $$\left( {\begin{array}{c}2n n\end{array}}\right) +2$$ bands. This shows that there are n-dimensional partially ordered vector spaces that have more bands than an n-dimensional Archimedean vector lattice when $$n\ge 4$$ .

A notion of disjointness in arbitrary partially ordered vector spaces is intro- duced by calling two elements x and y disjoint if the set of all upper bounds of x + y and −x − y equals the set of all upper bounds of x − y and −x + y. Several elementary properties are easily observed. The question whether the disjoint complement of a subset is a linear subspace appears to be more difficult. It is shown that in directed Archime- dean spaces disjoint complements are always subspaces. The proof relies on theory on order dense embedding in vector lattices. In a non-Archimedean directed space even the disjoint complement of a singleton may fail to be a subspace. According notions of dis- jointness preserving operator, band, and band preserving operator are defined and some of their basic properties are studied. Mathematics Subject Classifications 2000: 46A40, 06F20.

Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains

In a partly ordered space the orthogonality relation is defined by incomparability. We define integrally open and integrally semi-open ordered real vector spaces. We prove: if an ordered real vector space is integrally semi-open, then a complete lattice of double orthoclosed sets is orthomodular. An integrally open concept is closely related to an open set in the Euclidean topology in a finite dimensional ordered vector space. We prove: if V is an ordered Euclidean space, then V is integrally open and directed (and is also Archimedean) if and only if its positive cone, without vertex 0, is an open set in the Euclidean topology (and also the family of all order segments \( \{ z \in V:a < z < b\} \), a < b, is a base for the Euclidean topology).

This chapter is about metric spaces, an abstract generalization of the real line that allows discussion of open and closed sets, limits, convergence, continuity, and similar properties. The usual distance function for the real line becomes an example of a metric. The other notions are defined in terms of the metric. The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples. Section 1 gives the definition of metric space and open set, and it lists a number of important examples, including Euclidean spaces and certain spaces of functions. Sections 2 through 4 develop properties of open and closed sets, continuity, and convergence of sequences that are simple generalizations of known facts about $\mathbb{R}$. Section 5 shows how a subset of a metric space can be made into a metric space so that the restriction of a continuous function from the whole space to the subset remains continuous. It also shows that three natural metrics for the product of two metric spaces lead to the same open sets, continuous functions, and convergent sequences. Section 6 shows that any metric space is “Hausdorff,” “regular,” and “normal,” and it goes on to exhibit three different countability hypotheses about a metric space as equivalent. A metric space with these properties is called “separable.” Section 7 concerns compactness and completeness. A metric space is defined to be “compact” if every open cover has a finite subcover. This property is equivalent to the condition that every sequence has a convergent subsequence. The Heine–Borel Theorem says that the compact sets of $\mathbb{R}^n$ are exactly the closed bounded sets. A number of the results early in Chapter I that were proved by the Bolzano–Weierstrass Theorem in the context of the real line are seen to extend to any compact metric space. A metric space is “complete” if every Cauchy sequence is convergent. A metric space is compact if and only if it is complete and “totally bounded.” Section 8 concerns connectedness, which is an abstraction of the property of an interval of the line that accounts for the Intermediate Value Theorem. Section 9 proves a fundamental result known as the Baire Category Theorem. A sample consequence of the theorem is that the pointwise limit of a sequence of continuous complex-valued functions on a complete metric space must have points where it is continuous. Section 10 studies the spaces of real-valued and complex-valued continuous functions on a compact metric space. A generalization of Ascoli's Theorem from the setting of Chapter I provides a characterization of compact sets in either of these spaces of continuous functions. A generalization of the Weierstrass Approximation Theorem, known as the Stone–Weierstrass Theorem, gives sufficient conditions for a subalgebra of either of these spaces of continuous functions to be dense. One consequence is that these spaces of continuous functions are separable. Section 11 constructs the “completion” of a metric space out of Cauchy sequences in the given space. The result is a complete metric space and a distance-preserving map of the given metric space into the completion such that the image is dense.

This paper proposes a modified Gram-Schmidt algorithm for dimension reduction of a document vector space in order to do classification. We also evaluate the performance of the proposed algorithm by comparing with commonly known algorithms such as the centroid-based algorithm and latent semantic indexing. Further, to measure efficiency, a modified Gram-Schmidt training set is applied in the centroid-based algorithm and latent semantic indexing as well. Performance measurement was based on two different parameters, viz., classification accuracy and closeness of similarity with the corresponding training set. The results shows that modified Gram-Schmidt algorithm is indeed an effective method for dimension reduction prior classification. Moreover, it is easy to code and computationally inexpensive.KeywordsCentroidDocument ClassificationDimension ReductionGram-SchmidtLSIVector Space Model