An unpleasant set in a non-locally-convex vector lattice

1973 ◽  
Vol 18 (3) ◽  
pp. 229-233
Author(s):  
J. D. Pryce
Keyword(s):  
Convex Hull ◽  
Vector Lattice ◽  
Linear Topological Space ◽  
Real Vector ◽  
Vector Lattices ◽  
Köthe Spaces ◽  
The Common ◽  
Solid Hull ◽  
Image Position ◽  
Locally Convex

In a linear topological space E one often carries out various “ smoothing ” operations on a subset A, such as taking the convex hull co A and the closure A-. If E is also a (real) vector lattice, the solid hullis also a natural “ smoothing out ” of A. If sol A = A then A is called solid, and if E has a base of solid neighbourhoods of 0 as do all the common topological vector lattices such as C(X), Lp, Köthe spaces and so on—then E is called a locally solid space.

scholarly journals A note on vector lattices

1967 ◽  
Vol 7 (1) ◽  
pp. 32-38 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Vector Lattice ◽  
Vector Lattices ◽  
Image Position

Let E be a vector lattice in the sense of Birkhoff [1]. We use the following notations:

scholarly journals Convexity conditions for non-locally convex lattices

1984 ◽  
Vol 25 (2) ◽  
pp. 141-152 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Real Vector Space ◽  
Constant Independent ◽  
Real Vector ◽  
Image Position ◽  
Locally Convex ◽  
Convexity Conditions

First we recall that a (real) quasi-Banach space X is a complete metrizable real vector space whose topology is given by a quasi-norm satisfyingwhere C is some constant independent of x1 and x2. X is said to be p-normable (or topologically p-convex), where 0 < p ≤ l, if for some constant B we havefor any x1, …, xn, є X. A theorem of Aolci and Rolewicz (see [18]) asserts that if in C = 21/p-1, then X is p-normable. We can then equivalently re-norm X so that in (1.4) B = 1.

scholarly journals Countable vector lattices

1974 ◽  
Vol 10 (3) ◽  
pp. 371-376 ◽  
Author(s):  
Paul F. Conrad
Keyword(s):  
Vector Space ◽  
Vector Lattice ◽  
Real Vector Space ◽  
Vector Spaces ◽  
Real Vector ◽  
Real Numbers ◽  
Vector Lattices ◽  
Direct Sum ◽  
Ordered Vector Spaces ◽  
Countable Dimension

In his paper “On the structure of ordered real vector spaces” (Publ. Math. Debrecen 4 (1955–56), 334–343), Erdös shows that a totally ordered real vector space of countable dimension is order isomorphic to a lexicographic direct sum of copies of the group of real numbers. Brown, in “Valued vector spaces of countable dimension” (Publ. Math. Debrecen 18 (1971), 149–151), extends the result to a valued vector space of countable dimension and greatly simplifies the proof. In this note it is shown that a finite valued vector lattice of countable dimension is order isomorphic to a direct sum of o–simple totally ordered vector spaces. One obtains as corollaries the result of Erdös and the applications that Brown makes to totally ordered spaces.

The order-bound topology on Riesz spaces

1970 ◽  
Vol 67 (3) ◽  
pp. 587-593 ◽  
Author(s):  
Yau-Chuen Wong
Keyword(s):  
Convex Sets ◽  
Vector Lattice ◽  
Positive Cone ◽  
Riesz Space ◽  
Hausdorff Topology ◽  
Riesz Spaces ◽  
Neighbourhood Base ◽  
Solid Hull ◽  
Locally Convex

1. Introduction. Let (X, C) be a Riesz space (or vector lattice) with positive cone C. A subset B of X is said to be solid if it follows from |x| ≤ |b| with b in B that x is in B (where |x| denotes the supremum of x and − x). The solid hull of B (absolute envelope of B in the terminology of Roberts (2)) is denoted to be the smallest solid set containing B, and is denoted by SB. A locally convex Hausdorff topology on (X, C) is called a locally solid topology if admits a neighbourhood-base of 0 consisting of solid and convex sets in X; and (X, C, ), where is a locally solid topology, is called a locally convex Riesz space.

Banach Envelopes of Non-Locally Convex Spaces

1986 ◽  
Vol 38 (1) ◽  
pp. 65-86 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Convex Hull ◽  
Bounded Linear Operator ◽  
Locally Convex Spaces ◽  
Minkowski Functional ◽  
Bounded Linear ◽  
Banach Envelope ◽  
Image Position ◽  
Locally Convex

Let X be a quasi-Banach space whose dual X* separates the points of X. Then X* is a Banach space under the normFrom X we can construct the Banach envelope Xc of X by defining for x ∊ X, the normThen Xc is the completion of (X, ‖ ‖c). Alternatively ‖ ‖c is the Minkowski functional of the convex hull of the unit ball. Xc has the property that any bounded linear operator L:X → Z into a Banach space extends with preservation of norm to an operator .

scholarly journals Components in vector lattices and extreme extensions of quasi-measures and measures

1993 ◽  
Vol 35 (2) ◽  
pp. 153-162 ◽  
Author(s):  
Z. Lipecki
Keyword(s):  
Convex Set ◽  
Vector Lattice ◽  
Convex Combination ◽  
Extreme Points ◽  
Main Body ◽  
Projection Property ◽  
Vector Lattices ◽  
Definition Of ◽  
Image Position ◽  
Theoretical Results

We develop some ideas contained in the author's paper [8] which was, in turn, inspired by Bierlein and Stich [5]. The main body of the present paper is divided into three sections. Section 2 is concerned with some vector-lattice-theoretical results. They are then applied to extensions of quasi-measures and measures in Sections 3 and 4, respectively.Let X be a vector lattice, let x ε X+ and let S be a non-empty set. Theorems 1 and 2 describe some properties of the convex set(see Section 2 for the definition of the sum above). The extreme points of Dx,s are characterized in terms of the components of x. It is also shown that if X has the principal projection property and S is countable, then extr Dx,s is, in some sense, large in Dx,s. Furthermore, for finite S, each point in Dx,s is then a sσ-convex combination of extreme ones.

Optimal selection of stochastic intervals under a sum constraint

1987 ◽  
Vol 19 (2) ◽  
pp. 454-473 ◽  
Author(s):  
E. G. Coffman ◽  
L. Flatto ◽  
R. R. Weber
Keyword(s):  
Decision Rule ◽  
Selection Process ◽  
Random Variables ◽  
Optimal Threshold ◽  
Optimal Selection ◽  
Expected Number ◽  
Common Distribution ◽  
The Common ◽  
Image Position ◽  
Selection Of

We model a selection process arising in certain storage problems. A sequence (X1, · ··, Xn) of non-negative, independent and identically distributed random variables is given. F(x) denotes the common distribution of the Xi′s. With F(x) given we seek a decision rule for selecting a maximum number of the Xi′s subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c > 0, and (2) the Xi′s must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected.We prove first that there exists such a rule of threshold type, i.e. the ith element inspected is accepted if and only if it is no larger than a threshold which depends only on i and the sum of the elements already accepted. Next, we prove that if F(x) ~ Axα as x → 0 for some A, α> 0, then for fixed c the expected number, En(c), selected by an optimal threshold is characterized by Asymptotics as c → ∞and n → ∞with c/n held fixed are derived, and connections with several closely related, well-known problems are brought out and discussed.

scholarly journals Minimal vector lattice covers

1971 ◽  
Vol 5 (3) ◽  
pp. 331-335 ◽  
Author(s):  
Roger D. Bleier
Keyword(s):  
Vector Lattice ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Vector Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Archimedean Lattice

We show that each archimedean lattice-ordered group is contained in a unique (up to isomorphism) minimal archimedean vector lattice. This improves a result of Paul F. Conrad appearing previously in this Bulletin. Moreover, we show that this relationship between archimedean lattice-ordered groups and archimedean vector lattices is functorial.

scholarly journals Nonsmooth analysis and optimization on partially ordered vector spaces

1992 ◽  
Vol 15 (1) ◽  
pp. 65-81 ◽  
Author(s):  
Thomas W. Reiland
Keyword(s):  
Vector Lattice ◽  
Vector Spaces ◽  
Range Space ◽  
Topological Vector Spaces ◽  
Complete Vector Lattice ◽  
Lipschitz Mappings ◽  
Ordered Vector Spaces ◽  
Partially Ordered Vector Spaces ◽  
Complete Vector ◽  
Locally Convex

Interval-Lipschitz mappings between topological vector spaces are defined and compared with other Lipschitz-type operators. A theory of generalized gradients is presented when both spaces are locally convex and the range space is an order complete vector lattice. Sample applications to the theory of nonsmooth optimization are given.

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