Notes

Description:

This note contains the definitions necessary to understand the forms in Brunner [1984d], that is, Form 126.

Content:

This note contains the definitions necessary to understand the forms in Brunner [1984d], that is, Form 126.

Definition

  1. A \(B\)-lattice is a vector lattice (see below for the definition of vector lattice) together with a Frechet-complete (Cauchy sequences are convergent) Riez-norm \(\left\Vert{\cdot }\right\Vert\) (i.e.,(i) \(-y \le  x \le  y \rightarrow  \left\Vert{x}\right\Vert \le\left\Vert{y}\right\Vert\)(ii)  if \(\Vert x\Vert \le 1\) there is a \(y \ge  0\) such that \(-y\le  x \le  y\) and  \(\Vert y \Vert  \le 1\).)
  2. A positive linear functional on \(X\) is a linear mapping \(f : X \rightarrow  {\Bbb R}\) such that \(f(x) \ge  0\) when \(x \ge   0.\)
  3. A wedge \(W\) in a real linear space \(L\) is a convex set  such that \(t\subseteq W\) for every \(t\ge 0\) (\(t\in {\Bbb R})\). Acone is a wedge which contains no line through \(0\).
  4. An ordered linear space (OSL) is  a  linear  space \(L\) in which some wedge \(W\), called the positive wedge has been, specified. The partial ordering \(\le \) on \(L\) is defined by \(y\le x\) iff \(x - y \in W\).  (Note that this differs from the definition of ordered vector space in Note 31.)
    (Facts:
    1. \(\le \) is reflexive & transitive
    2. translation and multiplication by positive numbers preserves order, multiplication by negative numbers reverses order.
    3. \((x\ge y\) and \(y\ge x\) implies \(y= x)\) if and only if the wedge is a cone.
    4. \(x\ge 0\) iff \(x\in W\).
      If a relation satisfies (a) and (b) then (d) defines a wedge.)
  5. Let \(V\) be a partially ordered linear space with a positive wedge (defined). \(V\) is a vector lattice if the positive wedge in \(V\) is a cone and if each set \(\{x,x'\}\) of two elements has a least upper bound \(x\vee x'\).  These definitions are taken primarily from Day [1973].

Howard-Rubin number: 16

Type: Definitions

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