Notes

Description:

Definitions for forms [14 W], [70 A], [52 H] through [52 L], Form 372 and [372 A] through [372 D] .These are modifications of definitions from Schechter [1996a] and Schechter [1996b].

Content:

Definitions for forms [14 W], [70 A], [52 H] through [52 L], Form 372 and [372 A] through [372 D] .These are modifications of definitions from Schechter [1996a] and Schechter [1996b].

  1. A pre-order is a reflexive, transitive relation.
  2. A directed set \(({\Bbb D},\le)\) is a non-empty, preordered set in which each finite subset has an upper bound.
  3. A net in a set \(X\) is any function from a directed set into \(X\).  \((x : {\Bbb D} \rightarrow  X\), usually denoted \((x(\delta ) :\delta \in {\Bbb D})\) or \((x_{\delta }))\).
  4. If \(x:{\Bbb D} \rightarrow X\) is a net and \(S\) is any set we say \(x\) is eventually in \(S\) if \((\exists \delta _{o} \in {\Bbb D})(\forall \delta\in {\Bbb D})(\delta\ge\delta _{o} \rightarrow x(\delta)\in S)\) and we say \(x\) is frequently in \(S\) if \((\forall \delta \in {\Bbb D})(\exists \delta' \in {\Bbb D})(\delta' \ge \delta\) and \(x(\delta') \in S)\)
  5. \(x\) is frequently constant if for some \(c\), \(x\) is frequently in \(\{c\}\).
  6. Assume \((x(\alpha):\alpha \in {\Bbb A})\) and \((y(\beta ):\beta \in {\Bbb B})\) are two nets in \(X\), then \((y(\beta))\) is a subnet of \((x(\alpha))\) (in the sense of Aarnes and Andenees - (This is not the usual definition of subnet.)) if \((\forall S\subseteq X)\)(If \((x(\alpha ))\) is eventually in \(S\), then \((y(\beta))\) is eventually in \(S\)). (Which is equivalent to \((\forall S\subseteq X)\)( If \((y(\beta ))\) is frequently in \(S\) then \((x(\alpha ))\) is frequently in \(S\)).)
  7. A universal net (or ultranet) on a set \(X\) is a net \((x(\delta): \delta\in\Bbb D)\) with the property that \((\forall S\subseteq X)\) (\(x\) is eventually in \(S\) or \(x\) is eventually in \(X\backslash S\)).
  8. A partially ordered set is Dedekind complete if each subset that has an upper bound has a least upper bound.  (Equivalently, if each subset that has a lower bound has a greatest lower bound.)
  9. An ordered vector space is a real vector space \(Z\) equipped with a partial ordering \(\preccurlyeq\) which also satisfies:
    • \(x\preccurlyeq y\) implies \(x+z \preccurlyeq y + z\)
    • \(x\succcurlyeq 0\) and \(r\ge 0\) imply \(rx\succcurlyeq 0\).
    (Note:  This differs from the definition of ordered linear space in Note 16.)
    In definitions 10 through 13, \(X\) is a vector space,\((Z,\preccurlyeq)\) is an ordered vector space and \(p:X\to Z\).
  10. \(p\) is sublinear if \(p(x+y) \preccurlyeq p(x) + p(y)\) and for all \(c\in [0,+\infty)\), \(p(cx) = cp(x)\).
  11. \(p\) is convex if \(p(cx+(1-c)y) \preccurlyeq cp(x) + (1-c)p(y)\) for all \(c\in [0,1]\).
  12. \(p\) is concave if \(p(cx+(1-c)y) \succcurlyeq cp(x) + (1-c)p(y)\) for all \(c\in [0,1]\).
  13. \(p\) is affine if \(p(cx+(1-c)y) = cp(x) + (1-c)p(y)\) for all \(c\in [0,1]\).
  14. Let \((\Delta,\sqsubseteq)\) be a directed set and \((Z,\preccurlyeq)\) be a Dedekind complete ordered vector space. Let \(B(\Delta,Z)\) be the set of functions \(f\) from \(\Delta\) to \(Z\) such that the range of \(f\) has both an upper bound and a lower bound. For \(f\in B(\Delta,Z)\) let
    • \(\lim\inf(f) = \sup_{\alpha\in\Delta}\inf_{\alpha\sqsubseteq \beta} f(\beta)\)  and
    • \(\lim\sup(f) = \inf_{\alpha\in\Delta}\sup_{\alpha\sqsubseteq \beta} f(\beta)\)
    (Since \(Z\) is Dedekind complete both \(\lim\inf(f)\) and \(\lim\sup(f)\) exist in \(Z\) and \(\lim\inf(f) \preccurlyeq\lim\sup(f)\).)   We define a \(Z\)-valued Banach limit on \(\Delta\)to be a linear map LIM\(: B(\Delta,Z) \to Z\) which satisfies \[(\forall f\in B(\Delta,Z))(\lim\inf(f) \preccurlyeq \hbox{LIM}(f)\preccurlyeq \lim\sup(f))\]

Howard-Rubin number: 31

Type: Definitions

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