Infinitesimals in ordered vector spaces

Nonstandard analysis has contributed to various branches of mathematics (see, for example, [1, 5, 6]). The infinitesimal methods are rather useful in problems related with compactness and ultrafilters. In the present paper we apply nonstandard methods to ordered vector spaces. We consider some questions on the order/uniform convergence, the Archimedean property, and the order completeness in ordered spaces. Our terminology and notations follow to [2, 7, 9, 10], and [8] in what is related to ordered vector spaces and to [1, 5], and [4] in what is related to nonstandard analysis.

Here we collect some basic preliminary facts that will be used throughout the paper. We consider a superstructure V (S) = _n V _n (S) over a set S which includes several algebraic systems such as real numbers, necessarily vector spaces, etc. (cf. [4, p. 164–165]). Dealing with some superstructure in the sequel, we do not specify the basic set over which it is constructed. This set is usually chosen to be sufficiently substantive for questions under consideration. We denote the superstructure under consideration by M . We suppose that for our enlargement ^∗ S of S, the natural embedding ∗ : V (S) , → V ( ^∗ S) (or, simply, ∗ : M , → ^∗ M) satisfies the following principles (cf. [4, p. 165–166]).

Extension Principle. The set S is a proper subset of ^∗ S in the sense that ^∗ x = x for every x ∈ S. Moreover, ^∗ S is equipped with the same family of operations and relations as S is.

Transfer Principle. Let ψ(x 1 , x 2 , . . . , x _n ) be a bounded formula of the superstructure M (i. e., a formula of the language L M [4, p. 165]), and let A 1 , A 2 , . . . , A _n be elements of the superstructure M.

Then the assertion ψ(A 1 , A 2 , . . . , A _n ) about elements of M is true if and only if the assertion ψ( ^∗ A 1 , ^∗ A 2 , . . . , ^∗ A _n ) about elements of ^∗ M is true.

Let ^∗ M be a nonstandard enlargement of a superstructure M. An element x ∈ ^∗ M is called: standard if x = ^∗ X for some X ∈ M ; internal if x ∈ ^∗ X for some X ∈ M ; external if x is not internal. Note that every standard element is internal, and every element of an internal set is internal too (cf. [1, 1.2.5]). The following technical principle is an easy consequence of the transfer principle:

Internal Definition Principle. Let ψ(x, x 1 , . . . , x _n ) be a formula of L M , and let A, A 1 , . . . , A _n be internal sets. Then the set { x ∈ A : ψ(x, A 1 , . . . , A _n ) } is internal too.

It is well known (cf. [1, 2.1.4]) that a nonstandard enlargement ^∗ M of the superstructure M can be chosen so that the following principle (our last one) holds:

General Saturation Principle. For every family { X γ } γ _∈ Γ of internal sets which has standard cardinality (i. e., card(Γ) < card(M)) and has the finite intersection property, the condition _{γ ∈ Γ} X γ 6 = ∅ is valid.

In the sequel, we deal only with nonstandard enlargements satisfying the general saturation principle. These nonstandard enlargements are called poly-saturated . The following assertion (see, for example, [4, Lemma 4.0.5]) yields in poly-saturated enlargements:

Proposition 1. For every directed set (Θ, ≺ ) ∈ M , there is an (infinitely) remote element a ∈ ^a Θ := { ξ ∈ ^∗ Θ : ( ∀ τ ∈ Θ)τ ≺ ξ }.

An equivalent form of this proposition is that in a poly-saturated enlargement of M , any set X of card(X) < card(M) is a subset of a hyperfinite set Y (i. e. Y is in one-to-one internal correspondence with some ν ∈ ^∗ N).

In the paper we deal with real vector spaces only. A nonempty subset K of a vector space V is said to be a cone if K + K C K , XK C K for all scalars A > 0, and K П ( — K) = { 0 } . A cone K is said to be generating if K — K = V. A cone K defines the order relation >  к on V by x > k y if x — y E K . The pair (V, K ) is called ordered vector space, or simply ordered space. An ordered space (V, K ) is denoted also by (V, > k ), or by (V, >), or just by V when there is no chance of ambiguity. We denote the cone of V by V + . By [x, y] for x, y ∈ V we denote the order interval {z E V : y > z >  x } . An ordered space V is called Archimedean if for any x ∈ V + , inf _{n ∈} N ( n ^-1 x ) = 0. The Archimedean property is equivalent (see, for example, [10, Theorem I.3.1]) to the fact that [(x, y E V) Л (( V n E N)(y >  nx))] ^ [0 >  x]. An ordered space V is called almost Archimedean [2, p. 12] if it follows from y ∈ _n ¹ [ - x, x] for all n ∈ N and some x, y ∈ V that y = 0. It can be shown that any Archimedean ordered space is almost Archimedean, and that any almost Archimedean vector lattice is Archimedean, but in general the Archimedean property is stronger then almost Archimedean one (see, for example, [9, p. 254]). The following very useful criterion of the Archimedean property is well known for vector lattices (see, for example, [8, Theorem 22.5]). Since we did not find an appropriate reference for this criterion in the setting of ordered spaces, we include it with a proof.

Proposition 2. Let V be an ordered space. The following conditions are equivalent :

• (a)    V is Archimedean ;

• (b)    Given any decreasing net x _T ^ > d in V , and writing L = {y E V : ( V т E { t } ) [x _T > y] }, then inf _{τ ∈} {_τ}_, (x _τ - y) = 0 .

y ∈ L

• <1    (a) ^ (b): Let x _T ^ > d and L = {y E V : ( V t E { t } ) [x _T > y] } . Assume that x _T — y > z for all т and y E L. Since x _T — y >  0, the proof will be complete if we show that 0 > z. As x _T > y + z holds for all т and all y E L, we get y + z E L for every y E L. It follows by induction that y + nz E L for every y E L and n E N, in particular, x _T 0 > d + nz (and hence, x _T 0 - d > nz) for some to E { t } and all n E N. Since V is Archimedean, the last condition implies that 0 > z, what is required.

• (b)    ^ (a): Given arbitrary x > 0, we have to show that inf n^ N (n ^-1 x) = 0. Let y be such that n ^-1 x >  y >  0 for all n E N. Then x > ny for all n E N, and hence, — ny ^ > — x. Denote L = {z E V : ( V n E N) [ — ny — z >  0] } . Let u E L, n E N, then —ny — u = ( — (n + 1)y — u) + y > y. Since u E L, n E N are arbitrary, and inf (—ny — u) = 0 by the

  n ∈ N ,u ∈ L

  
  

• 2.    Some external vector spaces associated with an ordered space

hypothesis, we get 0 = inf ( — ny — u) >  y >  0. Thus y = 0, and hence inf (n 1 x) = 0. B n ∈ N ,u ∈ L n ∈ N

An ordered space for which every decreasing bounded from below net has an infimum is called Dedekind complete [2, p. 10] (shortly, x _T V> y ^ x _T ^ z). Any Dedekind complete ordered space is Archimedean (for example, by Proposition 2). For further information on ordered spaces we refer to [2, 8, 10].

Let V = (V, V + ) be an ordered space. Given κ ∈ ^∗ V , we denote by U(κ) the set { x ∈ V : x > к } of standard upper bounds of к, and by L(к) the set — U( — к) = { x E V : к >  x } of standard lower bounds of κ. Consider the following external sets (cf. [4, p. 184], [3] for the vector lattice setting):

fin( * V) := { к E * V : ( 3 x,y E V) к E [x,y] } , o-pns( * V) := { к E * V : i n f(U(к) — L(к)) = 0 } , n( * V) := {к E * V : i n f U (к) = supL(к) = 0 } , A( * V) := { к E * V : ( 3 y E V)( V n E N) пк E [ — y,y] } .

Here, by inf V / sup _V we denote the infimum/supremum calculated in V . It is easy to see that ab ove defined sets are external vector spaces over R satisfying η( ^∗ V ) ⊆ o-pns( ^∗ V ) ⊆ fin( ^∗ V ), V ⊆ o-pns( ^∗ V ), and λ( ^∗ V ) ⊆ fin( ^∗ V ). In general, V ∩ λ( ^∗ V ) may contain nonzero elements (take R ² with lexicographical ordering), and V + η( ^∗ V ) may be proper subspace of o-pns( ^∗ V ) (cf. [4, Theorem 4.4.2]). Moreover, the sets above are external ordered spaces with respect to the ordering inherited from ^∗ V . Clearly a subset A ⊆ V is order bounded iff ^∗ A ⊆ fin( ^∗ V ) iff ( ∃ ν ∈ ^∗ N \ N) _ν ^{1 ∗} A ⊂ λ( ^∗ V ) iff ( ∃ ν ∈ ^∗ N) _ν ^{1 ∗} A ⊂ fin( ^∗ V ). For the order convergence (the (o)-convergence, sf. [10, p. 14]) of monotone nets in V , there is the following simple nonstandard condition (cf. [4, 4.3.2], [3] in the case of vector lattice V ). Since its proof is slightly different, we include it below.

Proposition 3. Let (x _α ) _{α ∈} Ξ be monotone net in an ordered space V . Then the following conditions are equivalent:

• (a)    (x _α ) _α order converges to 0;

• (b)    x β ∈ η( ^∗ V ) for all remote elements β ∈ ^a Ξ;

• (c)    x β ∈ η( ^∗ V ) for some remote element β ∈ ^a Ξ .

<1 We consider only the case of decreasing net, x _a ^ . For (a) ^ (b), assume x _a ^ 0. Then, x _a >  x e > 0 for every в E a 5, a E 5. Hence, inf у U (x e ) = 0. But clearly, sup y L(x e ) > 0. Consequently, sup _V L(x β ) = 0, and x β ∈ η( ^∗ V ). The implication (b) ⇒ (c) is obvious. For (c) ⇒ (a), take some β ∈ ^a Ξ, x β ∈ η( ^∗ V ) (by Proposition 1). In particular, inf V U(x β ) = 0. Since x _a >  x e for every a E 5, one gets x _a V> 0. Let y E V be such that x _a > y > 0 for all a E 5. Then, by the transfer principle, x _a > y > 0 for all a E * 5, in particular, x e > y. It is possible only if y = 0. So, x _a ^ 0. B

It can be seen easily that implications (a) ⇒ (b) ⇒ (c) are true for an arbitrary net (x _α ) _{α ∈} Ξ . The implication (c) ⇒ (a) may be false without monotonicity condition (see, for example, [4, p. 185]). For the uniform convergence (i. e. (r)-convergence [8, 10]) of monotone sequences, the following proposition (cf. [4, 4.3.3.]) is true. Its proof is quite similar, so we omit it (a remark similar to that one after Proposition 3 holds too).

Proposition 4. Let (x _n ) _n be monotone sequence in an ordered space V . Then the following conditions are equivalent:

• (a)    (x _n ) _n (r) -converges to 0;

• (b)    x ν ∈ λ( ^∗ V) for all ν ∈ ^∗ N \ N;

• (c)    x ν ∈ λ( ^∗ V ) for some ν ∈ ^∗ N \ N.

We continue with the following auxiliary assertion (cf. [3, Lemma 4.3.4.]).

Lemma 1. Let V be an ordered space, u ∈ V , and ν ∈ ^∗ N \ N . Then either u = 0 or νu 6 ∈ o-pns( ^∗ V ) .

• <1 Let u = 0. Take x G U (vu) and y G L(vu). Then x >  vu > y. By the transfer principle, there exists n G N such that x >  nu >  y. By the convexity of order intervals, x >  mu > y for all m G * N satisfying n 6 m 6 v (indeed, mu = anu + (1 — a)vu, for 0 6 a = VV-Em 6 1). In particular, x > (n + 1)u > y. Thus, nu — y G [0,x — y] C [y — x, x — y] and ( n + 1) u - y ∈ [0 , x - y ] ⊆ [ y - x, x - y ]. Therefore nu - y ∈ [ y - x, x - y ] &  y - ( n + 1) u ∈ [ y - x, x - y ]. By the convexity of order intervals again, we get

  - ₂ u = - ₂ (( n + 1) - n ) u = ₂( y - ( n + 1) u ) + ₂ ( nu - y ) ∈ [ y - x,x - y ] .     (1)

It follows from (1), that

± ¹ ₂ u ∈ [ y - x,x - y ] .                                  (2)

Suppose now that νu ∈ o-pns( ^∗ V ), that is inf _V ( U ( νu ) - L ( νu )) = 0. Since x ∈ U ( νu ) and y G L(vu) in (2) were chosen arbitrary, we obtain 0 >  ± 2 u, that is u = 0, which is a contradiction to our initial assumption. Hence, vu G o-pns( * V), what is required. B

Clearly, it may be happened for some u = 0 in a non-Archimedean space V that νu ∈ fin( ^∗ V ) for all ν ∈ ^∗ N. The following theorem (cf. [3, Theorem 3.4.5] in the vector lattice setting) characterizes the almost Archimedean property in terms of external ordered spaces associated with our space.

Theorem 1. Let V be an ordered space. Then the following conditions are equivalent:

• (1)    V is almost Archimedean ;

• (2)    λ ( ^∗ V ) ∩ V = { 0 } ;

• (3)    λ ( ^∗ V ) ⊆ η ( ^∗ V );

• (4)    λ ( ^∗ V ) ⊆ o-pns( ^∗ V ) .

• < (1) ^ (2): It follows directly from the definition of A( * V).

• (2)    ⇒ (3): Suppose that κ ∈ λ ( ^∗ V ) \ η ( ^∗ V ). Then nκ ∈ [ - y, y ] holds for some y ∈ V and all n ∈ N. Hence, for every n ∈ N, - _n ¹ y ∈ L ( κ ), and _n ¹ y ∈ U ( κ ). Since κ ∈ η ( ^∗ V ), there

exists an element 0 = z G V such that U (к) >  z >  L ( k ). In particular, z G A( * V), which contradicts (2).

• (3)    ⇒ (4): It follows from η ( ^∗ V ) ⊆ o-pns( ^∗ V ).

• (4)    ^ (1): Take u,v G V such that u > nv > —u for all n G N. Let v G * N \ N. By the transfer principle, u > mvv > —u for all m G N, and hence, vv G A( * V). Then, by hypothesis, vv G o-pns( * V). Lemma 1 implies that v = 0. So, V is almost Archimedean. B

Our second theorem gives a nonstandard condition for an Archimedean ordered space to be Dedekind complete. Remark that for Archimedean vector lattices this condition is also necessary [4, Theorem 4.4.2] for the Dedekind completeness.

Theorem 2. Let V be an Archimedean ordered space satisfying o-pns( * V) = V + n( * V) , then V is Dedekind complete.

• <1    It suffices to show that every decreasing bounded from below net in V is order convergent. Let (x _a ) _a V> d G V. By Proposition 2,

inf (x _a — z) = 0,                                     (3)

aG{a}, zeL where L = {y G V : (Va 6 {a}) [xa > y]}. Fix a remote element в G a{a}. Since {u 6 V : xa V> u} = L(xe) and {xa : a G {a}} C U(xe), it follows from (3) that infv(U(xe) — L(xe)) = 0, or, in other words, xe G o-pns(*V) = V + n(*V)• Let x G V be such that xe — x G n(*V)• Thus, by Proposition 3, the decreasing net (xa — x)a is order convergent to 0, that is x = infae{a} xa. Hence, V is Dedekind complete. B