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Владикавказский математический журнал 2017, Том 19, Выпуск 3, С. 21-30
ORDER BORNOLOGICAL LOCALLY CONVEX LATTICE CONES
D. Ayaseh, A. Ranjbari
In this paper, we introduce the concepts of us-lattice cones and order bornological locally convex lattice cones. In the special case of locally convex solid Riesz spaces, these concepts reduce to the known concepts of seminormed Riesz spaces and order bornological Riesz spaces, respectively. We define solid sets in locally convex cones and present some characterizations for order bornological locally convex lattice cones.
Mathematics Subject Classification (2010): 46A03, 46A40. Key words: locally convex lattice cones, order bornological cones.
1. Introduction
The theory of locally convex cones as developed in [5] and [11], uses an order theoretical concept or a convex quasiuniform structure to introduce a topological structure on a cone. Examples of locally convex cones contain classes of functions that take infinite values and families of convex subsets of vector spaces. These type of structures are not vector space and also may not even be embedded into a larger vector spaces in order to apply technics from topological vector spaces. These structures are studied in the general theory of locally-convex cones. The class of bornological locally convex spaces is an important class of locally-convex spaces which are introduced by* Mackey in 1946. Every bounded linear operator on a bornological space is continuous. These structures have an advantage which they can be written as an inductive limit of seminormed spaces. Therefore every complete HausdorfF bornological locally convex space is the inductive limit of Banach spaces. We establish these results for locally convex cones in [3]. Also, We investigated the bornological convergence for cones in [2]. In the case of locally convex lattice cones, we want to study the order bornological locally convex lattice cones. The investigating of these structure is interesting, since these structures are the order inductive limit of us-lattice cones which are the extensions of seminormed Riesz spaces. We note that in the case of vector lattices the concept of separated us-lattice cones reduces to the concept of normed Riesz spaces and the concept of symmetric complete separated us-lattice cones reduces to the concept of Banach lattices, which have many applications in Economics. This research can be useful for researchers in mathematical economic theory. For recent researches see [2-4, 6, 9].
A cone is a set P endowed with an addition and a scalar multiplication for nonnegative real numbers. The addition is assumed to be associative and commutative, and there is a neutral element 0 G P. For the scalar multiplication the usual associative and distributive properties hold, that is a(pa) = (afi)a, (a + P)a = aa + pa, a(a + b) = aa + ab, la = ^d 0a = 0 for all a,b G P and a, P ^ 0.
Let P be a cone. A collection U of convex sub sets U C P2 = P x P is called a convex
P
©2017 Ayaseh D., Ranjbari A.
(Ui) A C U for every U G U (A = {(a, a) : a G P}); (U2) for all U, V G U there is a W G U such that W C U n V; (U3) AU o pU C (A + p)U for all U G ^d A, p > 0; (U4) aU G U for all U G ^d a > 0.
Here, for U, V C P^y U o V we mean the set of all (a, b) G P2 such that there is some c G P with (a, c) G ^d (c, b) G V.
P U P ( P , U)
a locally convex cone if
(U5) for each a G P and U G U there is some p > 0 such that (0, a) G pU.
UP
a
U(a) = {b G P : (b, a) G U}, resp. (a)U = {b G P :(a,b) G U}, U G U.
The common refinement of the upper and lower topologies is called symmetric topology.
a G P
Let U and W be convex quasiuniform structures on P. We say that U is finer than W if W C 11. _
The extended real number system i = IU {+°o} is a cone endowed with the usual algebraic operations, in particular a + oo = +oo for all a G R, a ■ (+oo) = +oo for all a > 0 and 0 ■ (+to) = 0. We set V = {e : e > 0}, where
Then f is a convex quasiuniform structure on R and (R, Y) is a locally convex cone. For a G R the intervals (-to, a + e] are the upper and the intervals [a — e, are the lower neighborhoods, while for a = +oo the entire cone R is the only upper neighborhood, and {+to} is open in the lower topology. The symmetric topology is the usual topology on R with as an isolated point
For cones P and Q a mapping T : P ^ Q is called a linear operator if T(a + b) = T(a) + T(b) Mid T(aa) = aT(a) hold for all a, b G ^d a ^ 0. If both (P, U) Mid (Q, W) are locally convex cones, the operator T is called (um/orm/y) continuous if for every W G W one can find U G 11 such that (T x T)(U) C W.
\ linear functional on P is a linear operator p : P —» R. The dual cone P* of a locally ( P , U) P
neighborhood U G U is defined as folows:
Let U be a convex quasiuniform structure on P. The subset B of U is called a base for U, whenever for every U G U there are n G N Ui,..., Un G B and Ai,..., An > 0 such that Ai Ui n ■ ■ ■ n AnUn C U.
Suppose that (P,U) is a locally convex cone. We shall say that F C PMs u-bounded (uniformly-bounded) if it is absorbed by each U G U. A subset A of P is called bounded above (below) whenever A x {0} (res. {0} x A) is u-bounded (see [3]).
sets
U(a)U = U(a) n (a)U, U G U.
U° = {p G P* : p(a) < p(b) + 1, V (a, b) G U}.
2. Solid sets and us-lattice cones
Locally convex lattice cones as a generalization of locally solid Riesz spaces has been introduced by Walter Roth in [10]. Here, we use the definition of this structure which have been presented in the terms of convex quasiuniform structures. We define solid sets in locally convex lattice cones and use them for our aim.
Definition 1. Let P be a cone and ^ be a reflexive, transitive and antisymmetric order on P (P is an ordered cone). We shall say that P is a V (or A)-lattice cone whenever
(1) a, b £ P implies that a V b £ P (or a Л b £ P);
(2) for a,b,c £ P, (a + c) V (b + c) = a V b + c(or (a + c) Л (b + c) = a Л b + c).
The cone P is called a lattice cone if it is a V mid Л-lattice cone.
Let P and Q be V (or Л)-1аШсе cones. The linear operator T : P ^ Q is called
V (or Л)-1аШсе homomorphism whenever T(a V b) = T(a) V T(b) (or T(a Л b) = T(a) Л T(b)) for a, b £ P.
Let E be a Riesz фасе. A subset A of E is called solid whenever Щ ^ |a| and a £ A imply-that b £ A (see [1]). We note that for a £ E, |a| = a V (-a). Now, we present a definition of solid sets in lattice cones.
Definition 2. Let P be a V (or Л)-1аШсе cone. We shall say that a subset Б of P2, is
V (or Л)-solid, whenever
(1) a ^ b implies that (a, b) £ Б;
(2) (a,b) £ аБ and (c,b) £ вБ imply that (a V c,b) £ (a + в)Б (or (a,b) £ аБ and (a, c) £ в Б imply th at (a, b Л c) £ (a + в)Б).
If P is a lattice cone, the subset Б is called solid whenever it is V-solid and Л-solid.
The V (or ^^^^^^^d hull of a subset Б of P2 is the smallest (with respect to the set inclusion)
V (or ^^^^^^^d subset of P2, which contains Б, we denote it by вку(Б) (or вНА(Б)). Also we denote the solid hull of Б by вк(Б).
If E is a Riesz space and A С E is solid (in the sense of the Riesz spaces) and convex, then A = {(c,b) £ E2 : 3 a £ A, c ^ b + a} is solid in the sense of lattice cones. Indeed, if a ^ b for a, b £ E, then (a, b) £ A, since 0 £ A mid a ^ b + 0. Now, let (a, b) £ yA and (c, b) £ \A for y,A > 0. Then we have a ^ b + Yt and c ^ b + At' for some t, t' £ A. Now, we have t V 0,t' V 0 £ A, since A is solid. Then a ^ b + Y(t V 0) + X(t' V 0) and c ^ b + Y(t V 0) + X(t' V 0). This shows that a V c ^ b + Y(t V 0) + X(t' V 0). Sinee A is convex, we conclude that y(t V 0) + \(t' V 0) £ (y + A)A. Therefore (a V c,b) £ (y + A) A. Similarly, we can prove that A is Л-solid.
Definition 3. Let P be an ordered cone and H be a convex quasiuniform structure on P. We shall say that H is compatible with the order structure of P whenever a ^ b implies that (a, b) £ U for all U £ H for a,b £ P.
Definition 4. Let P be a V (or Л)-lattice cone and H be a compatible convex quasiuniform structure on P such th at (P, H) is a locally convex cone. Then we shall say that (P, H) is a locally convex V (or cone, whenever H has a base of V (or sets. If H has a
base of V (or ^^^^^^^d sets, then it is called V (or Л)-solid convex quasiuniform structure. If P is a lattice cone, then the convex quasiuniform structure H is called solid whenever it has a base of solid sets. The locally convex cone (P, H) is called locally convex lattice cone if H has a base of solid sets.
Example 1. Let (E,t ) be a locally convex so lid Riesz space. Then т has a base V of solid, convex and balanced subsets. For V £ V,wesetV = {(a, b) £ E2 : 3 v £ V, a ^ b + v}.
Then V = {F : V G V} is a solid convex quasiuniform structure on E Therefore (E, V) is a locally convex lattice cone.
Let P be a cone. A subset B of P2 is called uniformly convex whenever it has the properties (Ui^d (U3). The locally convex cone (P,U) is called a uc-cone whenever U = {aU : a > 0} for some U G U (see [3]). If P is a V (or A)-lattice cone and U is V (or A)-solid, then (P,U) is called Vus (or Aus)-lattice cone. In the case that P is a lattice cone and U is solid, (P,il) is called us-lattice cone. For example normed Riesz spaces and Banach lattices are us-lattice cones as locally convex cones. Also the locally convex cone (R, Y) is a us-lattice cone. We note that every us-lattice cone is a locally convex lattice cone.
Let P be a V (or A)-lattice cone and B C P2. We denote the smallest uniformly convex and V (or A)-solid subset of P2, which contains B by usv(B) (or usA(B)), and we call it the uniformly convex V (or A)-so^d hull of B.\f P be a lattice cone, then we denote the uniformly convex solid hull of B, by us(B).
V ( A) V ( A) u
u
< Let (P, U) be a locally convex V (or A)-lattice cone and B be a u-bounded subset of P2. Let B be a base of V (or A)-solid sets for U. For every U G B there is A > 0 such that B C AU This shows that usv(B) C usv(AU) = AU (or usA(B) C usA(AU) = AU), since U is V (or A)-solid. Therefore usv(B) (or usA(B)) is u-bounded. >
uu
unclecl.
Proposition 2. Let P be a. V (or A)-iattice cone and (PY, UY)7er be a family of locally convex V (or A)-lattice cones. Also, let for every 7 G r, : P ^ PY is a V (or A)-lattice homomorphism. Then the coarsest convex quasiuniform structure U on P, which makes all continuous, is V (or A)-solid and (P, U) is a locally convex V (or A)-lattice cone.
< It is enough to show that for every 7 G ^d V (or A)-solid UY G U7, (gY x )-i(UY) V A V A P P
a V-lattice cone. Indeed, let a ^ b for a, b G P. Then (a) ^ (b) for each 7 G r, since is a V-lattice homomorphism for each 7 G r. This implies that (gY(a),gY(b)) G U7, since UY is V-solid for each 7 G r. Then (a, b) G (gY x )-i(UY). Now, let (a, b) G a(gY x )-i(UY) and (c, b) G x )-i(UY) for a, b, c G P and 7 G r. Then (gY(a),gY(b)) G aUY and (gY(c),gY(b)) G ^U7. Now, since UY is V-solid an #7 is V-lattice homomorphism, we conclude that (gY(a Vc),gY(b)) = (gY(a) V(c),gY(b)) G (a + ^)U7. Therefore (a Vc,b) G (a + ^)(g7 x
£y)-i(U7)• >
( P , U) V A
locally convex V (or A)-lattice cones (PY,UY)7er by the V (or A)-lattice homomorphisms g7, 7 G r. Similarly, the concept of order projective limit can be defined.
V ( A) V ( A)
limit of some Vus (or Aus)-lattice cones.
< Let (P, U) be a locally co nvex V (or A)-lattice cone. T hen U has a base B of V (or A)-solid sets. For B G B, we set UB = {aB : a > 0}. Then Us is a V (or A)-solid convex quasiuniform structure on P and (P, UB) is a locally co nvex V (or A)-lattice cone for each B G B. Now, it is easy to see that (P,U) is the V (or A)-order projective limit of (P, UB)BeB by the identity mappings. >
us
lattice cones.
In the special case of locally convex solid Riesz spaces, Proposition 3 yields that every (Hausdorff) locally convex solid Riesz space is the projective limit of (normed) seminormed Riesz spaces.
3. Order bornological locally convex lattice cones
Suppose that (P, U) and (Q, W) are locally convex cones and T : P ^ Q is a linear operator. We shall say T is u-bounded if (T x T)(F) is u-bounded in Q2 for every u-bounded subset F of P2. We shall say (P, U) is a bornological cone if every u-bounded linear operator ( P , U)
Let (P, U) and (Q, W) be locally convex cones. The linear operator T : P ^ Q is called TP
Q ( P , U)
( P , U) b
PU
subset of P. We set Pu = {a G P : 3 \> 0, (0, a) G \U} and Uu = {aU : a> 0}. Then (Pu,UU) is a locally convex cone (a uc-cone). In [3], we proved that there is the finest convex quasiuniform structure Ur (or Ubr) № locally convex cone (P, U) such that P2 (or P) has the same u-bounded (or bounded below) sub sets under U and UT(or Ubr). The locally convex cone (P, Ur) is the inductive limit of the uc-cones (PU, UU)UeB, where B is the collection of all uniformly convex u-bounded subsets of P2. Also (P, Ubr) is the inductive limit of the uc-cones (PU,UU)UeB, where B = {uch({0} x B) : B is bounded below}. If (P,U) is bornological or b-bornological, then U is equivalent to Ur or Ubr, respectively.
Definition 5. We shall say that the locally convex V (or A)-lattice cone (P,U) is V (or A)-order bornological whenever every u-bounded V (or A)-lattice homomorphism from (P, U) in VA
V A V A
bornological. For example, every Vus (or Aus)-lattice cone is V (or A)-order bornological. Also us
cone which its convex quasiuniform structure has countable base is bornological. This shows
( P , U) U ( P , U)
bornological.
Example 2. Let X be a topological space, and let P be the cone of all R+-valued continuous functions on X, where R+ is endowed with the usual, that is the one-point compactification topology. We consider on P the pointiwise order. For each e > 0, we set £ = {(f,g) G P2 : Vx G X, f(x) ^ g(x) + e}. Then for each e > 0 e is a solid set and U = {e : e > 0} is a solid convex qusiuniform structure. Then (P,U) is a locally convex
( P , U)
us P
vector space.
Theorem 1. Let (PY, UY)7er be a family of locally convex V (or A)-lattice cones. Also let P be a V (or A)-lattice cone and for each y g r, fY : PY ^ P be a V (or A)-lattice hornoniorphisni such that P = span (U7er fY(PY)). Then P endowed with the convex quasiuniform structure U created by the sets of the form usv(U7er(fY x fY)(UY)) (or usA(U7er(fY x fY)(UY))), where UY G U7, is a locally convex V (or A)-lattice cone.
< We consider the case that (PY,UY)7er ^re locally convex ^^^^^^e conh. Firstly, we prove that the elements of P are bounded below with respect to the sets usv(U^p fY(UY))■
Let a G P. Then there are n G N Yb ..., Yn G ^^d aYi G P7i, i = 1,..., n, such that a = ^n=i fn^n)• There are Aj > 0 i = 1,... n, such that (0, aYi) G AiU7i. This shows that (0, a) G Aus(U7er f*7(UY)), where A = maxi^i^n Aj. Then (P,U) is a locally convex cone. Since the sets usv(U7er fY(UYV-solid, we conclude that (P,U) is a locally convex V-lattice cone. A similar Moment yields our claim for the case that (PY,UY)7er are locally A>
The projective and inductive limits had been investigated for topological vector spaces and locally convex cones in [8] and [7], respectively. Under the assumptions of Theorem 1, ( P , U) V A V A
(PY,UY), under the V (or A)-lattice homomorphisms fY : PY ^ P. Similarly, the concept of order inductive limit can de defined.
Corollary 3. An order inductive limit of locally convex lattice cones is a locally convex lattice cone.
Corollary 4. An order inductive limit of locally convex solid Riesz spaces is a locally convex solid Riesz space.
( P , U) V ( A )
V (or A)-lattice cones (PY,UY)y G r, under lattice homomorphisms fY : PY ^ P, y G r, and (Q, W) be a. locally convex cone. Then the linear mapping T : P ^ Q is continuous if and only if for every y G T, TofY is continuous.
< The mapping T is continuous if and only if for each W G W, (T x T)-i(W) G U. By Theorem 1, this holds if and only if for every y G r
f x f)-i((T x T)-i(W)) = (Tof7 x Tof7)-i(W) G U7.
In the other words, we require the continuity of each Tof7 for each y G r. >
V ( A) V ( A)
V ( A)
< Let (P,U) be the V (or A)-order inductive limit of V (or A)-order bornological locally-convex lattice cones (PY, UY) by the V (or A)-lattice homomorphisms f7, y G r. Also suppose
T u V A ( P , U)
convex V (or A)-lattice co ne (Q, W). Then for every y G r, To f7 is a u-bounde d V (or A ^lattice homomorphism on (PY,UY). Since (PY,UY) is V (or A)-order bornological, we conclude
that Tof7 is continuous for each y G r, by Proposition 4. Therefore T is continuous by >
Similarly, one can prove that an order inductive limit of order bornological locally convex lattice cones is order bornological.
( P , U) V ( A )
finest V (or A)-solid convex quasiuniform structure UjT| (or UjjT|) on P under which P2 has the same u-bounded subsets as under U. Under the convex quasiuniform structure UjT| (or UAT |), P is a. V (or A)-order bornological cone, the V (or A)-order inductive limit of a family of Vus (or Aus )-sub!attice cones of P. The locally convex cone (P,U) is V (or A)-order bornological if and only if U and UjT| (or UjT|) are equivalent.
< We prove the theorem for the case that (P,U) is a locally convex V-lattice cone. Let B
u V P2 B G B
Pb = {a G P : 3 A > 0 s.t. (0, a) G AB^d Us = {aB : a > 0}.
We consider on Pb the order induced by the original order of P. Since B is V-solid, it is easy to see that (PB,UB) is a locally convex Vus-lattice cone. We have P = (JBeB PB. Indeed, for a G P, Let B' be the smallest uniformly convex V-solid subset of P2, which contains {(0, a)}. Then B' G B and a G PB<- Now let (P,U|T|) be the V-order inductive limit of (PB,UB)bgb under the inclusion mappings IB : PB ^ P. Then (P,U|T|) is a locally convex V-lattice cone by Theorem 1. The u-boundedness of B G B shows that IB : (Pb,UB) ^ (P,U) is continuous. Now, we conclude that U|T| is finer than U by the definition of V-order inductive limit. Then u-boundedness in U|T| implies u-boundedness in U. On the other hand, if F C PMs u-bounded with respect to U, then it is u-bounded in (Pp,Up), where F = usv(F). Now, the continuity of Ip yields that F is u-bounded with respect to U^Also (P, UJT|) V
theorem, we note that the identity mapping I : (P,U) ^ (P,U|T|) is a u-bounded V-lattice
( P , U) I U
finer than U|T On the other hand U|r | is finer th an U. Then they are equivalent. Similarly we can prove the theorem for the case that (P, U) is a locally convex A-lattice cone. >
( P , U) V ( A ) V ( A )
cone, then U = U|T| (or U = Uj^|).
( P , U)
quasiuniform structure U|r| on P, under which P2 has the same u-bounded subsets as under
U U| r| P
inductive limit of a family of us-sublattice cones of P. The locally convex cone (P, U) is
U U| r|
(P, U) Ur U| r|
< Sinee P2 has the same u-bounded subsets under ^d U|r|, and Ut is the finest convex quasiuniform structure that has this property, we conclude that Ur is finer th an U|r |. >
Proposition 7. Every bornological locally convex lattice cone is order bornological.
< Let (P, U) be a bornological locally convex lattice cone. Then we have U = Ur. Now, since U C U|r | C U^, we conclude that U = U|r ^ >
VA
VA
Theorem 3. For locally convex V (or A)-lattice cone (P, U) the followings are equivalent:
(a) (P,U) is V(or A)-order bornological;
(b) for every uniformly convex V (or A)-solid subset V of P2 that absorbs all u-bounded subsets, there is U G U such th at U C V;
(c) every u-bounded V (or A)-lattice homomorphism from P into any Vus (or Aus)-lattice cone is continuous.
(d) (P, U) is the V ^ A)-order inductive limit of some us-lattice subcones of (P, U).
< The statements (a) and (d) are equivalent by Proposition 5 and Theorem 2.
(a ^ b): Let (a) holds and V be a uniformly convex V (or A)-solid subset of P2, that absorbs all u-bounded subsets. We set V = {aV : a > 0}. Then (P, V) is a locally convex V (or A)-lattice cone. The identity mappings I : (P, U) ^ (P, V) is u-bounded, since V absorbs all u-bounded subsets. On the other hand I is a V (or A)-lattice homomorphism. (a) I U G U U C V
(b ^ a): Let (b) holds and T be a u-bounded V (or A)-lattice ^^^^morphism from (P,U) into another locally convex V (or A)-lattice co ne (Q, W). Le t W G W be V (or A)-solid. Then
(T x T)-i(W) V A P2 T V A
tice homomorphism. Now there is U G U such that U C (T x T)-i(W) by (b). Then T is continuous.
(a ^ c): The proof is clear.
(c ^ a): Suppose that (c) holds and T is a u-bounded V (or A)-lattice homomorphism from (P,U) into another locally convex V (or A)-lattice cone (Q, W). For W G W, we set Ww = {aW : a > 0} Clearly for every W G W, T : (P, U) ^ (Q, Ww) is a u-bounded V A ( c) W G W
have W G WW. Therefore the re is U G U such th at (T x T )(U) C W. >
Corollary 7. As a special case in locally convex solid Riesz spaces, Theorem 3 (c) yields that a locally convex solid Riesz space E is order bornological if and only if every bounded lattice homomorphism from E into a seminormed Riesz space is continuous.
Definition 6. Let (P,U) and (Q, W) be locally convex V (or A)-lattice cones. We shall say that T is |t|v (or |t|A)-continuous whenever T : (P,UjT|) ^ (Q, Wj^) (or T : (P, UjT|) ^ (Q, WjTj| ^ is continuous. Similarly, we can define the concept of |t|-continuity.
Proposition 8. Let (P, U) and (Q, W) be locally con vex V (or A)-lattice cones. Then the V (or A)-lattice homomorphism T : (P, U) ^ (Q, W) is u-bounded if and only if T is |t|v (or |t|A)-continuous.
< Let (P,U) and (Q, W) be locally convex V-lattice cones and T : (P,U) ^ (Q, W) be a u-bounded V-lattice homomorphism. Then T : (P,UjT|) ^ (Q, Wt|) is u-bounded, since P2 has the same u-bounded subsets under U and UjT|- Now, since (P, UjjT|) is V-order bprnological, we conclude that T is |t|v-continuous. One the other hand if T is |t|v-continuous, then T : (P,UjT|) ^ (Q, is u-bounded. This implies that T : (P,U) ^ (Q, W) is u-bounded. On can prove the assertion for the case that (P,U^d (Q, W) are locally
A>
Corollary 8. Let (P, U) and (Q, W) be locally convex lattice cones. Then the lattice homomorphism T : (P, U) ^ (Q, W) is u-bounded if and only if it is |t|-coniinuous.
Corollary 9. Every continuous V (or A)-lattice homomorphism is |t|v (or |t|a)-continuo-us. Also, every continuous lattice homomorphism is |t|-coniinuous.
( P , U) V A
UvT| UjT| V A
( P , U) V ( A )
(PY,UY)7er under the V (or A)-lattice homomorphisms fY : PY ^ P, y G T. Then (P,U|T|) is the V (or A)-V (or A)-order inductive limit of locally convex lattice cones (PY, UjjT|Y)7er (or (PY,UjT|Y)Yer) under V (or A)-lattice homomorphisms fY : PY ^ P, Y G r.
< We prove the theorem for the case that (P, U) is the V-order inductive limit of locally convex lattice cones (PY,UY)7gr- For every y G r, fY : (PY,Uj^.|Y) ^ (P,U|T|) is continuous
(P, W) V
(PY,UvT|Y)7er under V-lattice homomorphisms fY : PY ^ P, Y G T. Then Uj^|Y C W, by
V P2 u
W Uv B u Uv u U
|t |Y |t |Y'
that for y G r, (fY x fY)-i(B) is u-bounded under U7, and then in U^.|7- Therefore B is u-bounded un der W. Now, this shows that the identity mapping I : (P, Uv) ^ (P, W) is
abounded. Then it is continuous, since (P, UjT|) is V-order bornological. Then W = UjT A similar argument yields our claim for the other case. >
Corollary 10. The convex quasiuniform structure U|T| is stable under the order inductive limit.
In [3], we introduced the weak convex quasiuniform structure on a locally convex cone. Here, we define the absolute weak convex quasiuniform structure on a locally convex V (or A)-lattice cone.
Let (P,ll) be a locally convex V (or A)-lattice cone and let Lv(or LA) is the set of all continuous V (or A)-lattice homomorphism from (P,ll) into (R, V). We denote by ll^ (P, Lv) (or U\a\(P,La)) the coarsest convex quasiuniform structure on P, that makes all y £ LV (or y £ La), continuous. In fact (P, U\CT\(P ,LV)) (or (P, U\CT\ (P ,La))) is the V (or A)-order projective limit of (R, Y) under all ¡j, £ Lv (or ¡j, £ LA). This shows that (P,il\a\(P, Lv)) (or (P,U\a\(P, La))) is a locally convex V (or A)-lattice cone. If (P,U) is a locally convex lattice cone and L is the set of all continuous lattice homomorphism from (P,il) into (R, V), then we can define the solid convex quasiuniform structure on U\a\ (P, L) in a similar way.
For a locally convex lattice cone (P, U), it is easy to see that Ua (P, P*) is finer than U\a\ (P,L). But for some locally convex lattice cones, these convex quasiuniform structures are equivalent.
Example 3. Let 1+ = R+ U {+oo}. We set U = {(a, b) £ (R+)2 : a ^ 6} and 11 = {[/}. If we consider usual order on R+, then 11 is a solid convex quasiuniform structure on R+ and (R+,ll) is a locally convex lattice cone (a us-lattice cone). The dual cone of (R+,ll) consists of all nonnegative reals an functionals 0 and +oo acting as:
f+oo, a = +oo, _ lO, a = 0,
0(a) = <! and +oo(a) =
10 else Ielse,
respectively. Since all elements of R+ are lattice homomorphism, we conclude that il|0.|(R+,L) = llo-(R+,R^). In fact, we have L = P*. It is easy to see that 11 is strictly-finer than HH (R+ ,L).
Definition 7. Let (P, U) and (Q, W) be locally co nvex V (or A)-lattice cones. The linear operator T : P ^ Q is called |^-continuous, whenever T : (P, U^ (P,LV))) ^ (Q, W\o\ (Q, Lv)) (or T :(P, UH (P,La))) ^ (Q, W\a\ (Q, La))) is continuous.
(P, U) (Q, W) V ( A)
every continuous V (or A)-lattice homomorphism from (P, U) into (Q, W) is lal-continuous.
< We prove the assertion for the case that (P, U^d (Q, W) be locally convex V-lattice cones. We denote the sets of all continuous V-lattice homomorphisms on Q and P by L'V and LV, respectively. Let T : (P,U) ^ (Q, W) be a continuous V-lattice homomorphism and Wk\ £ W^\ (Q, LV). Then there are n £ N Mid .. £ L'V such that
HA"1 (1) C
i= 1
where Ai = ßi x fa, for i = 1,... ,n. We have faoT £ Lv for i = l,...,n, since T is a continuous V-lattice homomorphism. We set ri = faoT x faoT, for i = 1,...,n. Then U\„\ = nn=i r-1(1) £ UW(P,Ly) and we have (T x T)(UW\) C W^. >
Theorem 5. Let (P,U) be a V (or A)-order bornological locally convex lattice cone and (Q, W) be a locally convex V (or A)-lattice cone which has the same u-bounded subsets under W and Woj (Q,Lv)) (or W°1 (Q, Lv))). Then every \<j{-continuous V (or A)-lattice ho-mornorphism from P into Q is continuous.
< Let T be a \ j\-continuous V (or A)-lattice homomorphism from P into Q. Since every u-bounded subset is weakly u-bounded, we conclude that T is u-bounded by the assumptions. Now, since (P, U) is order bornological, T is continuous. >
Corollary 11. Let (P,U) be a V (or A)-order bornological locally convex lattice cone.
Then every linear functional on (P, U), which is V (or A)-lattice homomorphism, is continuous |<|
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Received 21 November, 2015 Davood Ayaseh
Department of Pure Mathematics, University of Tabriz 29 Bahman Blvd., Tabriz, Iran E-mail: [email protected]
Asghar Ranjbari
Department of Pure Mathematics, University of Tabriz 29 Bahman Blvd., Tabriz, Iran E-mail: [email protected]
ПОРЯДКОВО БОРНОЛОГИЧЕСКИЕ ЛОКАЛЬНО ВЫПУКЛЫЕ РЕШЕТОЧНЫЕ КОНУСЫ
Айазех Д., Ранджбари А.
В статье вводятся понятия us-решеточного конуса и норядково борнологического локально выпуклого решеточного конуса. В специальном случае локально солидного (= нормального) пространства Рисса (= векторной решетки) эти понятия сводятся к хорошо известным понятиям полунормированного пространства Рисса и порядково борнологического пространства Рисса, соответственно. Вводится также понятие солидного множества в локально выпуклом конусе и даются некоторые характеризации порядково борнологических локально выпуклых решеточных конусов.
Ключевые слова: локально выпуклый решеточный конус, порядково борнологический конус.
