Mathematics Scientia Bruneiana Vol. 16 2017
An Introduction to Locally Convex Cones
Walter Roth*
Department of Mathematics, Faculty of Science, Universiti Brunei Darussalam, Jalan Tungku Link, Gadong, BE 1410, Brunei Darussalam
*corresponding author email: [email protected]
Abstract This survey introduces and motivates the foundations of the theory of locally convex cones which aims to generalize the well-established theory of locally convex topological vector spaces. We explain the main concepts, provide definitions, principal results, examples and applications. For details and proofs we generally refer to the literature.
Index Terms: cone-valued functions, locally convex cones, Korovkin type approximation
1. Introduction shall give an outline of the principal concepts of Endowed with suitable topologies, vector spaces this emerging theory. We survey the main results yield rich and well-considered structures. Locally including some yet unpublished ones and provide convex topological vector spaces in particular primary examples and applications. However, we permit an effective duality theory whose study shall generally refrain from supplying technical provides valuable insight into the spaces details and proofs but refer to different sources themselves. Some important mathematical instead. settings, however – while close to the structure of vector spaces – do not allow subtraction of their 2. Ordered cones and monotone linear elements or multiplication by negative scalars. functionals Examples are certain classes of functions that may A cone is a set 푃 endowed with an addition take infinite values or are characterized through inequalities rather than equalities. They arise (푎, 푏) → 푎 + 푏 naturally in integration and in potential theory. Likewise, families of convex subsets of vector and a scalar multiplication spaces which are of interest in various contexts do not form vector spaces. If the cancellation law (훼, 푎) → 훼푎 fails, domains of this type may not even be embedded into larger vector spaces in order to for 푎 ∈ 푃 and real numbers 훼 ≥ 0. The addition is apply results and techniques from classical supposed to be associative and commutative, and functional analysis. They merit the investigation there is a neutral element 0 ∈ 푃, that is: of a more general structure. (푎 + 푏) + 푐 = 푎 + (푏 + 푐) for all 푎, 푏, 푐 ∈ 푃 The theory of locally convex cones as developed 푎 + 푏 = 푏 + 푎 for all 푎, 푏 ∈ 푃 in [7] admits most of these settings. A topological 0 + 푎 = 푎 for all 푎 ∈ 푃 structure on a cone is introduced using order- theoretical concepts. Staying reasonably close to For the scalar multiplication the usual associative the theory of locally convex spaces, this approach and distributive properties hold, that is: yields a sufficiently rich duality theory including Hahn-Banach type extension and separation 훼(훽푎) = (훼훽)푎 for all 훼, 훽 ≥ 0 and theorems for linear functionals. In this article we 푎 ∈ 푃
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(훼 + 훽)푎 = 훼푎 + 훽푎 for all 훼, 훽 ≥ 0 and 퐴 + 퐵 = {푎 + 푏| 푎 ∈ 퐴 and 푏 ∈ 퐵} 푎 ∈ 푃 훼(푎 + 푏) = 훼푎 + 훼푏 for all 훼 ≥ 0 and for 퐴, 퐵 ∈ 퐶표푛푣(푃), and 푎, 푏 ∈ 푃 1푎 = 푎 for all 푎 ∈ 푃 푎퐴 = {훼푎| 푎 ∈ 퐴} 0푎 = 0 for all 푎 ∈ 푃 for 퐴 ∈ 퐶표푛푣(푃) and 훼 ≥ 0, it is easily verified Unlike the situation for vector spaces, the that 퐶표푛푣(푃) is again a cone. Convexity is condition 0푎 = 0 needs to be stated independently required to show that (훼 + 훽)퐴 equals 훼퐴 + 훽퐴. for cones, as it is not a consequence of the The set inclusion defines a suitable order on preceding requirements (see [6]). The 퐶표푛푣(푃) that is compatible with these algebraic cancellation law, stating that operations. The cancellation law generally fails for 퐶표푛푣(푃). (C) 푎 + 푐 = 푏 + 푐 implies that 푎 = 푏 (c) Let 푃 be an ordered cone, 푋 any non-empty set. however, is not required in general. It holds if and For 푃-valued functions on 푋 the addition, scalar only if the cone 푃 can be embedded into a real multiplication and order may be defined vector space. pointwise. The set 퐹(푋, 푃) of all such functions again becomes an ordered cone for which the A subcone 푄 of a cone 푃 is a non-empty subset of cancellation law holds if and only if it holds for 푃. 푃 that is closed for addition and multiplication by non-negative scalars. A linear functional on a cone 푃 is a mapping 휇: 푃 → ℝ̅ such that An ordered cone 푃 carries additionally a reflexive transitive relation ≤ that is compatible with the 휇(푎 + 푏) = 휇(푎) + 휇(푏) and 휇(훼푎) = 훼휇(푎) algebraic operations, that is holds for all 푎, 푏 ∈ 푃 and 훼 ≥ 0. Note that linear 푎 ≤ 푏 implies that 푎 + 푐 ≤ 푏 + 푐 and 훼푎 ≤ 훼푏 functionals take only finite values in invertible elements of 푃. If 푃 is ordered, then 휇 is called for all 푎, 푏, 푐 ∈ 푃 and 훼 ≥ 0. As equality in 푃 is monotone if obviously such an order, all our results about ordered cones will apply to cones without order 푎 ≤ 푏 implies that 휇(푎) ≤ 휇(푏). structures as well. We provide a few examples: In various places the literature deals with linear 2.1 Examples. (a) In ℝ̅ = ℝ ∪ {+∞} we functionals on cones that take values in ℝ ∪ {−∞} consider the usual order and algebraic operations, (see [6]) instead. In vector spaces both approaches in particular 훼 + ∞ = +∞ for all 훼 ∈ ℝ̅, 훼 ∙ coincide, as linear functionals can take only finite (+∞) = +∞ for all 훼 > 0 and 0 ∙ (+∞) = 0. values there, but in applications for cones the value +∞ arises more naturally. (b) Let 푃 be a cone. A subset 퐴 of 푃 is called convex if The existence of sufficiently many monotone 훼푎 + (1 − 훼)푏 ∈ 퐴 linear functionals on an ordered cone is guaranteed by a Hahn-Banach type sandwich whenever 푎, 푏 ∈ 퐴 and 0 ≤ 훼 ≤ 1.We denote by theorem whose proof may be found in [13] or in a 퐶표푛푣(푃) the set of all non-empty convex subsets rather weaker version in [7]. It is the basis for the of 푃. With the addition and scalar multiplication duality theory of ordered cones. In this context, a defined as usual by sublinear functional on a cone 푃 is a mapping 푝 ∶ 푃 → ℝ̅ such that
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푝(훼푎) = 훼푝(푎) and 푝(푎 + 푏) ≤ 푝(푎) + 푝(푏) (U1) 푣 is convex. (U2) If 푎 ≤ 푏 for 푎, 푏 ∈ 푃, then (푎, 푏) ∈ 푣. holds for all 푎, 푏 ∈ 푃 and 훼 ≥ 0. Likewise, a (U3) If (푎, 푏) ∈ 푣 and (푏, 푐) ∈ 𝜌푣 for superlinear functional on 푃 is a mapping 푞 ∶ 푃 → , 𝜌 > 0, then (푎, 푐) ∈ ( + 𝜌)푣. ℝ̅ such that (U4) For every 푏 ∈ 푃 there is ≥ 0 such that (0, 푏) ∈ 푣. 푞(훼푎) = 훼푞(푎) and 푞(푎 + 푏) ≥ 푞(푎) + 푞(푏) holds for all 푎, 푏 ∈ 푃 and 훼 ≥ 0. Note that Any subset 푣 of 푃2 with the above properties (U1) superlinear functionals can assume only finite to (U4) qualifies as a uniform neighborhood for 푃, values in invertible elements of 푃. and any family 푉 of such neighborhoods fulfilling the usual conditions for a quasiuniform structure, It is convenient to use the pointwise order relation that is: for functions 푓, 푔 on 푃; that is we shall write 푓 ≤ 푔 to abbreviate 푓(푎) ≤ 푔(푎) for all 푎 ∈ 푃. (U5) For 푢, 푣 ∈ 푉 there is 푤 ∈ 푉 such that 푤 푢 ∩ 푣. 2.2 Sandwich Theorem (algebraic). Let 푃 be an (U6) 푣 ∈ 푉 for all 푣 ∈ 푉 and > 0. ordered cone and let 푝 ∶ 푃 → ℝ̅ be a sublinear and 푞 ∶ 푃 → ℝ̅ a superlinear functional such that generates a locally convex cone (푃, 푉) as elaborated in [7]. More specifically, 푉 creates 푞(푎) ≤ 푝(푏) whenever 푎 ≤ 푏 for 푎, 푏 ∈ 푃. three hyperspace topologies on 푃 and every 푣 ∈ 푉 defines neighborhoods for an element 푎 ∈ 푃 by There exists a monotone linear functional 휇: 푃 → ℝ̅ such that 푞 ≤ 휇 ≤ 푝. 푣(푎) = {푏 ∈ 푃| (푏, 푎) ∈ 푣 for all > 1} in the upper topology Note that the above condition for 푞 and 푝 is (푎)푣 = {푏 ∈ 푃| (푎, 푏) ∈ 푣 for all > 1} fulfilled if 푞 ≤ 푝 and if one of these functionals is in the lower topology monotone. The superlinear functional 푞 may 푣(푎)푣 = 푣(푎) ∩ (푎)푣 however not be omitted altogether (or in the symmetric topology equivalently, replaced by one that also allows the value −∞) without further assumptions. (see However, it is convenient to think of a locally Example 2.2 in [13].) convex cone (푃, 푉) as a subcone of a full locally convex cone 푃̃, i.e. a cone that contains the 3. Locally convex cones neighborhoods 푣 as positive elements (see [7], Ch. Because subtraction and multiplication by I). negative scalars are generally not available, a topological structure for a cone should not be Referring to the order in 푃̃, the relation 푎 ∈ 푣(푏) expected to be invariant under translation and may be reformulated as 푎 ≤ 푏 + 푣. This leads to a scalar multiplication. There are various equivalent second and equivalent approach to locally convex approaches to locally convex cones as outlined in cones that uses the order structure of a larger full [7]. The use of convex quasiuniform structures is cone in order to describe the topology of 푃 (for motivated by the following features of relations between order and topology we refer to neighborhoods in a cone: With every ℝ̅-valued [9]). Let us indicate how this full cone 푃̃ may be monotone linear functional 휇 on an ordered cone constructed (for details, see [7], Ch. I.5): For a 푃 we may associate a subset fixed neighborhood 푣 ∈ 푉 set
푣 = {(푎, 푏) ∈ 푃2| 휇(푎) ≤ 휇(푏) + 1} 푃̃ = {푎 훼푣| 푎 ∈ 푃, 0 ≤ 푎 < +∞}. of 푃2 with the following properties:
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We use the obvious algebraic operations on 푃̃ and topology is the usual topology on ℝ with +∞ as the order an isolated point.
푎 훼푣 ≤ 푏 훽푣 (b) For the subcone ℝ̅ + = {푎 ∈ ℝ̅| 푎 ≥ 0} of ℝ̅ we may also consider the singleton neighborhood if either 훼 = 훽 and 푎 ≤ 푏, or 훼 < 훽 and (푎, 푏) ∈ system 푉 = {0}. The elements of ℝ̅ + are 푣 for all > 훽 − 훼. The embedding 푎 → 푎 0푣 obviously bounded below even with respect to the preserves the algebraic operations and the order of neighborhood 푣 = 0, hence ℝ̅ + is a full locally 푃. The procedure for embedding a locally convex convex cone. For 푎 ∈ ℝ̅ the intervals (−∞, 푎] and cone (푃, 푉) into a full cone (푃̃, 푉) that contains a [푎, +∞] are the only upper and lower whole system 푉 of neighborhoods as positive neighborhoods, respectively. The symmetric elements is similar and elaborated in Ch. I.5 of [7]. topology is the discrete topology on ℝ̅ +. The quasiuniform structure of 푃 may then be recovered through the subsets (c) Let (퐸, 푉, ≤) be a locally convex ordered topological vector space, where 푉 is a basis of {(푎, 푏) ∈ 푃2| 푎 ≤ 푏 + 푣} 푃2 closed, convex, balanced and order convex neighborhoods of the origin in 퐸. Recall that corresponding to the neighborhoods 푣 ∈ 푉. equality is an order relation, hence this example will cover locally convex spaces in general. In We shall in the following use this order-theoretical order to interpret 퐸 as a locally convex cone we approach: We may always assume that a given shall embed it into a larger full cone. This is done locally convex cone (푃, 푉) is a subcone of a full in a canonical way: Let 푃 be the cone of all non- locally convex cone (푃̃, 푉) that contains all empty convex subsets of 퐸, endowed with the neighborhoods as positive elements, and we shall usual addition and multiplication of sets by non- use the order of the latter to describe the topology negative scalars, that is of 푃. The above conditions (U1) to (U6) for the quasiuniform structure on 푃 equivalently translate 훼퐴 = {훼푎| 푎 ∈ 퐴} and into conditions involving the order relation of 푃̃ as 퐴 + 퐵 = {푎 + 푏| 푎 ∈ 퐴 and 푏 ∈ 퐵} follows: for 퐴, 퐵 ∈ 푃 and 훼 ≥ 0. We define the order on (V1) 푣 ≥ 0 for all 푣 ∈ 푉. 푃 by (V2) For 푢, 푣 ∈ 푉 there is 푤 ∈ 푉 such that 퐴 ≤ 퐵 if 퐴 ↓ 퐵 = 퐵 + 퐸− 푤 ≤ 푢 and 푤 ≤ 푣. (V3) 푣 ∈ 푉 whenever 푣 ∈ 푉 and > 0. where 퐸− = {푥 ∈ 퐸| 푥 ≤ 0} is the negative cone (V4) For 푣 ∈ 푉 and every 푎 ∈ 푃 there is ≥ 0 in 퐸. The requirements for an ordered cone are such that 0 ≤ 푎 + 푣. easily checked. The neighborhood system in 푃 is given by the neighborhood basis 푉 푃. We Condition (V4) states that every element 푎 ∈ 푃 is observe that for every 퐴 ∈ 푃 and 푣 ∈ 푉 there is bounded below. 𝜌 > 0 such that 𝜌푣 ∩ 퐴 ≠ ∅. This yields 0 ∈ 퐴 + 𝜌푣. Therefore {0} ≤ 퐴 + 𝜌푣, and every element 3.1 Examples. (a) The ordered cone ℝ̅ endowed 퐴 ∈ 푃 is indeed bounded below. Thus (푃, 푉) is a with the neighborhood system 푉 = {휀 ∈ ℝ| 휀 > full locally convex cone. Via the embedding 푥 → 0} is a full locally convex cone. For 푎 ∈ ℝ the {푥} ∶ 퐸 → 푃 the space 퐸 itself is a subcone of 푃. intervals (−∞, 푎 + 휀] are the upper and the This embedding preserves the order structure of 퐸, intervals [푎 − 휀, +∞] are the lower and on its image the symmetric topology of 푃 neighborhoods, while for 푎 = +∞ the entire cone coincides with the given vector space topology of ℝ̅ is the only upper neighborhood, and {+∞} is 퐸. Thus 퐸 is indeed a locally convex cone, but not open in the lower topology. The symmetric a full cone.
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(d) The preceding procedure can be applied to In this case we consider the subcone 퐹푏푦(푋, 푃) of locally convex cones in general. Let (푃, 푉) be a all functions in 퐹(푋, 푃) that are uniformly locally convex cone and let 퐶표푛푣(푃) denote the bounded below on the sets in 푌. Together with the cone of all non-empty convex subsets of 푃, neighborhood system 푉푌, it forms a locally convex endowed with the canonical order, that is cone. (퐹푏푦(푋, 푃), 푉푌) carries the topology of uniform convergence on the sets in 푌. 퐴 ≤ 퐵 if for every 푎 ∈ 퐴 there is 푏 ∈ 퐵 such that 푎 ≤ 푏 (f) For 푥 ∈ ℝ̅ denote 푥+ = max {푥, 0} and 푥− = min {푥, 0}. For 1 ≤ 푝 ≤ +∞ and a sequence for 퐴, 퐵 푃. The neighborhood 푣 ∈ 푉 is defined 푝 (푥푖)푖∈ℕ in ℝ̅ let ‖푥푖‖푝 denote the usual 푙 norm, as a neighborhood for 퐶표푛푣(푃) by that is ∞ (1/푝) 퐴 ≤ 퐵 + 푣 if for every 푎 ∈ 퐴 there is 푏 ∈ 퐵 ‖(푥 )‖ = (∑|푥 |푝) ∈ ℝ̅ such that 푎 ≤ 푏 + 푣 푖 푝 푖 푖=1
The requirements for a locally convex cone are for 푝 < +∞, and easily checked for (퐶표푛푣(푃), 푉), and (푃, 푉) is identified with a subcone of (퐶표푛푣(푃), 푉). Other ‖(푥 )‖ = sup{|푥 || i ∈ ℕ} ∈ ℝ̅. subcones of 퐶표푛푣(푃) that merit further 푖 ∞ 푖 investigation are those of all closed, closed and Now let 푝 be the cone of all sequences in bounded, or compact convex sets in 퐶표푛푣(푃), 퐶 (푥푖)푖∈ℕ ̅ respectively. Details on these and further related ℝ such that ‖(푥푖)‖푝 < +∞ . We use the pointwise 푝 examples may be found in [7] and [17]. order in 퐶 and the neighborhood system 푉푝 = {𝜌푣푝| 𝜌 > 0}, where (e) Let (푃, 푉) be a locally convex cone, 푋 a set and let 퐹(푋, 푃) be the cone of all 푃-valued (푥푖)푖∈ℕ ≤ (푦푖)푖∈ℕ + 𝜌푣푝 functions on 푋, endowed with the pointwise ̅ + operations and order. If 푃 is a full cone containing means that ‖(푥푖 − 푦푖) ‖푝 ≤ 𝜌. (In this expression both 푃 and 푉 then we may identify the elements the 푙푝 norm is evaluated only over the indices 𝑖 ∈ 푣 ∈ 푉 with the constant functions 푥 → 푣 for all ℕ for which 푦푖 < +∞.) It can be easily verified 푝 푥 ∈ 푋, hence 푉 is a subset and a neighborhood that (퐶 , 푉푝) is a locally convex cone. In fact system for 퐹(푋, 푃̅). A function 푓 ∈ 퐹(푋, 푃̅) is 푝 (퐶 , 푉푝) can be embedded into a full cone uniformly bounded below, if for every 푣 ∈ 푉 there following a procedure analogous to that in 2.1 (c). is 𝜌 ≥ 0 such that 0 ≤ 푓 + 𝜌푣. These functions The case for 푝 = +∞ is of course already covered ̅ form a full locally convex cone (퐹푏(푋, 푃), 푉), by Part (d). carrying the topology of uniform convergence. As ̅ a subcone, (퐹푏(푋, 푃), 푉) is a locally convex cone. 4. Continuous linear functionals and Hahn- Alternatively, a more general neighborhood Banach type theorems system 푉푌 for 퐹(푋, 푃) may be created using a A linear functional 휇 on a locally convex cone suitable family 푌 of subsets 푦 of 푋, directed (푃, 푉) is said to be (uniformly) continuous with downward with respect to set inclusion, and the respect to a neighborhood 푣 ∈ 푉 if neighborhoods 푣푦 for 푣 ∈ 푉 and 푦 ∈ 푌, defined for functions 푓, 푔 ∈ 퐹(푋, 푃) as 휇(푎) ≤ 휇(푏) + 1 whenever 푎 ≤ 푏 + 푣.
푓 ≤ 푔 + 푣푦 if 푓(푥) ≤ 푔(푥) + 푣 Continuity implies that the functional 휇 is for all 푥 ∈ 푦. monotone, even with respect to the global preorder ≲, and takes only finite values in bounded elements 푏 ∈ ℬ (see Section 5 below).
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The set of all linear functionals 휇 on 푃 which are some 푐 ∈ 퐶. A convex set 퐶 푃 such that 0 ∈ 퐶 continuous with respect to a certain neighborhood is called left-absorbing if for every 푎 ∈ 푃 there are ○ 푣 is called the polar of 푣 in 푃 and denoted by 푣푃 푐 ∈ 퐶 and ≥ 0 such that 푐 ≤ 푎. (or 푣○ for short). Endowed with the canonical addition and multiplication by non-negative 4.2 Corollary. Let 푃 be an ordered cone. For a scalars, the union of all polars 푣○ for 푣 ∈ 푉 forms sublinear functional 푝 ∶ 푃 → ℝ̅ there exists a the dual cone 푃∗ of 푃. monotone linear functional 휇 ∶ 푃 → ℝ̅ such that 휇 ≤ 푝 if and only if 푝 is bounded below on some We may now formulate a topological version of increasing left-absorbing convex set 퐶 푃. the sandwich theorem (Theorem 3.1 in [13]) for linear functionals: Generalizing our previous An ℝ̅-valued function 푓 defined on a convex notion we define an extended superlinear subset 퐶 of a cone 푃 is called convex if functional on 푃 as a mapping 푓(푐1 + (1 − )푐2) ≤ 푓(푐1) + (1 − )푓(푐2) 푞: 푃 → ℝ̅ = ℝ ∪ {+∞, −∞} holds for all 푐1, 푐2 ∈ 퐶 and ∈ [0,1]. Likewise, an such that 푞(훼푎) = 훼푞(푎) holds for all 푎 ∈ 푃 and ℝ̅-valued function 푔 on 퐶 is concave if 훼 ≥ 0 and 푔(푐1 + (1 − )푐2) ≥ 푔(푐1) + (1 − )푔(푐2) 푞(푎 + 푏) ≥ 푞(푎) + 푞(푏) whenever 푞(푎), 푞(푏) > −∞ holds for all 푐1, 푐2 ∈ 퐶 such that 푔(푐1), 푔(푐2) > −∞ and ∈ [0,1]. An affine function ℎ ∶ 퐶 → ℝ̅ (We set 훼 + (−∞) = −∞ for all 훼 ∈ ℝ ∪ {−∞}, is both convex and concave. A variety of 훼 ∙ (−∞) = −∞ for all 훼 > 0 and 0 ∙ (−∞) = 0 extension results for linear functionals may be in this context.) derived from Theorem 4.1 in [13]. We cite:
4.1 Sandwich Theorem (topological). Let 4.3 Extension Theorem. Let (푃, 푉) be a locally (푃, 푉) be a locally convex cone, and let 푣 ∈ 푉. convex cone, 퐶 and 퐷 non-empty convex subsets For a sublinear functional 푝 ∶ 푃 → ℝ̅ and an of 푃, and let 푣 ∈ 푉. Let 푝 ∶ 푃 → ℝ̅ be a sublinear extended superlinear functional 푞 ∶ 푃 → ℝ̅ there and 푞 ∶ 푃 → ℝ̅ an extended superlinear exists a linear functional 휇 ∈ 푣○ such that 푞 ≤ functional. For a convex function 푓 ∶ 퐶 → ℝ̅ and 휇 ≤ 푝 if and only if a concave function 푔 ∶ 퐷 → ℝ̅ there exists a monotone linear functional 휇 ∈ 푣○ such that 푞(푎) ≤ 푝(푏) + 1 holds whenever 푎 ≤ 푏 + 푣 푞 ≤ 휇 ≤ 푝, 푔 ≤ 휇 on 퐷 and 휇 ≤ 푓 on 퐶 Recall that every monotone linear functional 휇 on an ordered cone 푃 gives rise to a uniform if and only if 2 neighborhood 푣 = {(푎, 푏) ∈ 푃 | 휇(푎) ≤ 휇(푏) + 1} which in turn may be used to define a locally 푞(푎) + 𝜌푔(푑) ≤ 푝(푏) + 𝜎푓(푐) + 1 holds convex structure on 푃. Thus, the condition for 푝 whenever 푎 + 𝜌푑 ≤ 푏 + 𝜎푐 + 푣 and 푞 in Theorem 4.1 for some neighborhood 푣 is necessary and sufficient for the existence of a for 푎, 푏 ∈ 푃, 푐 ∈ 퐶, 푑 ∈ 퐷 and 𝜌, 𝜎 > 0 such that monotone linear functional 휇 on 푃 such that 푞 ≤ 푞(푎), 𝜌푔(푑) > −∞. 휇 ≤ 푝. The generality of this result allows a wide range Citing from [13] we mention a few corollaries. A of special cases. If 푔 ≡ −∞, for example, we have set 퐶 푃 is called increasing resp. decreasing, if to consider the condition of Theorem 4.3 only for 푎 ∈ 퐶 whenever 푐 ≤ 푎 resp. 푎 ≤ 푐 for 푎 ∈ 푃 and 𝜌 = 0, if 푓 ≡ +∞ only for 𝜎 = 0, and if both 푔 ≡
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−∞ and 푓 ≡ +∞, then Theorem 4.3 reduces to the 휇(푐) ≤ 훼 ≤ 휇(푑) for all 푐 ∈ 퐶 and 푑 ∈ 퐷 previous Sandwich Theorem 4.1. Another case of particular interest occurs when 퐶 = 퐷 and 푓 = 푔 if and only if is an affine function, resp. a linear functional if 퐶 is a subcone of 푃. The latter, with the choice of 훼𝜌 ≤ 훼𝜎 + 1 whenever 𝜌푑 ≤ 𝜎푐 + 푣 푝(푎) = +∞ and 푞(푎) = −∞ for all 0 ≠ 푎 ∈ 푃 yields the Extension Theorem II.2.9 from [7]: for all 푐 ∈ 퐶, 푑 ∈ 퐷 and 𝜌, 𝜎 ≥ 0.
4.4 Corollary. Let (퐶, 푉) be a subcone of the 5. The weak preorder and the relative locally convex cone (푃, 푉). Every continuous topologies linear functional on 퐶 can be extended to a We also consider a (topological and linear) closure continuous linear functional on 푃; more precisely: of the given order on a locally convex cone, called ○ ○ For every 휇 ∈ 푣퐶 there is 휇̃ ∈ 푣푃 such that 휇̃ the weak preorder ≼ which is defined as follows coincides with 휇 on 퐶. (see I.3 in [17]): We set
The range of all continuous linear functionals that 푎 ≼ 푏 + 푣 for 푎, 푏 ∈ 푃 and 푣 ∈ 푉 are sandwiched between a given sublinear and an extended superlinear functional is described in if for every 휀 > 0 there is 1 ≤ 훾 ≤ 1 + 휀 such that Theorem 5.1 in [13]. 푎 ≤ 훾푏 + (1 + 휀)푣, and set 푎 ≼ 푏 4.5 Range Theorem. Let (푃, 푉) be a locally convex cone. Let 푝 and 푞 be sublinear and if 푎 ≼ 푏 + 푣 for all 푣 ∈ 푉. This order is clearly extended superlinear functionals on 푃 and weaker than the given order, that is 푎 ≤ 푏 or 푎 ≤ suppose that there is at least one linear functional 푏 + 푣 implies 푎 ≼ 푏 or 푎 ≼ 푏 + 푣. Importantly, 휇 ∈ 푃∗ satisfying 푞 ≤ 휇 ≤ 푝. Then for all 푎 ∈ 푃 the weak preorder on a locally convex cone is we have entirely determined by its dual cone 푃∗, that is 푎 ≼ 푏 holds if and only if 휇(푎) ≤ 휇(푏) for all 휇 ∈ 푃∗, sup휇∈푃∗,푞≤휇≤푝휇(푎) and 푎 ≼ 푏 + 푣 if and only 휇(푎) ≤ 휇(푏) + 1 for ○ = sup푣∈푉inf{푝(푏) − 푞(푐)| 푏, 푐 all 휇 ∈ 푣 (Corollaries I.4.31 and I.4.34 in [17]). ∈ 푃, 푞(푐) ∈ ℝ, 푎 + 푐 ≤ 푏 + 푣} If endowed with the weak preorder (푃, 푉) is again a locally convex cone with the same dual 푃∗. For all 푎 ∈ 푃 such that 휇(푎) is finite for at least one 휇 ∈ 푃∗ satisfying 푞 ≤ 휇 ≤ 푝 we have While all elements of a locally convex cone are bounded below, they need not be bounded above. inf휇∈푃∗,푞≤휇≤푝휇(푎) An element 푎 ∈ 푃 is called bounded (above) (see [7], I.2.3) if for every 푣 ∈ 푉 there is > 0 such = inf푣∈푉sup{푞(푐) − 푝(푏)| 푏, 푐 ∈ 푃, 푝(푏) ∈ ℝ, 푐 ≤ 푎 + 푏 + 푣} that 푎 ≤ 푣. By ℬ we denote the subcone of 푃 containing all bounded elements. ℬ is indeed a As another consequence of the Extension face of 푃, as 푎 + 푏 ∈ ℬ for 푎, 푏 ∈ 푃 implies that Theorem 4.3 we obtain the following result both 푎, 푏 ∈ ℬ. Clearly all invertible elements of 푃 (Theorem 4.5 in [13]) about the separation of are bounded, and bounded elements satisfy a convex subsets by monotone linear functionals: modified version of the cancellation law (see [17], I.4.5), that is 4.6 Separation Theorem. Let 퐶 and 퐷 be non- empty convex subsets of a locally convex cone (C) 푎 + 푐 ≼ 푏 + 푐 for 푎, 푏 ∈ 푃 and 푐 ∈ ℬ (푃, 푉). Let 푣 ∈ 푉 and 훼 ∈ ℝ. There exists a implies 푎 ≼ 푏 monotone linear functional 휇 ∈ 푣○ such that We quote Theorem I.3.3 from [17]:
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5.1 Representation Theorem. A locally convex (푃, 푉) are bounded relative to each other if for cone (푃, 푉) endowed with its weak preorder can every 푣 ∈ 푉 there are 훼, 훽, , 𝜌 ≥ 0 such that both be represented as a locally convex cone of ℝ̅- valued functions on some set 푋, or equivalently as 푎 ≤ 훽푏 + 푣 and 푏 ≤ 훼푎 + 𝜌푣 a locally convex cone of convex subsets of some locally convex ordered topological vector space. This notion defines an equivalence relation on 푃 and its equivalence classes ℬ푠(푎) are called the The previously introduced upper, lower and (symmetric) boundedness components of 푃. symmetric locally convex cone topologies for a Propositions 5.3, 5.4, 5.6 and 6.1 in [16] state: locally convex cone (푃, 푉) prove to be too restrictive for the concept of continuity of 푃- 6.1 Proposition. The boundedness components of valued functions, since for unbounded elements a locally convex cone (푃, 푉) are closed for even the scalar multiplication turns out to be addition and multiplication by strictly positive discontinuous (see I.4 in [17]). This is remedied scalars. They satisfy a version of the cancellation by using the coarser (but somewhat cumbersome) law, that is relative topologies on 푃 instead. These topologies 푎 + 푐 ≼ 푏 + 푐 are defined using the weak preorder on 푃: for elements 푎, 푏 and 푐 of the same boundedness The upper, lower and symmetric relative component implies that topologies on a locally convex cone (푃, 푉) are generated by the neighborhoods 푣휀(푎), (푎)푣휀 and 푎 ≼ 푏. 푠 푣휀 (푎) = 푣휀(푎) ∩ (푎)푣휀, respectively, for 푎 ∈ 푃, 푣 ∈ 푉 and 휀 > 0, where 6.2 Proposition. The boundedness components of a locally convex cone (푃, 푉) furnish a partition of 푣휀(푎) = {푏 ∈ 푃| 푏 ≤ 훾푎 + 휀푣 for some 푃 into disjoint convex subsets that are closed and 1 ≤ 훾 ≤ 1 + 휀} connected in the symmetric relative topology. (푎)푣휀 = {푏 ∈ 푃| 푎 ≤ 훾푏 + 휀푣 for some They coincide with the connectedness components 1 ≤ 훾 ≤ 1 + 휀} of 푃.
The relative topologies are locally convex but not If the neighborhood system 푉 consists of the necessarily locally convex cone topologies in the positive multiples of a single neighborhood, 푃 is sense of Section 3 (for details see I.4 in [17]), since locally connected and its connectedness the resulting uniformity need not be convex. components are also open. These topologies are generally coarser, but locally coincide on bounded elements with the given 7. Continuous linear operators upper, lower and symmetric topologies on 푃 and For cones 푃 and 푄 a mapping 푇 ∶ 푃 → 푄 is called render the scalar multiplication (with scalars other a linear operator if than zero) continuous. The symmetric relative topology is known to be Hausdorff if and only if 푇(푎 + 푏) = 푇(푎) + 푇(푏) and the weak preorder on 푃 is antisymmetric 푇(훼푎) = 훼푇(푎) (Proposition I.4.8 in [17]). If 푃 is a locally convex topological vector space, then all of the above hold for all 푎, 푏 ∈ 푃 and 훼 ≥ 0. If both 푃 and 푄 topologies coincide with the given topology. are ordered, then 푇 is called monotone if
6. Boundedness and connectedness components 푎 ≤ 푏 implies 푇(푎) ≤ 푇(푏). The details for this section can be found in [16]. Two elements 푎 and 푏 of a locally convex cone
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If both (푃, 푉) and (푄, 푊) are locally convex refrain from supplying the details. We cite the cones, then 푇 is said to be (uniformly) continuous main result, which generalizes the classical if for every 푤 ∈ 푊 one can find 푣 ∈ 푉 such that uniform boundedness theorem:
푇(푎) ≤ 푇(푏) + 푤 whenever 푎 ≤ 푏 + 푣 7.1 Uniform Boundedness Theorem. Let (푃, 푉) and (푄, 푊) be locally convex cones, and let 푇̂ be for 푎, 푏 ∈ 푃. A set 푇̂ of linear operators is called a family of u-continuous linear operators from 푃 equicontinuous if the above condition holds for to 푄. Suppose that for every 푏 ∈ 푃 and 푤 ∈ 푊 every 푤 ∈ 푊 with the same 푣 ∈ 푉 for all 푇 ∈ 푇̂. there is 푣 ∈ 푉 such that for every 푎 ∈ 푣(푏) ∩ Uniform continuity for an operator implies (푏)푣 there is > 0 such that monotonicity with respect to the global preorders on 푃 and on 푄 that is: if 푇(푎) ≤ 푇(푏) + 푤 for all 푇 ∈ 푇̂
푎 ≤ 푏 + 푣 for all 푣 ∈ 푉, then If (푃, 푉) is barreled and (푄, 푊) has the strict 푇(푎) ≤ 푇(푏) + 푤 for all 푤 ∈ 푊 separation property [that is, (푄, 푊) satisfies Theorem 4.6)], then for every internally bounded In this context, a linear functional is a linear set ℬ 푃, every 푏 ∈ ℬ and 푤 ∈ 푊 there is 푣 ∈ 푉 operator 휇 ∶ 푃 → ℝ̅, and the above notion of and > 0 such that continuity conforms to the preceding one (see Section 4). Moreover, for two continuous linear 푇(푎) ≤ 푇(푏) + 푤 for all 푇 ∈ 푇̂ operators 푆 and 푇 from 푃 into 푄 and for ≥ 0, the sum 푆 + 푇 and the multiple 푇 are again linear and all 푎 ∈ 푣(푏′) ∩ (푏′′)푣 for some 푏′, 푏′′ ∈ ℬ. and continuous. Thus the continuous linear operators from 푃 into 푄 again form a cone. The 8. Duality of cones and inner products adjoint operator 푇∗ of 푇 ∶ 푃 → 푄 is defined by We excerpt and augment the following from Ch.II.3 in [7]: A dual pair (푃, 푄) consists of two (푇∗())(푎) = (푇(푎)) ordered cones 푃 and 푄 together with a bilinear map, i.e. a mapping for all ∈ 푄∗ and 푎 ∈ 푃. Clearly 푇∗() ∈ 푃∗, and 푇∗ is a linear operator from 푄∗ to 푃∗; more (푎, 푏) → 〈푎, 푏〉 ∶ 푃 × 푄 → ℝ̅ precisely: If for 푣 ∈ 푉 and 푤 ∈ 푊 we have 푇(푎) ≤ 푇(푏) + 푤 whenever 푎 ≤ 푏 + 푣, then 푇∗ which is linear in both variables and compatible maps 푤○ into 푣○. with the order structures on both cones, satisfying
While some concepts from duality and operator 〈푎, 푦〉 + 〈푏, 푥〉 ≤ 〈푎, 푥〉 + 〈푏, 푦〉 whenever theory of locally convex vector spaces may be 푎 ≤ 푏 and 푥 ≤ 푦. readily transferred to the more general context of locally convex cones, others require a new Let us denote by approach and offer insights into a far more elaborate structure. The concept of completeness, 푃+ = {푎 ∈ 푃| 0 ≤ 푎} and for example, does not lend itself to a 푄+ = {푎 ∈ 푄| 0 ≤ 푎} straightforward transcription. It is adapted to locally convex cones in [12] in order to allow a the respective subcones of positive elements in 푃 reformulation of the uniform boundedness and 푄. The above condition guarantees that all principle for Fréchet spaces. The approach uses elements 푥 ∈ 푄+, via 푎 → 〈푎, 푥〉 define monotone the notions of internally bounded subsets, weakly linear functionals on 푃, and vice versa. cone complete and barreled cones. These definitions turn out to be rather technical and we
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If we endow the dual cone 푃∗ of a locally convex 푣○ 푃∗ of all neighborhoods 푣 ∈ 푉. The resulting cone (푃, 푉) with the canonical order polar topology on 푃 coincides with the original one. This shows in particular that every locally 휇 ≤ if = 휇 + 𝜎 for some 𝜎 ∈ 푃∗, convex cone topology satisfying (SP) may be considered as a polar topology. then all elements 휇 ∈ 푃∗ are positive. With the evaluation as its canonical bilinear form, (푃, 푃∗) Two specific topologies on 푄, denoted 푤(푄, 푃) forms a dual pair. and 푠(푄, 푃), are of particular interest: Both are topologies of pointwise convergence for the Dual pairs give rise to polar topologies in the elements of 푃 considered as functions on 푄 with following way: A subset 푋 of 푄+ is said to be 𝜎- values in ℝ̅. For 푤(푄, 푃), ℝ̅ is considered with its bounded below if usual (one-point compactification) topology, whereas +∞ is treated as an isolated point for inf{〈푎, 푥〉| 푥 ∈ 푋} > −∞ 푠(푄, 푃). A typical neighborhood for 푥 ∈ 푄, defined via a finite subset 퐴 = {푎1, … , 푎푛} of 푃, is for all 푎 ∈ 푃. Every such subset 푋 푄+ defines given in the topology 푤(푄, 푃) by 2 a uniform neighborhood 푣푋 ∈ 푃 by 푊퐴(푥) 푣 = {(푎, 푏) ∈ 푃2| 〈푎, 푥〉 ≤ 〈푏, 푥〉 + 1 |〈푎 , 푦〉 − 〈푎 , 푥〉| ≤ 1, if 〈푎 , 푥〉 < +∞ 푋 = {푦 ∈ 푄| 푖 푖 푖 } for all 푥 ∈ 푋} 〈푎푖, 푦〉 > 1, if 〈푎푖, 푥〉 = +∞ and any collection of 𝜎-bounded below subsets and in the topology 푠(푄, 푃) by of 푄 satisfying: 푆퐴(푥) (P1) 푋 ∈ whenever 푋 ∈ and > 0. |〈푎 , 푦〉 − 〈푎 , 푥〉| ≤ 1, if 〈푎 , 푥〉 < +∞ = {푦 ∈ 푄| 푖 푖 푖 } (P2) For all 푋, 푌 ∈ there is some 푍 ∈ such 〈푎푖, 푦〉 = +∞, if 〈푎푖, 푥〉 = +∞ that 푋 ∪ 푌 푍. In general, 푠(푄, 푃) is therefore finer than 푤(푄, 푃), defines a convex quasiuniform structure on 푃. If but both topologies coincide if the bilinear form we denote the corresponding neighborhood on 푃 × 푄 attains only finite values. system by 푉 = {푣푋| 푋 ∈ }, then (푃, 푉) ○ becomes a locally convex cone. The polar 푣푋 of In analogy to the Alaoglu-Bourbaki theorem in the neighborhood 푣푋 ∈ 푉 consists of all linear locally convex vector spaces (see [18], III.4), we functionals 휇 on 푃 such that for 푎, 푏 ∈ 푃 obtain (Proposition 2.4 in [7]):
〈푎, 푥〉 ≤ 〈푏, 푥〉 + 1 for all 푥 ∈ 푋 implies that 8.2 Theorem. Let (푃, 푉) be a locally convex ○ 휇(푎) ≤ 휇(푏) + 1. cone. The polar 푣 of any neighborhood 푣 ∈ 푉 is ∗ a compact convex subset of 푃 with respect to the ∗ All elements of 푋 푄, considered as linear topology 푤(푃 , 푃). ○ functionals on 푃, are therefore contained in 푣푋. Likewise, a Mackey-Arens type result is available 8.1 Examples. (a) Let be the family of all finite for locally convex cones (Theorem 3.8 in [7]): subsets of 푄+. The resulting polar topology on 푃 is called the weak*-topology 𝜎(푃, 푄). 8.3 Theorem. Let (푃, 푄) be a dual pair of ordered cones, and let 푋 푄 be the union of finitely many + (b) Let (푃, 푉) be a locally convex cone with the 푠(푄, 푃)-compact convex subsets of 푄 . Then for ○ strict separation property (SP). Consider the dual every linear functional 휇 ∈ 푣푋 on 푃 there is an pair (푃, 푃∗) and the collection of the polars element 푥 ∈ 푄 such that
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휇(푎) = 〈푎, 푥〉 for all 푎 ∈ 푃 with element 푎 ∈ 푃 which may exist in 푃, and the 휇(푎) < +∞. element (−1)푎 ∈ 푃. Both need not coincide.
The last theorem applies is particular to the We define the modular order ≼푚 for elements weak*-topology 𝜎(푃, 푄) which is generated by 푎, 푏 ∈ 푃 by the finite subsets of 푄. 푎 ≼푚 푏 if 훾푎 ≤ 훾푏 for all 훾 ∈ An inner product on an ordered cone 푃 may be defined as a bilinear form on 푃 × 푃 which is The basic properties of an order relation are easily commutative and satisfies checked. Likewise the relation ≼푚 is seen to be compatible with the extended algebraic operations 2〈푎, 푏〉 ≤ 〈푎, 푎〉 + 〈푏, 푏〉 for all 푎, 푏 ∈ 푃 in 푃, i.e.
Investigations on inner products yield Cauchy- 푎 ≼푚 푏 implies 푎 ≼푚 푏 Schwarz and Bessel-type inequalities, concepts and 푎 + 푐 ≼푚 푏 + 푐 for orthogonality and best approximation, as well as an analogy for the Riesz representation theorem for all ∈ 핂 and 푐 ∈ 푃. Obviously for continuous linear functionals. For details we refer to [14]. 푎 ≼푚 푏 implies that 푎 ≤ 푏.
9. Extended algebraic operations Indeed, our version of the distributive law entails Example 2.1 (b) suggests that the scalar that multiplication in a cone might be canonically (훼 + 훽)푎 ≼푚 훼푎 + 훽푎 extended for all scalars in ℝ or ℂ, but only a holds for all 푎 ∈ 푃 and 훼, 훽 ∈ 핂. weakened version of the distributive law holds for non-positive scalars. For details of the following Using the modular order we define an equivalence we refer to [11]. Let 핂 denote either the field of relation ~푚 on 푃 by the real or the complex numbers, and 푎~푚푏 if 푎 ≼푚 푏 and 푏 ≼푚 푎 = {훿 ∈ 핂| |훿| ≤ 1}, resp. = {훾 ∈ 핂| |훾| = 1} An element 푎 ∈ 푃 is called 푚̃-invertible if there is 푏 ∈ 푃 such that 푎 + 푏~푚0. Any two 푚̃-inverses the closed unit disc, resp. unit sphere in 핂. of the same element 푎 are equivalent in the above sense. We summarize a few observations (Lemma An ordered cone 푃 is linear over 핂 if the scalar 2.1 in [11]): multiplication is extended to all scalars in 핂 and in addition to the requirements for an ordered cone 9.1 Lemma. Let 푃 be an ordered cone that is satisfies linear over 핂. Then
훼(훽푎) = (훼훽)푎 for all 푎 ∈ 푃 and (a) 훼0 = 0 for all 훼 ∈ 핂. 훼, 훽 ∈ 핂 (b) 0 ≼푚 푎 + (−1)푎 for all 푎 ∈ 푃. 훼(푎 + 푏) = 훼푎 + 훼푏 for all 푎, 푏 ∈ 푃 and (c) If 푎 ∈ 푃 is 푚̃-invertible, then 훼 ∈ 핂 (훼 + 훽)푎~푚훼푎 + 훽푎 holds for all 훼, 훽 ∈ 핂, (훼 + 훽)푎 = 훼푎 + 훽푎 for all 푎 ∈ 푃 and and (−1)푎~푚푏 for all 푚̃-inverses 푏 of 푎. 훼, 훽 ∈ 핂 (d) If both 푎, 푏 ∈ 푃 are 푚̃-invertible, then 푎 ≼푚 푏 implies 푎~푚푏. It is necessary in this context to distinguish carefully between the additive inverse – 푎 of an
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If (푃, 푉) is a locally convex cone and 푃 is linear for 훼 ∈ 핂 and 퐴 ∈ 푃, and the addition and order over 핂, then the neighborhoods 푣 ∈ 푉 and the as in 3.1 (c), that is modular order on 푃 give rise to corresponding modular neighborhoods 푣푚 ∈ 푉푚 in the following 퐴 ≤ 퐵 if 퐴 ↓ 퐵 way: For 푎, 푏 ∈ 푃 and 푣 ∈ 푉 we define Thus 푃 is linear over 핂. Considering the modular order on 푃, for 퐴 ∈ 푃 we denote by 푎 ≼푚 푏 + 푣푚 ↓ 퐴 = ⋂(훾̅ ↓ (훾퐴)) if 훾푎 ≼ 훾푏 + 푣 for all 훾 ∈ . Clearly 푎 ≼ 푏 + 푚 푚 푚 훾∈ 푣푚 implies that 푎 ≼푚 푏 + ||푣푚 for all ∈ 핂. We denote the system of modular neighborhoods (for 핂 = ℝ this is just the order interval generated on 푃 by 푉푚. If we require that every element 푎 ∈ by 퐴). Thus 푃 is also bounded below with respect to these modular neighborhoods, i.e. if for every 푣 ∈ 푉 퐴 ≼푚 퐵 if 퐴 ↓푚 퐵 there is > 0 such that As in 3.1 (c), the abstract neighborhood system in 0 ≤ 훾푎 + 푣 for all 훾 ∈ , 푃 is given by a basis 푉 푃 of closed absolutely convex neighborhoods of the origin in 퐸. Every then (푃, 푉푚) with the modular order is again a element 퐴 ∈ 푃 is seen to be m-bounded below, locally convex cone. In this case we shall say that thus fulfilling the last requirement for a locally (푃, 푉) is a locally convex cone over 핂. The convex cone over 핂. respective (upper, lower and symmetric) modular topologies on 푃 are finer than those resulting from The case 퐸 = 핂 with the order from 9.2 (a), i.e. the original neighborhoods in 푉. 푎 ≤ 푏 if ℜ(푎) ≤ ℜ(푏), is of particular interest for the investigation of linear functionals: For 퐴, 퐵 ∈ ̅ 9.2 Examples. (a) Let 푃 = 핂 = 핂 ∪ {∞} be 퐶표푛푣(핂) we have 퐴 ≤ 퐵 if sup{ℜ(푎)| 푎 ∈ 퐴} ≤ endowed with the usual algebraic operations, in sup{ℜ(푏)| 푏 ∈ 퐵} and 퐴 ≼ 퐵 if 퐴 퐵. For 휀 > ̅ 푚 particular 푎 + ∞ = ∞ for all 푎 ∈ 핂, 훼 ∙ ∞ = ∞ 0 the neighborhood 휀 ∈ 푉 is determined by ̅ for all 0 ≠ 훼 ∈ 핂 and 0 ∙ ∞ = 0. The order on 핂 is defined by 퐴 ≤ 퐵 휀 if sup{ℜ(푎)| 푎 ∈ 퐴} ≤ sup{ℜ(푏)| 푏 ∈ 퐵} + 휀, 푎 ≤ 푏 if 푏 = ∞ or ℜ(푎) ≤ ℜ(푏). and With the neighborhood system 푉 = {휀 > 0}, 핂̅ is 퐴 ≼푚 퐵 휀푚 if 퐴 퐵 휀 a full locally convex cone. It is easily checked that ̅ ̅ 핂 is linear over 핂. The modular order on 핂 is (c) Let 푃 consist of all ℝ̅-valued functions 푓 on identified as 푎 ≼푚 푏 if either 푏 = ∞ or 푎 = 푏. For [−1, +1] that are uniformly bounded below and 푣 = 휀 ∈ 푉 we have 푎 ≼푚 푏 + 푣푚 if either 푏 = ∞ satisfy 0 ≤ 푓(푥) + 푓(−푥) for all 푥 ∈ [−1, +1]. or |푎 − 푏| ≤ 휀. Endowed with the pointwise addition and multiplication by non-negative scalars, the order (b) We augment our Example 3.1 (c) as follows: 푓 ≤ 푔 if 푓(푥) ≤ 푔(푥) for all 0 ≤ 푥 ≤ 1, and the Let (퐸, ≤) be a locally convex ordered topological neighborhood system 푉 consisting of the (strictly) vector space over 핂. For 퐴 ∈ 푃 = 퐶표푛푣(퐸) we positive constants, 푃 is a full locally convex cone. define the multiplication by any scalar 훼 ∈ 핂 by We may extend the scalar multiplication to negative reals 훼 and 푓 ∈ 푃 by 훼퐴 = {훼푎| 푎 ∈ 퐴} (훼푓)(푥) = (−훼)푓(−푥)
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Mathematics Scientia Bruneiana Vol. 16 2017 for all 푥 ∈ [−1, +1]. Thus 푃 is seen to be linear is obviously void. For 핂 = ℂ, however, the m- over ℝ. The modular order on 푃 is the pointwise continuous elements form a subcone of 푃 which order on the whole interval [−1, +1]. we shall denote by 퐶푚. Obviously ℬ푚 퐶푚. A ∗ functional 휇 ∈ 푃푚 is called regular if For a locally convex cone over 핂 we shall denote by ℬ푚 the subcone of all m-bounded elements, i.e. 휇(푎) = sup{휇(푐)| 푐 ∈ 퐶푚, 푐 ≼푚 푎} those elements 푎 ∈ 푃 such that for every 푣 ∈ 푉 there is > 0 such that 푎 ≼푚 푣푚. Clearly holds for all 푎 ∈ 푃. For 핂 = ℝ, of course, as all ∗ ℬ푚 . ℬ. We cite Theorem 2.3 from [11]: elements 푎 ∈ 푃 are m-continuous, every 휇 ∈ 푃푚 is ∗ regular. For a regular linear functional 휇 ∈ 푃푚 and 9.3 Theorem. Every locally convex cone (푃, 푉) every 푎 ∈ 푃 we may define a corresponding set- can be embedded into a locally convex cone (푃̃, 푉) valued function 휇푐 ∶ 푃 → 퐶표푛푣(핂) by over 핂. The embedding is linear, one-to-one and preserves the global preorder and the 휇푐(푎) = {푎 ∈ 핂| ℜ(훾훼) ≤ 휇(훾훼) neighborhoods of 푃. All bounded elements 푎 ∈ 푃 for all 훾 ∈ 핂} are mapped onto m-bounded elements of 푃̃ and are 푚̃-invertible in 푃̃. The regularity of 휇 entails (see [11]) that 휇푐(푎) is non-empty, closed and convex in 핂, and that Let (푃, 푉) be a locally convex cone over 핂. Endowed with the corresponding modular 휇(훾훼) = sup{ℜ(훾훼)| 훼 ∈ 휇푐(푎)} neighborhood system, (푃, 푉푚) is again a locally convex cone. We denote the dual cone of (푃, 푉푚) holds for all 훾 ∈ 핂. The latter shows in particular ∗ by 푃푚 and refer to it as the modular dual of 푃. As that the correspondence between 휇 and 휇푐 is one- continuity with respect to the given topology to-one. For 핂 = ℝ the values of 휇푐 are closed implies continuity with respect to the modular intervals in ℝ; more precisely: ∗ ∗ ○ topology we have 푃 푃푚. By 푣푚 we denote the (modular) polar of the neighborhood 푣푚 ∈ 푉푚, i.e. 휇푐(푎) = [−휇((−1)푎), 휇(푎)] ∩ ℝ. ∗ the set of all linear functionals 휇 ∈ 푃푚 such that The mapping 휇푐 ∶ 푃 → 퐶표푛푣(핂) is additive and 휇(푎) ≤ 휇(푏) + 1 holds whenever 푎 ≼푚 푏 + 푣푚 homogeneous with respect to the multiplication by all scalars in 핂. More precisely: Monotone linear functionals in 휇 ∶ 푃 → ℝ̅ are required to be homogeneous only with respect to 9.4 Lemma. Let 휇 ∶ 푃 → ℝ̅ be a regular the multiplication by positive reals. For negative monotone linear functional. For 휇푐 ∶ 푃 → reals 훼 < 0 the relation 훼푎 + (−훼)푎 ≥ 0 yields 퐶표푛푣(핂) the following hold: 휇(훼푎) ≥ 훼휇(푎). But for complex numbers 훼 in general we fail to recognize any obvious relation (a) 휇푐(푎) is a non-empty closed convex subset between 휇(훼푎) and 훼휇(푎). This may be remedied, of 핂. ∗ at least for a large class of functionals in 푃푚, by (b) 휇푐(푎 + 푏) = 휇푐(푎) 휇푐(푏) for all 푎, 푏 ∈ 푃. the following procedure: An element 푎 ∈ 푃 is (c) 휇푐(훼푎) = 훼휇푐(푎) for all 푎 ∈ 푃 and 훼 ∈ 핂. called m-continuous if the mapping (d) If 푎 ∈ 푃 is 푚̃-invertible then 휇푐(푎) is a singleton subset of 핂. 훾 → 훾푎 ∶ → 푃 (e) 휇푐 is continuous with respect to the modular topologies on 푃 and 퐶표푛푣(핂); more ○ is uniformly continuous with respect to the upper precisely: if 휇 ∈ 푣푚 then, for 푎, 푏 ∈ 푃, topology on 푃, i.e. if for every 푣 ∈ 푉 there is 휀 > ′ 0 such that 훾푎 ≤ 훾 푎 + 푣 holds for all 훾, 훾′ ∈ 푎 ≼푚 푏 + 푣푚 implies that ′ satisfying |훾 − 훾 | ≤ 휀. For 핂 = ℝ this condition 휇푐(푎) 휇푐(푏) ,
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Mathematics Scientia Bruneiana Vol. 16 2017 where denotes the closed unit disc in ℂ. In the case of a locally convex cone over ℝ, where = {−1, +1}, this result simplifies considerably. ∗ 9.5 Examples. Reviewing our Example 9.2 (b), Every linear functional 휇 ∈ 푃푚 is regular then, and i.e. the locally convex cone 푃 = 퐶표푛푣(퐸) over 핂, the integral representation in Theorem 7.6 reduces where (퐸, ≤) denotes a locally convex ordered to a sum of two functionals. topological vector space, we realize that for every 핂-valued continuous linear functional 푓 on 퐸, the 9.7 Corollary. Let (푃, 푉) be a locally convex ∗ mapping 휇: 푃 → ℝ̅ such that cone over ℝ. For every linear functional 휇 ∈ 푃푚 ∗ there exist 휇1, 휇2 ∈ 푃 such that 휇(퐴) = sup{ℜ(푓(푎))| 푎 ∈ 퐴} 휇(푎) = 휇1(푎) + 휇2((−1)푎) for all 푎 ∈ 푃. ∗ is linear, an element of 푃푚 and obviously regular. 10. Application: Korovkin type approximation The corresponding set-valued functional 휇푐 ∶ 푃 → 퐶표푛푣(핂) is given by Locally convex cones provide a suitable setting for a rather general approach to Korovkin type theorems, an extensively studied field in abstract 휇푐(퐴) = 푓(퐴) = {푓(푎)| 푎 ∈ 퐴}. approximation theory. For a detailed survey on However, in the complex case, even for 퐸 = ℂ, this subject we refer to [2]. Approximation one can find examples of non-regular linear schemes may often be modeled by sequences (or ∗ nets) of linear operators. For a sequence (푇 ) functionals in 푃푚. 푛 푛∈ℕ of positive linear operators on 퐶([0,1]), However, in the complex case, even for 퐸 = ℂ, Korovkin's theorem (see [8]) states that 푇푛(푓) one can find examples of non-regular linear converges uniformly to 푓 for every 푓 ∈ 퐶([0,1]), ∗ whenever 푇 (푔) converges to 푔 for the three test functionals in 푃푚. 푛 functions 푔 = 1, 푥, 푥2. This result was It is possible to construct a decomposition for subsequently generalized for different sets of test ∗ ∗ regular functionals 휇 ∈ 푃푚 into functionals in 푃 . functions 푔 and different topological spaces 푋 In a locally convex ordered topological vector replacing the interval [0,1]. Classical examples space over ℝ every continuous linear functional include the Bernstein operators and the Fejér sums may be expressed as a difference of two positive which provide approximation schemes by ones (see [18], IV.3.2). A similar decomposition polynomials and trigonometric polynomials, is available in the complex case. The more general respectively. Further generalizations investigate setting of locally convex cones, however, requires the convergence of certain classes of linear the use of Riemann-Stieltjes type integrals instead operators on various domains, such as positive of sums. In this instance we refrain from supplying operators on topological vector lattices, the detailed arguments and notations for this rather contractive operators on normed spaces, technical procedure. They may be found in [11]. multiplicative operators on Banach algebras, The main result is: monotone operators on set-valued functions, monotone operators with certain restricting 9.6 Theorem. Let (푃, 푉) be a locally convex cone properties on spaces of stochastic processes, etc. over 핂. For every regular linear functional 휇 ∈ Typically, for a subset 푀 of a domain 퐿 one tries ∗ ∗ to identify all elements 푓 ∈ 퐿 such that 푃푚 there exists a 푃 -valued m-integrating family
(휗퐸)퐸∈ℝ on the unit circle in ℂ such that 푇훼(푔) → 푔 for all 푔 ∈ 푀 implies that 푇훼(푓) → 푓, 휇 = ∫ 훾 푑휗 whenever (푇훼)훼∈퐴 is an equicontinuous net (generalized sequence) in the restricted class of
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Mathematics Scientia Bruneiana Vol. 16 2017 operators on 퐿. Locally convex cones allow a superharmonic in all functionals of the 푤(푃∗, 푃)- unified approach to most of the above mentioned closure of the set of extreme points of 푣○. Then for cases. Various restrictions on classes of operators every equicontinuous net (푇훼)훼∈퐴 of linear may be taken care of by the proper choice of operators on 푃 domains and their topologies alone and approximation results may be obtained through 푇훼(푏) ↑ 푏 for all 푏 ∈ 푄 implies that the investigation of continuous linear operators 푇훼(푎) ↑ 푎. between locally convex cones. We proceed to outline a few results that may be found in Chapters Let us mention just one of the many well-known III and IV of [7]: Korovkin type theorems that may be derived using Theorems 10.1 and 10.2: Let 푋 be a locally Let 푄 be a subcone of the locally convex cone compact Hausdorff space, 푃 = 퐶0(푋) the space of (푃, 푉). The element 푎 ∈ 푃 is said to be 푄- all continuous real-valued functions on 푋 that superharmonic in 휇 ∈ 푃∗ if 휇(푎) is finite and if vanish at infinity, and let 푉 consist of all positive for all ∈ 푃∗, constant functions. With the pointwise order and algebraic operations, (푃, 푉) is a locally convex (푏) ≤ 휇(푏) for all 푏 ∈ 푄 implies that cone. Continuous linear operators on 푃 are (푎) ≤ 휇(푎) monotone and bounded with respect to the norm of uniform convergence on 퐶0(푋). The extreme This notation is derived from potential theory. We points of polars of neighborhoods are just the non- cite Theorem III.1.3 from [7] which is an negative multiples of point evaluations. Finally, immediate corollary to our Range Theorem 4.5 convergence 푓훼 → 푓 for a net of functions in with the following insertions: We choose 푞(푎) = 퐶0(푋) in the uniform topology means that both −∞ for all 푎 ≠ 0 and 푝(푎) = 휇(푎) for 푎 ∈ 푄, 푓훼 ↑ 푓 and (−푓훼) ↑ (−푓). We obtain a result due otherwise 푝(푎) = +∞, and obtain: to Bauer and Donner [4]:
10.1 Sup-Inf-Theorem. Let 푄 be a subcone of the 10.3 Theorem. Let 푋 be a locally compact ∗ locally convex cone (푃, 푉). Let 푎 ∈ 푃 and 휇 ∈ 푃 Hausdorff space, and let 푀 be a subset of 퐶0(푋). such that 휇(푎) is finite. Then 푎 is 푄- For a function 푓 ∈ 퐶0(푋) the following are superharmonic in 휇 if and only if equivalent:
휇(푎) = sup푣∈푉 inf{휇(푏)| 푏 ∈ 푄, 푎 ≤ 푏 + 푣}. (a) For every equicontinuous net (푇훼)훼∈퐴 of positive linear operators on 퐶0(푋) We shall cite only a simplified version of the main Convergence Theorem IV.1.13 in [7] for nets of 푇훼(푔) → 푔 for all 푔 ∈ 푀 implies that linear operators on a locally convex cone. It is 푇훼(푓) → 푓 however sufficient to derive the classical results for Korovkin type approximation processes. For a (Convergence is meant with respect to the net (푎훼)훼∈퐴 in 푃 we shall denote 푎훼 ↑ 푏 if topology of uniform convergence on 푋.) (푎훼)훼∈퐴 converges towards 푏 ∈ 푃 with respect to the upper topology, i.e. if for every 푣 ∈ 푉 there is (b) For every 푥 ∈ 푋 훼0 such that 푔 ∈ span(푀), 푓(푥) = sup휀>0inf {푔(푥)| } 푎훼 ≤ 푏 + 푣 for all 훼 ≥ 훼0. 푓 ≤ 푔 + 휀 푔 ∈ span(푀), = inf sup {푔(푥)| } 10.2 Convergence Theorem. Let 푄 be a subcone 휀>0 푔 ≤ 푓 + 휀 of the locally convex cone (푃, 푉). Suppose that for every 푣 ∈ 푉 the element 푎 ∈ 푃 is 푄-
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(c) For every 푥 ∈ 푋 and for every bounded For linear functionals 휇, ∈ 푃∗ we set positive regular Borel measure 휇 on 푋 휇 ≼ if 휇(푙) ≤ (푙) for all 푙 ∈ 퐿 and 휇(푔) = 푔(푥) for all 푔 ∈ 푀 implies that 휇(푢) ≥ (푢) for all 푢 ∈ 푈. 휇(푓) = 푓(푥) We write 휇~ if both 휇 ≼ and ≼ 휇, i.e. if the The General Convergence Theorem IV.1.13 in [7] functionals 휇 and coincide on 푈 and 퐿. Integrals allows a far wider range of applications, including on 푃 are the minimal functionals in this order and quantitative estimates for the order of (푃, 퐿, 푈) is called an integration cone. convergence for the approximation processes modeled by sequences or nets of linear operators. 11.1 Theorem. Let (푃, 퐿, 푈) be an integration cone. 11. Application: Topological integration theory (a) For every continuous linear functional ∗ A rather general approach to topological 휇0 ∈ 푃 there is an integral 휇 on 푃 such that integration theory using locally convex cones is established in [10]. It utilizes techniques originally 휇(푙) ≤ 휇0(푙) for all 푙 ∈ 퐿 and 휇(푢) ≥ developed for Choquet theory. Continuous linear 휇0(푢) for all 푢 ∈ 푈. functionals on a given locally convex cone 푃 are called integrals if they are minimal, resp. maximal (b) The linear functional 휇 ∈ 푃∗ is an integral if with respect to certain subcones of 푃. Their and only if properties resemble those of Radon measures on locally compact spaces. They satisfy convergence 휇(푙) = inf푣∈푉 sup{휇(푢)| 푢 ≤ 푙 + 푣, 푢 ∈ 푈} theorems corresponding to Fatou's Lemma and for all 푙 ∈ 퐿, Lebesgue's theorem about bounded convergence. and Depending on the choice of the determining subcones of 푃, one obtains a wide variety of 휇(푢) = inf{휇(푙)| 푢 ≤ 푙, 푙 ∈ 퐿} for all 푢 ∈ 푈. applications, including classical integration theory on locally compact spaces (see [5]), Choquet An element 푎 ∈ 푃 is said to be 휇-integrable with theory about integral representation (see [1]), H- respect to an integral 휇 if integrals on H-cones in abstract potential theory and monotone functionals on cones of convex ~휇 implies that (푎) = 휇(푎) sets. We shall outline some of the main concepts without supplying details and proofs which may for all ∈ 푃∗. For a given integral 휇 on 푃 the 휇- be found in [10]: integrable elements form a subcone of 푃 that contains both 퐿 and 푈. Let (푃, 푉) be a full locally convex cone, 퐿 and 푈 two subcones of 푃. 퐿 is supposed to be a full cone, 11.2 Theorem. Let 휇 be an integral on 푃. The whereas all elements of 푈 are supposed to be element 푎 ∈ 푃 is 휇-integrable if and only if bounded. The following two conditions hold:
inf푣∈푉 sup{휇(푢)| 푢 ≤ 푎 + 푣, 푢 ∈ 푈} = (U) For all 푎 ∈ 푃, 푙 ∈ 퐿, 푢 ∈ 푈 such that inf {휇(푙)| 푎 ≤ 푙, 푙 ∈ 퐿}. 푢 ≤ 푎 + 푙 and for every 푣 ∈ 푉 there is ′ 푢′ ∈ 푈 such that 푢 ≤ 푎 + 푣 and For a Lebesgue-type convergence theorem we ′ 푢 ≤ 푢 + 푙 + 푣. require a subset of special integrals that correspond to the point evaluations in classical (L) For all 푎 ∈ 푃, 푙 ∈ 퐿, 푢 ∈ 푈 such that integration theory. In this vein, for a neighborhood 푎 + 푢 ≤ 푙 and for every 푣 ∈ 푉 there is 푣 ∈ 푉 we define the integral boundary relative to ′ 푙′ ∈ 퐿 such that 푎 ≤ 푙′ and 푙 + 푢 ≤ 푙 + 푣. 푣 to be the set 푣 of all integrals 훿 on 푃 such that
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훿(푣) < +∞, satisfying the following property: If 휇(푢) = inf{휇(푐)| 푢 ≤ 푐, 푐 ∈ 퐶(푋)} for any two integrals 휇1, 휇2 on 푃 we have for all 푢 ∈ 푈.
훿(푣) = (휇1 + 휇2)(푣) and Following Theorem 11.2, a function 푓 ∈ 푃 is 휇- 훿(푢) ≤ (휇1 + 휇2)(푢) for all 푢 ∈ 푈 integrable if and only if then there are 1, 2 ≥ 0 such that 휇1~1훿 and sup{휇(푢)| 푢 ≤ 푓, 푢 ∈ 푈} 휇2~2훿. For a neighborhood 푣 ∈ 푉 we shall say = inf{휇(푙)| 푓 ≤ 푙, 푙 ∈ 퐿}. that a subset 퐴 of 푃 is uniformly 푣-dominated if there is 𝜌 ≥ 0 such that 푎 ≤ 𝜌푣 for all 푎 ∈ 퐴. The integrals of this theory, therefore are the positive Radon measures on the compact space 푋, We formulate the main convergence result and the above notion of integrability coincides (Theorem 4.3 in [11]) which is modeled after the with the usual one (see [5], IV.4, Théorème 3), Bishop de-Leeuw theorem from Choquet theory. except for the fact that we allow integrals to take the value +∞. Theorem 11.1 (a) implies that every 11.3 Theorem. Let 휇 be an integral on the positive linear functional on 퐶(푋) permits an integration cone (푃, 퐿, 푈). For a neighborhood extension to a positive Radon measure on 푋, 푣 ∈ 푉 let (푎푛)푛∈ℕ be a uniformly 푣-dominated which is the result of the Riesz Representation sequence of 휇-integrable elements in 푃. If Theorem. For a neighborhood 푣 ∈ 푉 the integral boundary relative to 푣 consists of positive lim sup푛∈ℕ훿(푎푛) ≤ 훿(푣) multiples of point evaluations in 푋. Thus Theorem 11.3 yields Lebesgue's convergence theorem. The for all 훿 ∈ 푣, then adaptation of this example for a locally compact Hausdorff space 푋 is rather more technical and lim sup푛∈ℕ휇(푎푛) ≤ 휇(푣). may be found in [10], Example 3.13 (c).
For detailed arguments in the following examples (b) Let 푋 be a compact convex subset of a locally we refer to Examples 1.1 and 3.13 in [10]. convex Hausdorff space, and let (푃, 푉) be as in (a). We choose the subcone of all ℝ̅-valued lower 11.4 Examples. (a) This example models semicontinuous concave functions for 퐿 and the topological integration theory on a compact real-valued upper semicontinuous convex Hausdorff space 푋 as presented in [5]: Let 푃 be functions for 푈. As the elements of the dual cone the cone of all bounded below ℝ̅-valued functions 푃∗ of 푃 when restricted to 퐶(푋) are positive on 푋, endowed with the pointwise algebraic Radon measures on 푋, our integrals on 푃 are just operations and order, and let 푉 consist of all the usual maximal representation measures from strictly positive constant functions on 푋. Then classical Choquet theory. The 휇-integrable (푃, 푉) is a full locally convex cone. We choose for elements of 푃 include all continuous functions on 퐿 the subcone of all ℝ̅-valued lower 푋. Theorem 11.2 yields Mokobodzki's semicontinuous functions and for 푈 all real- characterization of maximal measures in Choquet valued upper semicontinuous functions in 푃. As theory (Proposition 1.4.5 in [1]). The subspace required, 푉 퐿, and all functions in 푈 are 푈 ∩ 퐿 consists of the continuous affine functions bounded. For an integral 휇 ∈ 푃∗, condition 11.1 on 푋, and Theorem 11.1 (a) implies that every (b) implies that positive linear functional on this subspace (i.e. a positive multiple of a point evaluation on 푋) may 휇(푙) = sup{휇(푐)| 푐 ≤ 푙, 푐 ∈ 퐶(푋)} be represented by such a maximal measure. for all 푙 ∈ 퐿 Moreover, for every neighborhood 푣 ∈ 푉, the integral boundary 푣 consists of positive and multiples of evaluations in the extreme points of
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푋, hence Theorem 11.3 recovers the Bishop de- Mathematics, 1517, 1992, Springer Verlag, Leeuw theorem from classical Choquet theory Heidelberg-Berlin-New York. about the support of maximal measures. [8] P.P. Korovkin, “Linear operators and approximation theory,” Russian (c) Let (푃 = 퐶표푛푣(퐸), 푉) be the full locally Monographs and Texts on Advanced convex cone introduced in Example 3.1 (c). We Mathematics, vol. III, 1960, Gordon and choose 퐿 = 푃 and for 푈 the subcone of 푃 of all Breach, New York. singleton subsets of the space 퐸. Following [9] L. Nachbin, Topology and Order, 1965, Van Theorem 11.2 every integral 휇 on 푃 is already Nostrand, Princeton. determined by its values on the subcone 푈, that is [10] W. Roth, “Integral type linear functionals on by a monotone continuous linear functional 휇0 in ordered cones,” Trans. Amer. Math. Soc., the usual dual 퐸′ of the locally convex ordered 1996, vol. 348, no. 12, 5065-5085. topological vector space 퐸; that is [11] W. Roth, “Real and complex linear extensions for locally convex cones,” Journal of Functional Analysis, 1997, vol. 휇(퐴) = sup{휇0(푎)| 푎 ∈ 퐴} 151, no. 2, 437-454. for every 퐴 ∈ 푃. This describes a one-to-one [12] W. Roth, “A uniform boundedness theorem correspondence between the monotone for locally convex cones,” Proc. Amer. functionals in 퐸′ and the integrals on 푃. For a Math. Soc., 1998, vol. 126, no. 7, 83-89. neighborhood 푣 ∈ 푉 the integral boundary [13] W. Roth, “Hahn-Banach type theorems for relative to 푣 consists of those integrals on 푃 that locally convex cones,” Journal of the are induced by positive multiples of the extreme Australian Math. Soc. (Series A) 68, 2000 points of the usual polar of 푣 in 퐸′. no. 1, 104-125. [14] W. Roth, “Inner products on ordered cones,” References New Zealand Journal of Mathematics, 2001, [1] E. M. Alfsen, “Compact convex sets and 30, 157-175. boundary integrals,” Ergebnisse der [15] W. Roth, “Separation properties for locally Mathematik und ihrer Grenzgebiete, 1971, convex cones,” Journal of Convex Analysis, vol. 57, Springer Verlag, Heidelberg-Berlin- 2002, vol. 9, No. 1, 301-307. New York. [16] W. Roth, “Boundedness and connectedness [2] F. Altomare and M. Campiti, “Korovkin type components for locally convex cones,” New approximation theory and its applications,” Zealand Journal of Mathematics, 2005, 34, Gruyter Studies in Mathematics, 1994, vol. 143-158. 17, Walter de Gruyter, Berlin-New York. [17] W. Roth, “Operator-valued measures and [3] B. Anger and J. Lembcke, “Hahn-Banach integrals for cone-valued functions,” Lecture type theorems for hypolinear functionals,” Notes in Mathematics, vol. 1964, 2009, Math. Ann., 1974, 209, 127-151. Springer Verlag, Heidelberg-Berlin-New [4] H. Bauer and K. Donner, “Korovkin York. [18] H.H. Schäfer, “Topological vector spaces,” approximation in 퐶0(푋),” Math. Ann., 1978, 236, 225-237. 1980, Springer Verlag, Heidelberg-Berlin- [5] N. Bourbaki, Éléments de Mathématique, New York. Fascicule III, Livre VI, Intégration, 1965, Hermann, Paris. [6] B. Fuchssteiner and W. Lusky, “Convex cones,” North Holland Math. Studies, 1981, vol.56. [7] K. Keimel and W. Roth, “Ordered cones and approximation,” Lecture Notes in
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