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An Introduction to Locally Convex Cones 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 Locally including some yet unpublished ones and provide convex topological vector spaces in particular primary examples and applications.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 31 Mathematics Scientia Bruneiana Vol. 16 2017 (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 32 Mathematics Scientia Bruneiana Vol. 16 2017 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) ∈ 푣 for superlinear functional on 푃 is a mapping 푞 ∶ 푃 → , > 0, then (푎, 푐) ∈ ( + )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]).
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