bounds for linear functionals on convex sets

Math. Nachr. 284, No. 17–18, 2272 – 2286 (2011) / DOI 10.1002/mana.200910198

Bounds for linear functionals on convex sets

Walter Roth∗

Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam

Received 8 September 2009, revised 1 April 2010, accepted 4 May 2010Published online 26 August 2011

Key words Hahn-Banach type theorems, range of linear functionalsMSC (2010) Primary: 46A22, 46A03, 46A20; Secondary: 46A40, 46A55

We consider continuous monotone linear functionals on a locally convex ordered topological vector space thatare sandwiched between a given (R ∪ {+∞})-valued sublinear and an (R ∪ {−∞})-valued superlinear func-tional. We review conditions for the existence of such functionals and in our main results investigate their rangeof suprema and infima on a given convex subset. These yield effective versions of the Hahn-Banach theoremwhich give easy access to various applications including separation properties for convex sets, a non-Baire ap-proach to the Uniform Boundedness theorem, the notion of sub-and superharmonicity with respect to a subcone,and results for the extension of monotone affine functions.

c© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Let (E,V,≤) be a locally convex ordered topological vector space, that is a real vector space E endowed withan order relation ≤ and a basis V of balanced convex and order convex neighborhoods of the origin. (A subset Aof E is order convex if a ≤ b ≤ c for a, c ∈ V implies that b ∈ A as well.) The order is supposed to be reflexive,transitive and compatible with the algebraic operations, that is a ≤ b implies a + c ≤ b + c and αa ≤ αb for alla, b, c ∈ E and α ≥ 0. The positive cone in E is supposed to be closed. Since equality in E is obviously such anorder, our results will also apply to locally convex vector spaces without a given order structure.

By E∗ we denote the topological dual of E, that is the space of all continuous real-valued linear functionalson E. A functional μ ∈ E∗ is called monotone if a ≤ b implies μ(a) ≤ μ(b), or equivalently if μ(a) ≥ 0for all a ≥ 0. E∗

+ will denote the subcone of all monotone continuous real-valued linear functionals in E∗. An(extended ) sublinear functional on a vector space E is a mapping

p : E −→ R = R ∪ {+∞}

such that p(a + b) ≤ p(a) + p(b) and p(αa) = αp(a) for all a, b ∈ E and α ≥ 0. In R we consider the usualalgebraic operations, in particular α+∞ = +∞ for all α ∈ R, α · (+∞) = +∞ for all α > 0 and 0 · (+∞) = 0.Likewise, a functional

q : E −→ R = R ∪ {−∞}

is called (extended ) superlinear if −q is sublinear. We are interested in the properties of continuous mono-tone linear functionals μ ∈ E∗

+ that are sandwiched between given sub- and superlinear functionals p andq, that is q(a) ≤ μ(a) ≤ p(a) holds for all a ∈ E. We shall write q ≤ μ ≤ p for this. Versions of theHahn-Banach theorem of this type can be easily adapted to model a variety of extension and separation problems.The permission of infinite values for the functionals involved is crucial. The choice of q ≡ −∞, that is q(0) = 0and q(a) = −∞ for all a = 0, effectively removes this superlinear functional altogether. (We shall also writep ≡ +∞ for p(0) = 0 and p(a) = +∞ for a = 0.) But even if q ≡ −∞ and with the equality as the order onE the existence of a linear functional that is dominated by p is not guaranteed, provided that p is allowed to take

∗ e-mail: [email protected], Phone: +673 222 4662, Fax: +673 222 4662

c© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Math. Nachr. 284, No. 17–18 (2011) / www.mn-journal.com 2273

the value +∞ (see Example 3.2 below). Necessary and sufficient conditions for the existence of such a linearfunctional were established by Anger and Lembcke in [1]. ( R-valued sublinear functionals are called hypolinearfunctionals in their paper.) A later study by the author in [11] extends these results to locally convex cones, whichare generalizations of locally convex ordered topological vector spaces ([7] and [12]).

Section 2 begins with a review of some established results, mostly quoted from [1] and [11]. These will beused to establish several boundedness statements in Section 3 and an investigation about the range of supremaand infima suprema for the functionals concerned on a given convex subset. Section 4 contains examples andapplications.

2 Review of the basic results

In the following let (E,V,≤) be a locally convex ordered topological vector space and let p : E → R andq : E → R be sub- and superlinear functionals. We first quote Theorem 3.1 from [11] which was formulatedfor locally convex cones. For the adaptation to the more special situation of ordered vector spaces we use thefollowing notations: A subset A of E is called decreasing or increasing, respectively, if b ∈ A whenever b ≤ a,or a ≤ b, for some a ∈ A. The decreasing hull A↓ and the increasing hull A↑ of A are defined as

A↓ = {b ∈ E | b ≤ a for some a ∈ A} and

A↑ = {b ∈ E | a ≤ b for some a ∈ A},respectively. Note that (−A)↓ = −(A↑). The topological closure of a subset A of E is denoted by A. Thedecreasing, or increasing, hull of a convex set is again convex and its topological closure A↓ , or A↑ , is bothdecreasing, or increasing, and convex. The polar V ◦ of a neighborhood V ∈ V consists of all linear functionalsμ ∈ E∗ such that |μ(v)| ≤ 1 for all v ∈ V.

Theorem 2.1 (Sandwich Theorem) Let E be a locally convex ordered topological vector space, and let V ∈ V

be a neighborhood of the origin in E. For a sublinear functional p : E → R and a superlinear functionalq : E → R there exists a monotone linear functional μ ∈ V ◦ such that q ≤ μ ≤ p if and only if

q(a) ≤ p(b) + 1 whenever a ∈ b + V↓ .

For q ≡ −∞ this condition means that p is bounded below on the increasing hull of some neighborhoodV ∈ V (since every V ∈ V is balanced, we have V ↑ = −V↓ ), and in the non-ordered case (that is equality asthe order) it means that p is lower semicontinuous at 0 ∈ E. This formulation can be found in Theorem 1.8 of [1].

The non-topological, that is the algebraic case is obtained if V consists of all balanced convex and absorbingsubsets of E. If one of the functionals, say p , takes only finite values and if q(a) ≤ p(b) holds whenever a ≤ bfor a, b ∈ E, then the condition of Theorem 2.1 holds with the algebraic neighborhood V = {v ∈ E | p(v),p(−v) ≤ 1}.

Theorem 2.1 gives rise to a general extension theorem which yields a variety of known special cases. We quoteand adapt Theorem 4.1 from [11]. It deals with convex, concave and affine functions on convex subsets of E. Forthis recall that an R-valued function f defined on a convex subset C of E is called convex if

f(λc1 + (1 − λ)c2

)≤ λf(c1) + (1 − λ)f(c2)

holds for all c1 , c2 ∈ C and λ ∈ [0, 1]. Likewise, f : C → R is called concave if −f is convex. An affinefunction f : C → R is both convex and concave.

For functions f and g on C we shall denote f ≤C

g if f(c) ≤ g(c) for all c ∈ C.

Theorem 2.2 (Extension Theorem) Let E be a locally convex ordered topological vector space, C and Dnon-empty convex subsets of E, and let V ∈ V be a neighborhood of the origin in E. Let p : E → R be asublinear and q : E → R a superlinear functional. For a convex function f : C → R and a concave functiong : D → R there exists a monotone linear functional μ ∈ V ◦ such that

q ≤ μ ≤ p and g ≤D

μ ≤C

f

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2274 W. Roth: Bounds for linear functionals on convex sets

if and only if

q(a) + ρg(d) ≤ p(b) + σf(c) + 1 whenever (a + ρd) ∈ (b + σc) + V↓

for all a, b ∈ E, c ∈ C, d ∈ D and ρ, σ ≥ 0.

For C = D and f = g this is indeed an extension statement. For C = D = {0} and f = g = 0 it recoversTheorem 2.1.

3 Bounds on convex subsets

In this sections we shall investigate bounds from above or below on a given convex subset of E for linear func-tionals that are sandwiched between a sub- and a superlinear functional. These are our main results. For a convexset A ⊂ E note that 0 ∈ A↑ if and only if A ∩ V↓ = ∅ for all V ∈ V. Indeed, for every V ∈ V we haveA↑ ∩ V = ∅ if and only if there is a ∈ A such that a ≤ v for some v ∈ V that is A∩ V↓ = ∅. Likewise, 0 ∈ A↓holds if and only if A ∩ V ↑ = ∅ for all V ∈ V.

Theorem 3.1 Let E be a locally convex ordered topological vector space and let A be a non-empty convexsubset of E. Let p : E → R be a sublinear and q : E → R a superlinear functional.

(a) Suppose that 0 ∈ A↑ . Then there exists μ ∈ E∗+ such that q ≤ μ ≤ p and μ is bounded below on A if and

only if

supV ∈V

inf {p(b) − q(c) | b, c ∈ E, a ∈ A, (c + a) ∈ b + V↓ } > −∞.

(b) Suppose that 0 ∈ A↓ . Then there exists μ ∈ E∗+ such that q ≤ μ ≤ p and μ is bounded above on A if and

only if

infV ∈V

sup {q(c) − p(b) | b, c ∈ E, a ∈ A, c ∈ (b + a) + V↓ } < +∞.

P r o o f. Part (b) follows from (a) if we replace the set A by −A. Obviously, the existence of a lower boundfor a linear functional on −A corresponds to the existence of an upper bound on A. Moreover, 0 ∈ (−A)↑ if andonly if 0 ∈ A↓ . For Part (a) first assume that there is μ ∈ E∗

+ and α ∈ R such that q ≤ μ ≤ p and μ(a) ≥ α forall a ∈ A. Then μ ∈ V ◦ for some V ∈ V, hence

q(c) + α ≤ q(c) + μ(a) ≤ μ(c) + μ(a) = μ(c + a) ≤ μ(b) + 1 ≤ p(b) + 1

whenever (c + a) ∈ b + V↓ , that is c + a ≤ b + v for some v ∈ V, for a ∈ A and b, c ∈ E. This showsinf{p(b) − q(c) | b, c ∈ E, a ∈ A, (c + a) ∈ b + V↓ } ≥ α − 1 > −∞. For the converse let us assume thatthere is no functional μ ∈ E∗

+ satisfying the requirements of Part (a). In this case we shall use Theorem 2.2 withC = {0}, A in place of D and the functions f = 0 and g = −α for some α > 0. The condition in 2.2 thenmust fail for every choice of V ∈ V and α > 0. Now given a neighborhood V ∈ V and α > 0 we apply this tothe neighborhood (1/2α)V ∈ V instead of V. According to 2.2 there are a ∈ A, c, b ∈ E and ρ ≥ 0 such that(c + ρa) ∈ b + (1/2α)V↓ , but q(c) + ρ(−α) > p(b) + 1, that is p(b) − q(c) < −αρ − 1. If αρ ≥ 1/2, we have(

1ρ

c + a

)∈ 1

ρb +

12αρ

V↓ ⊂ 1ρ

b + V↓ ,

hence

inf{p(b) − q(c) | b, c ∈ E, a ∈ A, (c + a) ∈ b + V↓ } ≤ 1ρ

(p(b) − q(c)

)< −α − 1

ρ< −α.

If on the other hand 0 ≤ αρ < 1/2, following the remark at the beginning of this section we can find an elementa′ ∈ A ∩ (1/2)V↓ . Then a′′ = (αρ)a + (1 − αρ)a′ ∈ A as well, and

a′′ ∈ (αρ)a +1 − αρ

2V↓ ⊂ (αρ)a +

12

V↓ .

c© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mn-journal.com


Thus

(αc + a′′) ∈ αc + (αρ)a +12

V↓ ⊂(

αb +12

V↓)

+12

V↓ ⊂ αb + V↓ ,

and

inf{p(b) − q(c) | b, c ∈ E, a ∈ A, (c + a) ∈ b + V↓ } ≤ α(p(b) − q(c)) < −α2ρ − α ≤ −α

holds in this case as well. Because α > 0 was chosen independently of the neighborhood V, we infer that

inf{p(b) − q(c) | b, c ∈ E, a ∈ A, (c + a) ∈ b + V↓ } = −∞

holds for all V ∈ V. Our claim follows.

For A = {0} Theorem 3.1 recovers the Sandwich Theorem 2.1. By taking the negatives on both sides of theinequality, the condition in 3.1(a) can obviously be reformulated as

infV ∈V

sup {q(c) − p(b) | b, c ∈ E, a ∈ A, (c + a) ∈ b + V↓ } < +∞.

The condition in Part (b) can be restated in a similar way. The requirement that 0 ∈ A↑(or 0 ∈ A↓

)can generally

not be waived, as the following example will show. However, Theorem 3.1 applies to a non-empty subset A of Ein general, not necessarily convex, since boundedness of a linear functional on A and on the convex hull ofA ∪ {0} coincide.

Example 3.2 Let E be the vector space of all sequences in R with only finitely many non-zero elements,endowed with the equality as order and the l∞-norm of uniform convergence. For a = (αi)i∈N ∈ E set n(a) =max{i ∈ N | αi = 0}, if a = 0 and n(0) = 0. We define a sublinear functional p on E by

p(a) =

⎧⎪⎨⎪⎩

+∞, if at least one αi > 0,

n(a)∞∑

i=1

αi, if all αi ≤ 0 ,

and set q ≡ −∞. Assume that there is a linear functional μ ∈ E∗ such that q ≤ μ ≤ p and set εn = μ(en ), whereen denotes the n-th unit sequence in E. For any n ≥ 2 and λ ≥ 0 set an = −(λe1 + en ). Then μ(an ) = −(λε1 +εn ), and as μ(an ) ≤ p(an ) = −n(λ + 1), we conclude that λε1 + εn ≥ n(λ + 1), that is εn − n ≥ λ(n − ε1).But for an n > ε1 , this cannot hold true for all λ ≥ 0. Therefore there is no such μ ∈ E∗, and the respectiveconditions in Theorem 3.1 should fail for any choice for the set A. But if we choose A = {e1} and the half-unitball V ∈ V as the neighborhood, then β1 > 0 for every b = (βi)i∈N ∈ E such that e1 ∈ b+V. Thus p(b) = +∞and therefore

supV ∈V

inf {p(b) − q(c) | b, c ∈ E, a ∈ A, (c + a) ∈ b + V↓ } = +∞.

This observation contradicts 3.1(a), but the closure of A does not contain the zero element.

We proceed to investigate the largest lower and the smallest upper bound for linear functionals on a convexsubset.

Theorem 3.3 Let E be a locally convex ordered topological vector space and let A be a non-empty convexsubset of E. Let p : E → R be a sublinear and q : E → R a superlinear functional.

(a) If there is μ ∈ E∗+ such that q ≤ μ ≤ p and μ is bounded below on A, then

supμ∈E∗

+q≤μ≤p

infa∈A

μ(a) = supV ∈V

inf {p(b) − q(c) | b, c ∈ E, a ∈ A, (c + a) ∈ b + V↓ }.

(b) If there is μ ∈ E∗+ such that q ≤ μ ≤ p and μ is bounded above on A, then

infμ∈E∗

+q≤μ≤p

supa∈A

μ(a) = infV ∈V

sup {q(c) − p(b) | b, c ∈ E, a ∈ A, c ∈ (b + a) + V↓ }.



P r o o f. Part (b) follows from (a) if we replace the set A by −A. For Part (a) we use α and β to abbreviate theleft and the right-hand side of the equation concerned. We know from the assumption for (a) that α > −∞. Letμ ∈ E∗

+ such that q ≤ μ ≤ p. Given ε > 0 there is V ∈ V such that μ ∈ εV ◦. Then for (c + a) ∈ b + V↓ fora ∈ A and b, c ∈ E we have

q(c) + μ(a) ≤ μ(c) + μ(a) = μ(c + a) ≤ μ(b) + ε ≤ p(b) + ε,

hence μ(a) ≤ p(b) − q(c) + ε, and

infa∈A

μ(a) ≤ inf {p(b) − q(c) | b, c ∈ E, a ∈ A, (c + a) ∈ b + V↓ }.

This verifies α ≤ β, and we shall proceed to demonstrate that α ≥ β holds as well. For α = +∞ there is nothingto prove. Thus we may assume that α ∈ R, and we shall show that

α ≥ inf {p(b) − q(c) | b, c ∈ E, a ∈ A, (c + a) ∈ b + V↓ }

holds for all V ∈ V. Indeed, given a fixed neighborhood V ∈ V and 0 < ε ≤ 1/4 there is μ ∈ E∗+ such that

q ≤ μ ≤ p and μ(a) ≥ α − ε for all a ∈ A. We find a neighborhood W ∈ V such that μ ∈ (1/2)W 0 and mayassume that W ⊂ V. Now we shall use the Extension Theorem 2.2 for the convex sets C = {0} and D = A andthe functions f ≡ 0 and g ≡ α+ε. As there is no linear functional ν ∈ E∗

+ such that q ≤ ν ≤ p and ν(a) ≥ α+εfor all a ∈ A, the condition in Theorem 2.2 must fail for every neighborhood in V and in particular for W : Thereare a ∈ A,b, c ∈ E and ρ ≥ 0 such that (c + ρa) ∈ b + W↓ and

q(c) + ρ(α + ε) > p(b) + 1.

This implies in particular that p(b), q(c) ∈ R. As μ ∈ (1/2)W 0, we have

q(c) + ρ(α − ε) ≤ μ(c) + ρμ(a) = μ(c + ρa) ≤ μ(b) + (1/2) ≤ p(b) + (1/2)

as well. Combining the last two inequalities yields

p(b) + 1 ≤ p(b) + (1/2) + 2ρε and ρ ≥ 1/(4ε) ≥ 1.

Multiplying by 1/ρ ≤ 1, (c + ρa) ∈ b + W↓ yields (c′ + a) ∈ b′ + (1/ρ)W↓ ⊂ b′ + W↓ with c′ = (1/ρ)c andb′ = (1/ρ)b. Furthermore, since p(b′) = (1/ρ)p(b) and q(c′) = (1/ρ)q(c) we infer from the above that

p(b′) ≤ p(b′) + (1/ρ) = (1/ρ)(p(b) + 1) < q(c′) + (α + ε),

that is p(b′) − q(c′) ≤ α + ε. Summarizing, we obtain

α + ε ≥ inf {p(b) − q(c) | b, c ∈ E, a ∈ A, (c + a) ∈ b + W↓ }≥ inf {p(b) − q(c) | b, c ∈ E, a ∈ A, (c + a) ∈ b + V↓ },

for any choice of ε > 0. This yields α ≥ β and completes our argument.

For a singleton set A = {a} Theorem 3.3 requires only that there is μ ∈ E∗+ such that q ≤ μ ≤ p, and the

equations simplify to

supμ∈E∗

+q≤μ≤p

μ(a) = supV ∈V

inf {p(b) − q(c) | b, c ∈ E, (c + a) ∈ b + V↓ }

and

infμ∈E∗

+q≤μ≤p

μ(a) = infV ∈V

sup {q(c) − p(b) | b, c ∈ E, c ∈ (b + a) + V↓ }

in this case. Now taking the supremum or the infimum in these equations over a non-empty set A yields:



Theorem 3.4 Let E be a locally convex ordered topological vector space and let A be a non-empty convexsubset of E. Let p : E → R be a sublinear and let q : E → R be a superlinear functional. Suppose that there isμ ∈ E∗

+ such that q ≤ μ ≤ p. Then

(a) supμ∈E∗

+q≤μ≤p

supa∈A

μ(a) = supa∈AV ∈V

inf {p(b) − q(c) | b, c ∈ E, (c + a) ∈ b + V↓ },

(b) infμ∈E∗

+q≤μ≤p

infa∈A

μ(a) = infa∈AV ∈V

sup {q(c) − p(b) | b, c ∈ E, c ∈ (b + a) + V↓ }.

Remark 3.5 If there is no linear functional μ ∈ E∗+ such that q ≤ μ ≤ p and μ is bounded below on A, then

the expression on the left-hand side of the equation in Theorem 3.3 (a) evaluates as −∞. This need not be thecase for the right-hand side, as a review of Example 3.2 can show. Indeed, for p, q as in 3.2 and A = {e1} wehave

supV ∈V

inf {p(b) − q(c) | b, c ∈ E, a ∈ A, (c + a) ∈ b + V↓ } = +∞.

However, if 0 ∈ A↑ , and if there is no such bounded below functional, then Theorem 3.1 (a) yields

supV ∈V

inf {p(b) − q(c) | b, c ∈ E, a ∈ A, (c + a) ∈ b + V↓ } = −∞.

Thus the equation in 3.3 (a) holds in this case as well. A similar statement applies to 3.3 (b) if 0 ∈ A↓ .The conditions in Theorem 3.4 are not guaranteed even if 0 ∈ A in case that there is no μ ∈ E∗

+ suchthat q ≤ μ ≤ p. This can again be demonstrated by slight modification of Example 3.2. Indeed, chooseA = {λe1 | 0 ≤ λ ≤ 1} in the settings of this example. Then the left-hand sides of 3.4(a) and (b) evaluateas −∞ and +∞, respectively. The presence of 0 ∈ A does not yield the same values for the right-hand sides inthis case, which equal +∞ and −∞ instead.

It is important to realize that the suprema and infima in the preceding results are generally not maxima orminima. This observation will be illustrated in the following example.

Example 3.6 If the sublinear functional p takes the value +∞, then obviously the supremum on the left-handside of Theorem 3.3(a) need not be a maximum, even for a finite set A. Indeed, let E = {0} be any Hausdorfflocally convex topological vector space, and let p ≡ +∞ and q ≡ −∞. Set A = {a} for some a = 0. Then theexpression on the right-hand side in 3.3 equals +∞, but there is no linear functional on E which takes this valueat a. But even if the left-hand side of 3.3(a) is finite, it need not be a maximum. To illustrate this, let E = C(R),the space of all continuous real-valued functions on R, endowed with the usual operations, the pointwise orderand the topology of uniform convergence on compact sets. The dual E∗

+ of E consists of all positive regular Borelmeasures on R which have a compact support. Let p(f) = sup{f(x) | x ∈ R} and q ≡ −∞. Then q ≤ μ ≤ pfor μ ∈ E∗

+ if and only if μ(1) ≤ 1. Thus for the singleton set A = {a}, where a(x) = arctan(x) we have

supμ∈E∗

+q≤μ≤p

infa∈A

μ(a) = supμ∈E∗

+q≤μ≤p

μ(a) =π

2.

But μ(a) < π/2 for all μ ∈ E∗+ such that q ≤ μ ≤ p.

However, if there is a neighborhood U ∈ V such that q ≤ μ ≤ p for a linear functional μ ∈ E∗+ implies that

μ ∈ U◦, then the supremum and the infimum on the left-hand sides in the expressions of Theorem 3.3 (a) and (b)turn into a maximum and a minimum, respectively. Indeed, the set X = {μ ∈ E∗

+ | q ≤ μ ≤ p} is weak*-compact in this case, and the mappings μ �→ infa∈A μ(a) and μ �→ infa∈A μ(a) are weak* upper and lowersemicontinuous, respectively. They therefore attain there minimum and maximum values on X. The followinglemma provides a sufficient condition for this scenario.



Lemma 3.7 Let E be a locally convex ordered topological vector space. Let p : E → R be a sublinear andq : E → R a superlinear functional. Let U ∈ V. If

supu∈U

inf {p(b) − q(c) | b, c ∈ E, c + u ≤ b} ≤ 1, (U)

then every linear functional μ ∈ E∗+ such that q ≤ μ ≤ p is contained in U◦.

P r o o f. Assume that (U) holds, let μ ∈ E∗+ such that q ≤ μ ≤ p and let u ∈ U. Given ε > 0 there are

b, c ∈ E such that c+u ≤ b and p(b)− q(c) ≤ 1+ ε. This implies in particular that p(b), q(c) ∈ R. We infer that

q(c) + μ(u) ≤ μ(c) + μ(u) = μ(c + u) ≤ μ(b) ≤ p(b) ≤ q(c) + (1 + ε).

This shows μ(u) ≤ 1, since ε > 0 was arbitrarily chosen, and therefore μ ∈ U◦ as claimed.

Note that Condition (U) holds in particular if p ≤U 1. Indeed, for any u ∈ U choose c = 0 and b = u. Thenp(b) − q(c) = p(u) ≤ 1. Likewise, the condition holds if and if −1 ≤U q. (Choose c = −u ∈ U and b = 0 inthis case.)

Condition (U) also yields a simplification for the right-hand sides in the equations of Theorem 3.3. This willbe formulated in the last of our main results.

Theorem 3.8 Let E be a locally convex ordered topological vector space and let A be a non-empty convexsubset of E. Let p : E → R be a sublinear and q : E → R a superlinear functional, and suppose thatCondition (U) holds for some U ∈ V.

(a) If there is μ ∈ E∗+ such that q ≤ μ ≤ p and μ is bounded below onA, then

maxμ∈E∗

+q≤μ≤p

infa∈A

μ(a) = inf {p(b) − q(c) | b, c ∈ E, a ∈ A, c + a ≤ b}.

(b) If there is μ ∈ E∗+ such that q ≤ μ ≤ p and μ is bounded above onA, then

minμ∈E∗

+q≤μ≤p

supa∈A

μ(a) = sup {q(c) − p(b) | b, c ∈ E, a ∈ A, c ≤ b + a}.

P r o o f. Again, Part (b) follows from (a) with −A in place of A. Condition (U) turns the left-hand sideof 3.3(a) into the left-hand side of 3.8(a). This was argued before. Let us verify that the right-hand sides alsocoincide. Clearly,

supV ∈V

inf {p(b) − q(c) | b, c ∈ E, a ∈ A, (c + a) ∈ b + V↓ }

≤ inf {p(b) − q(c) | b, c ∈ E, a ∈ A, c + a ≤ b}

holds in any case. For the converse inequality let 0 < ε ≤ 1 and let a ∈ A and b, c ∈ E such that c+a ∈ b+εU↓ ,that is c + a ≤ b + εu for some u ∈ U. According to (U) there are b′, c′ ∈ E such that c′ + u ≤ b′ andp(b′) − q(c′) ≤ 1 + ε. Thus c + εc′ + a + εu ≤ b + εb′ + εu and (c + εc′) + a ≤ (b + εb′). We have

p(b + εb′) − q(c + εc′) ≤ (p(b) − q(c)) + ε(p(b′) − q(c′))≤ (p(b) − q(c)) + ε(1 + ε)≤ (p(b) − q(c)) + 2ε.

This shows

inf {p(b) − q(c) | b, c ∈ E, a ∈ A, c + a ≤ b}≤ inf {p(b) − q(c) | b, c ∈ E, a ∈ A, (c + a) ∈ b + εU↓ } + 2ε,



hence

inf {p(b) − q(c) | b, c ∈ E, a ∈ A, c + a ≤ b}≤ sup

V ∈V

inf {p(b) − q(c) | b, c ∈ E, a ∈ A, (c + a) ∈ b + V↓ },

our claim.

Unfortunately Condition (U) does generally not guarantee the transformation of the suprema and infima onthe left-hand sides in the expressions of Theorem 3.4 into maxima and minima. This will be illustrated in thefollowing example.

Examples 3.9

(a) As in Example 3.2 let E be the vector space of all sequences in R with only finitely many non-zeroelements, endowed with the equality as order and the l∞-norm of uniform convergence. Let p(a) =maxi∈N αi for a = (αi)i∈N ∈ E and q ≡ −∞. Note that p takes only finite values in this case and thatCondition (U) holds with U being the unit ball of E. The linear functionals μ ∈ E∗ such that q ≤ μ ≤ pare the sequences μ = (νi)i∈N such that νi ≥ 0,

∑∞i=1 νi ≤ 1 and μ(a) =

∑∞i=1 νiαi for a = (αi)i∈N ∈

E. Let an = (αni )i∈N, where αn

i = −n for i < n, αnn = n and αn

i = 0 for i > n, and let A be the convexhull of the elements {an}n∈N. For any ε > 0 and V = εB ∈ V, where B denotes the unit ball in E, wehave

inf{p(b) − q(c) | b, c ∈ E, (c + an ) ∈ b + V↓ } = inf{p(b) | ‖b − an‖ ≤ ε} ≥ n − ε.

Thus

supa∈AV ∈V

inf {p(b) − q(c) | b, c ∈ E, (c + a) ∈ b + V↓ } = +∞.

But for every functional μ = (νi)i∈N ∈ E∗ such that q ≤ μ ≤ p one calculates μ(an ) = n(νn −∑n−1

i=1 νi

). This shows limn→∞ μ(an ) = −∞ for every such μ = 0, and we conclude that there is no

μ ∈ E∗ such that q ≤ μ ≤ p and sup{μ(a) | a ∈ A} = +∞.

(b) The preceding example can be easily modified to cover the case of finite-valued suprema as well: In thedefinition of the elements an = (αn

i )i∈N ∈ E change the value of αnn to 1 − (1/n). Then

supa∈AV ∈V

inf {p(b) − q(c) | b, c ∈ E, (c + a) ∈ b + V↓ } = 1,

but sup{μ(a) | a ∈ A} < 1 for every μ ∈ E∗ such that q ≤ μ ≤ p.

4 Examples and applications

Special settings in the general results of the preceding section lead to a variety of applications for which weshall provide a few examples. If used for the balanced convex hull of a set A ⊂ E instead of A itself we obtaininformation about bounds for the absolute values for linear functionals. If used for A − A instead of A, thisinformation concerns the variation of the functional. Unsurprisingly, the statements of Section 3 can be used fora variety of separation results for convex sets. Next we investigate the notions of sub-and superharmonicity withrespect to a subspace of a locally convex ordered topological vector space. A final result deals with the range ofmonotone affine functions.

4.1 Bounds for absolute values

If A is a balanced (that is αa ∈ A whenever a ∈ a and |α| ≤ 1) convex subset of E, then a linear functionalμ ∈ E∗

+ is bounded above on A if and only if it is bounded below, hence if and only if it is bounded. The balancedconvex hull of a non-empty convex set A ⊂ E is given by

bc(A) = {ρa + ρ′a′ | a, a′ ∈ A, |ρ| + |ρ′| ≤ 1}



Obviously, the smallest (absolute) bound for a linear functional μ ∈ E∗+ on A equals its smallest upper bound

and the negative of its largest lower bound on bc(A). We shall denote

‖μ‖A = supa∈A

|μ(a)| = supa∈bc(A)

μ(a).

The functional μ is bounded on A if ‖μ‖A < +∞. Since 0 ∈ bc(A), Theorem 3.1 yields:

Proposition 4.1 Let E be a locally convex ordered topological vector space and let A be a non-empty convexsubset of E. Let p : E → R be a sublinear and q : E → R a superlinear functional. There exists μ ∈ E∗

+ suchthat q ≤ μ ≤ p and ‖μ‖A < +∞ if and only if

supV ∈V

inf {p(b) − q(c) | b, c ∈ E, a ∈ bc(A), (c + a) ∈ b + V↓ } > −∞.

Recall from the last observation in Remark 3.5 that for a convex set A which contains 0 ∈ E the equalities inTheorem 3.3 hold formally true even if there is no μ ∈ E∗

+ such that q ≤ μ ≤ p and such that μ is bounded belowor above on A. Both sides evaluate as +∞ or −∞ in this case. The corresponding requirement in Theorem 3.4can however not be dropped. We therefore obtain the following:

Proposition 4.2 Let E be a locally convex ordered topological vector space and let A be a non-empty convexsubset of E. Let p : E → R be a sublinear and q : E → R a superlinear functional. If there is μ ∈ E∗

+ such thatq ≤ μ ≤ p, then

(a) infμ∈E∗

+q≤μ≤p

‖μ‖A = infV ∈V

sup {q(c) − p(b) | b, c ∈ E, a ∈ bc(A), c − (b + a) ∈ V↓ }.

(b) supμ∈E∗

+q≤μ≤p

‖μ‖A = supa∈bc(A)

V ∈V

inf {p(b) − q(c) | b, c ∈ E, (c + a) − b ∈ V↓ }.

Theorem 3.8 yields:

Proposition 4.3 Let E be a locally convex ordered topological vector space and let A be a non-empty convexsubset of E. Let p : E → R be a sublinear and q : E → R a superlinear functional, and suppose thatCondition (U) holds for some U ∈ V. If there is μ ∈ E∗

+ such that q ≤ μ ≤ p and ‖μ‖A < +∞, then

minμ∈E∗

+q≤μ≤p

‖μ‖A = sup {q(c) − p(b) | b, c ∈ E, a ∈ bc(A), c ≤ b + a}.

The following result holds only for bounds of the absolute value and does not have a precursor in Section 3.

Proposition 4.4 Let E be a locally convex ordered topological vector space and let A be a non-empty convexsubset of E. Let p : E → R be a sublinear and q : E → R a superlinear functional, and suppose that there isμ ∈ E∗

+ such that q ≤ μ ≤ p. Then ‖μ‖A < +∞ for every such μ ∈ E∗+ if and only if

supa∈bc(A)

inf {p(b) − q(c) + λ | b, c ∈ E, λ ≥ 0, (c + a) ∈ b + λV↓ } < +∞

for every V ∈ V.

P r o o f. For V ∈ V and a ∈ E let us abbreviate

pV (a) = inf {p(b) − q(c) + λ | b, c ∈ E, λ ≥ 0, (c + a) − b ∈ λV↓ }.

First suppose that sup{pV (a) | a ∈ bc(A)} < +∞ for every V ∈ V and let μ ∈ E∗+ be such that q ≤ μ ≤ p.

Then μ ∈ V ◦ for some V ∈ V. For every choice of a, b, c ∈ E and λ ≥ 0 such that (c + a) − b ∈ λV↓ , that isc + a ≤ b + v for some v ∈ λV, we have

q(c) + μ(a) ≤ μ(c) + μ(a) = μ(c + a) ≤ μ(b + v) ≤ μ(b) + λ ≤ p(b) + λ.



Thus μ(a) ≤ p(b) − q(c) + λ, and we infer that μ(a) ≤ pV (a). This yields

‖μ‖A = supa∈A

|μ(a)| = supa∈bc(A)

μ(a) ≤ supa∈bc(A)

pV (a) < +∞ .

Hence every linear functional μ ∈ E∗+ such that q ≤ μ ≤ p is bounded on A.

For the converse suppose that sup{pV (a) | a ∈ bc(A)} = +∞ holds for some V ∈ V. We shall verify that inthis case there is a monotone linear functional μ ∈ V ◦ such that q ≤ μ ≤ p and ‖μ‖A = +∞. By our assumptionthere is at least one ν ∈ E∗

+ such that q ≤ ν ≤ p, that is ν ∈ U◦ for some U ∈ V. Since pV ≤ pW and ν ∈ W ◦

whenever W ⊂ U ∩ V, we may assume that U = V. In a first step we shall establish that for every α > 0 thereis μ ∈ E∗

+ ∩ V ◦ such that q ≤ μ ≤ p and μ(a) ≥ α for some a ∈ bc(A). Indeed, choose a ∈ bc(A) such thatpV (a) ≥ α. Let us verify the criterion in the Extension Theorem 2.2 with the convex sets D = {a}, C = {0}and the functions g ≡ α and f ≡ 0. For this let b, c ∈ E and ρ ≥ 0 be such that (c + ρa) − b ∈ V↓ . If ρ = 0,then c − b ∈ V↓ and q(c) ≤ p(b) + 1 by the Sandwich Theorem 2.1. This yields the requirement from 2.2. Ifρ > 0, we set λ = 1/ρ and have (λc + a) − λb ∈ λV↓ . Thus

p(λb) − q(λc) + λ ≥ pV (a) ≥ α

and therefore q(c) + ρα ≤ p(b) + 1, as required in 2.2. Now in a second step we shall construct sequencesof elements an ∈ bc(A), monotone linear functionals μn ∈ V ◦ such that q ≤ μn ≤ p, and 0 < αn ≤ 2−n

as follows: We set α1 = 1/2 and according to our first step select a1 ∈ bc(A) and μ1 ∈ E∗+ ∩ V ◦ such that

q ≤ μ1 ≤ p and μ1(a1) ≥ α1 . Now, successively for n = 2, 3, . . . , if ‖μn−1‖A = +∞, our claim has beenestablished. Otherwise, we choose 0 < αn ≤ 2−n such that αnai ∈ 2−nV for all i = 1, . . . , n − 1. Thenαnν(ai) ≥ −2−n holds for every ν ∈ E∗

+ ∩V ◦. Because all the functionals μ1 , . . . , μn−1 are bounded on A, but

sup{μ(a) | a ∈ bc(A), μ ∈ E∗+ ∩ V ◦, q ≤ μ ≤ p} = +∞,

by our first step we can find an ∈ bc(A) and μn ∈ E∗+ ∩ V ◦ such that q ≤ μn ≤ p and αnμn (an ) ≥ n −∑n−1

i=1 αiμi(an ), that is∑n

i=1 αiμi(an ) ≥ n. If we obtain infinite sequences (an )n∈N, (μn )n∈N and (αn )n∈N inthis way, we set 0 < α =

∑∞n=1 αi ≤ 1 and complete our argument as follows: For every c ∈ E we have c ∈ λV

for some λ ≥ 0, and as all the functionals μn are contained in the polar V ◦ of V, this implies |μn (c)| ≤ λ. Asαn ≤ 2−n , the series μ(c) = (1/α)

∑∞i=1 αiμi(c) converges in R and defines a monotone linear functional on

E. Clearly q ≤ μ ≤ p and μ is contained in E∗+ ∩ V ◦, as for c ∈ V we easily verify that

|μ(c)| ≤ 1α

∞∑i=1

αi |μi(c)| ≤1α

∞∑i=1

αi = 1.

On the other hand we compute for the elements an ∈ bc(A)

∞∑i=1

αiμi(an ) =n∑

i=1

αiμi(an ) +∞∑

i=n+1

αiμi(an ) ≥ n −∞∑

i=n+1

2−i ≥ n − 1.

This shows

‖μ‖A = supa∈bc(A)

μ(a) ≥ supn∈N

μ(an ) = +∞,

thus completing our argument.

The following well-known implication of Proposition 4.4 is generally derived using the Uniform BoundednessTheorem. It fact, Corollary 4.5 below can be used to furnish an alternative non-Baire proof of this fundamentaltheorem (see [10]).

Corollary 4.5 A subset of a locally convex topological vector space is bounded if and only if it is weaklybounded.



P r o o f. Clearly every bounded set is weakly bounded. For the converse, assume that A ⊂ E is weaklybounded. We may assume that A is convex and apply Proposition 4.4 with p ≡ +∞ and q ≡ −∞. Since‖μ‖A < +∞ for all μ ∈ E∗, we have

supa∈bc(A)

inf {p(b) − q(c) + λ | b, c ∈ E, λ ≥ 0, (c + a) ∈ b + λV }

= supa∈bc(A)

inf {λ | λ ≥ 0, a ∈ λV } < +∞

for all V ∈ V. But this means A ⊂ λV for every V ∈ V with some λ ≥ 0. Hence A is bounded.

4.2 Bounds for the variation

The variation of a linear functional μ ∈ E∗+ on a non-empty convex subset A of E is defined as

varA (μ) = supa,b∈A

(μ(a) − μ(b)

).

A functional μ is said to be of bounded variation if varA (μ) < +∞. Since A − A is balanced and convex, weinfer that

varA (μ) = supa∈(A−A)

μ(a) = ‖μ‖(A−A) .

Moreover, bc(A) ⊂ (A−A) ⊂ 2bc(A) yields that varA (μ) < +∞ if and only if ‖μ‖A < +∞. The statements ofPropositions 4.1 and 4.4 therefore hold with varA (μ) in place of ‖μ‖A . The actual bounds for the variation can beobtained from Propositions 4.2 and 4.3 if we replace ‖μ‖A by varA (μ), and bc(A) by A − A in all expressions.

4.3 Separation

There is a variety of separation results for convex subsets of locally convex topological vector spaces. (See forexample II.9 in [13], I.4 and I.6 in [5], or [9].) We shall discuss the following: Let B and C be two non-emptyconvex subsets of E. We are looking for conditions which guarantee that B is strongly separated or weaklyseparated from C, i.e., that there is μ ∈ E∗

+ such that

supb∈B

μ(b) < infc∈C

μ(c) or supb∈B

μ(b) ≤ infc∈C

μ(c),

respectively. For the second (weak) case we also require that μ = 0. This is of course guaranteed in the first case.

Proposition 4.6 Let E be a locally convex ordered topological vector space and let B and C be non-emptyconvex subsets of E. Then

(a) B can be strongly separated from C if and only if the closure of (C↑ − B↓ ) does not contain 0 ∈ E.

(b) B can be weakly separated from C if and only if⋃

λ>0 λ(C↑ − B↓ ) is not dense in E.

P r o o f. Let A = C − B. The definitions of strong or weak separability translate into infa∈A μ(a) > 0 orinfa∈A μ(a) ≥ 0, respectively, for some non-zero linear functional μ ∈ E∗

+ . For the proof of Part (a) we employTheorem 3.3 with p ≡ +∞ and q ≡ −∞. Theorem 3.3(a) states that

supμ∈E∗

+q≤μ≤p

infa∈A

μ(a) > 0

if and only if there is V ∈ V such that a /∈ V↓ for any choice of a ∈ A. (Otherwise we could choose b = c = 0 inthe expression on the right-hand side of 3.3(a) and obtain 0 for the infimum.) Thus strong separation holds if andonly if A ∩ V↓ = ∅ or equivalently, A↑ ∩ V = ∅ for some V ∈ V. Since A↑ = (C−B)↑ = C↑ −B↓ , our claimfollows. For Part (b) we set A = ∪λ>0 λA↑ = ∪λ>0 λ(C↑ − B↓ ). The condition in 4.6(b) is clearly necessaryfor weak separation, because the existence of a non-zero linear functional μ ∈ E∗

+ which is non-negative on A



(and therefore non-negative on A

)guarantees that any element e ∈ E such that μ(e) < 0 is an interior point of

the complement of A. For the sufficiency of the criterion in Part (b) suppose that A is not dense in E. Then thereis an element e ∈ E and a neighborhood V ∈ V such that (e + V ) ∩ A = ∅. This implies

(A − {e}

)∩ V = ∅

and also(A − {e}↓

)∩ V = ∅, since A is increasing. Now according to Part (a) the singleton set {e} can be

strongly separated from A, i.e., there is a non-zero linear functional μ ∈ E∗+ such that μ(e) < infa∈A μ(a). Let

a ∈ A. Then μ(e) < λμ(a) for all λ > 0. This implies μ(a) ≥ 0, hence infa∈A μ(a) ≥ 0, as claimed.

Examples and Remarks 4.7 The condition in 4.6(a) can be reformulated as C↑ ∩ (B↓ + V ) = ∅ for someV ∈ V. This criterion for strong separation is of course well-known, but derived from 3.3(a) with ease. Thecriterion in 4.6(b) appears to be new. It is not very handy, but sharp. We shall use it to derive some of the morepopular sufficient conditions for weak separation.

(a) We have sup{μ(a) | a ∈ A} = sup{μ(a) | a ∈ A↓

}for every μ ∈ E∗

+ and A ⊂ E. Indeed, μ ∈ V ◦

for some V ∈ V. Given a ∈ A↓ and ε > 0 there is a′ ∈ A such that a ∈ a′ + εV↓ . This yieldsμ(a) ≤ μ(a′) + ε. Similarly, one verifies and inf{μ(a) | a ∈ A} = inf

{μ(a) | a ∈ A↑

}. Therefore we

observe that the convex subsets B and C of E can be weakly separated if and only if the sets B↓ and C↑can be weakly separated.

(b) Now suppose that B↓ has a non-empty topological interior B =(B↓ )◦

and that B ∩ C↑ = ∅. SinceB = B↓ , the set A = ∪λ>0λ(C↑ − B↓ ) has a non-empty interior and does not contain 0 ∈ E. Thene + V ⊂ A for some e ∈ E and V ∈ V implies that −e + V is in the complement of A since otherwisewe had 0 ∈ A. But this shows that A is not dense in E, hence B and C can be weakly separated byProposition 4.6(b). Because the closure of B contains B, Part (a) yields that B and C can also be weaklyseparated.

(c) Now let us consider the case that E is finite dimensional. We begin by recalling a few facts:

(i) If a convex set A is dense in E, then A = E. For this we may assume that 0 ∈ A since the statement isinvariant with respect to translation. Indeed, if A is dense in E, so is its span, which therefore equals Ebecause every subspace of a finite dimensional space is closed. Thus A contains a basis {e1 , . . .+en}of E. The point e = (1/2n)(e1 + · · ·+ en ) is an interior point of A, since e + (1/2n)B ⊂ A, whereB = {λ1e1 + · · · + λnen | |λ1 | + · · · + |λn | ≤ 1}. Now assume that there is x ∈ E \ A and lety = 2x − e. Then y + (1/2n)B is in the complement of A, since the existence of an element a ∈ Ain y + (1/2n)B, that is a = y + b for some b ∈ (1/2n)B, would lead to x = (1/2)(a + (e− b)) ∈ Aby the convexity of A. But y + (1/2n)B ⊂ E \ A contradicts the assumption that A is dense in E.Thus E \ A = ∅, as claimed.

(ii) The intrinsic core (see I.2 in [5]) icr(A) of a convex set A ⊂ E consists of all points a ∈ A suchthat for every a′ ∈ A there is ε > 0 such that (1 + ε)a − εa′ ∈ A. In a finite dimensional space theintrinsic core of a non-empty convex subset A is nonempty and dense in A. Indeed, we may againassume that 0 ∈ A. Then A contains a basis {e1 , . . . , em} of its span, and it can be easily checkedthat the point e = (1/2m)(e1 + . . . , em ) is in icr(A). Moreover, icr(A) is dense in A, since for anya ∈ A and 0 ≤ λ < 1 the point λa + (1 − λ)e is also contained in icr(A).

(iii) Now let B,C be non-empty convex subsets of E such that icr(B↓ ) ∩ icr(C↑ ) = ∅. Then A =∪λ>0λ

(icr(C↑ )− icr(B↓ )

)does not contain 0 ∈ E, hence is not dense in E by (i). Therefore B and

C can be weakly separated by 4.6(b) together with our observation in Parts (a) and (c)(ii).

Complete treatments of the non-topological (algebraical) case, including Part (c) of the preceding remarks canbe found in the classical texts by Kothe [9] and Holmes [5].

4.4 Sub-and superharmonic elements

As an application of Theorem 3.3 we shall investigate the following question: Given a convex R-valued function fon a convex subset C of E, what is the supremum on a given element a ∈ E of the values μ(a) for all continuouslinear functionals μ that are dominated by f on C ? We suppose that μ ≤C f for at least one linear functional



μ ∈ E∗+ . This yields μ(a) ≤ λf(c) whenever a ≤ λc for a ∈ E, c ∈ C and λ ≥ 0. Hence

p(a) = inf{λf(c) | c ∈ C, λ ≥ 0, a ≤ λc}

defines a sublinear functional on E. (The above guarantees that p(a) ≥ μ(a) > −∞.) If we set q ≡ −∞, thenthere is at least one μ ∈ E∗

+ satisfying q ≤ μ ≤ p. Obviously, a linear functional μ ∈ E∗+ is dominated by

the convex function f on C if and only if it is dominated by p on all of E. The expression in Theorem 3.3(a)is therefore readily formulated for this case and yields the Part(a) of our next proposition. Part(b) is derived in asimilar fashion from 3.3(b).

Proposition 4.8 Let E be a locally convex ordered topological vector space, let C be a non-empty convexsubset of E and let a ∈ E.

(a) If f is a convex R-valued function on C such that μ ≤C f for at least one μ ∈ E∗+ , then

supμ∈E∗

+

μ≤C f

μ(a) = supV ∈V

inf {λf(c) | c ∈ C, λ ≥ 0, a ∈ λc + V↓ }.

(b) If g is a concave R-valued function on C such that μ ≥C g for at least one μ ∈ E∗+ , then

infμ∈E∗

+

μ≥C g

μ(a) = infV ∈V

sup {λg(c) | c ∈ C, λ ≥ 0, λc ∈ a + V↓ }.

Note that the Extension Theorem 2.2 states that there is μ ∈ E∗+ such that μ ≤C f if and only if there is a

neighborhood V ∈ V such that

0 ≤ σf(c) + 1 holds for all c ∈ C and σ ≥ 0 such that 0 ∈ σc + V↓ ,

or equivalently, since −V↓ = V ↑ , if

f(c) ≥ − inf{λ ≥ 0 | c ∈ λV ↑ } for all c ∈ C.

Proposition 4.8 is of particular interest in the special case when C is a subcone of E and f = g is the restrictionof a linear functional μ ∈ E∗

+ onto C. In this case we need to consider only λ = 1 in the conditions of the 4.8(a)and (b). According to [7], an element a ∈ E is said to be C-superharmonic at μ if for every ν ∈ E∗

+

ν(a) ≤ μ(a) holds whenever ν(c) ≤ μ(c) for all c ∈ C.

Similarly, a ∈ E is said to be C-subharmonic at μ if for every ν ∈ E∗+

ν(a) ≥ μ(a) holds whenever ν(c) ≥ μ(c) for all c ∈ C.

The element a ∈ E is C-harmonic at μ if it is both sub- and superharmonic. We obtain a well-known re-sult (c.f. [2], [7]) for a characterization of super- and subharmonicity from Proposition 4.8. Indeed, a ∈ E isC-superharmonic or C-subharmonic at μ if and only if

supν∈E∗

+

ν≤C μ

ν(a) = μ(a) or infν∈E∗

+

ν≥C μ

ν(a) = μ(a).

Thus Proposition 4.8 yields:

Corollary 4.9 Let E be a locally convex ordered topological vector space, μ ∈ E∗+ , and let C be a subcone

of E.



(a) An element a ∈ E is C-superharmonic at μ if and only if

μ(a) = supV ∈V

inf {μ(c) | c ∈ C, a ∈ c + V↓ }.

(b) A element a ∈ E is C-subharmonic at μ if and only if

μ(a) = infV ∈V

sup {μ(c) | c ∈ C, c ∈ a + V↓ }.

Example 4.10 For a concrete application of Corollary 4.9 let X be a compact Hausdorff space and E = C(X)the space of all continuous real-valued functions on X, endowed with the maximum norm and the pointwise order.The dual E∗

+ of E then consists of all positive regular Borel measures on X. Let C be a subspace of E. For x ∈ Xlet εx ∈ E∗

+ denote the point evaluation at x. A function f ∈ C(X) is C-superharmonic at x ∈ X (that is at εx) iffor every regular Borel measure μ on X,

∫X

g dμ ≤ g(x) for all g ∈ C implies that∫

Xf dμ ≤ f(x). According

to 4.9 this is equivalent to

f(x) = supε>0

inf {g(x) | g ∈ C, f ≤ g + ε}.

Likewise, f ∈ C(X) is C-subharmonic at x ∈ X if and only if

f(x) = infε>0

sup {g(x) | g ∈ C, g ≤ f + ε}.

A function f ∈ C(X) is therefore C-harmonic at all points of X if and only if it is contained in the uniformclosure of the vector sublattice generated by C in C(X).

4.5 Bounds for affine functions

Every real-valued monotone affine function h on a locally convex ordered topological vector space E can beexpressed as h = μ + r, where μ ∈ E∗

+ and r = h(0) ∈ R. We denote the cone of these functions by A+ . Avariation of Proposition 4.8 with the supremum resp. infimum on the left-hand sides of the equations taken overall functions in A+ instead of E∗

+ is available if we embed E via the (affine) mapping a �→ (a, 1) into the vector

space E = E × R, endowed with the product topology and the order induced by E, that is (a, α) ≤ (b, β) ifa ≤ b and α = β. The neighborhoods of the origin in E are given by the sets Vε = {(a, α) | a ∈ V, |α| ≤ ε} forV ∈ V and ε > 0. The dual E∗

+ of E consists of all pairs μ = (μ, r) for μ ∈ E∗+ and r ∈ R. They evaluate as

(a, α) �→ μ(a) + rα on E, hence as affine functions on the embedding of E in E. Any convex, concave or affinefunction on C ⊂ E canonically corresponds to a function of the same type on the embedding of C into E. Clearly,a convex R-valued function f on C dominates at least one affine function h ∈ A+ on C if and only if there isρ ∈ R such that f + ρ dominates a linear functional μ ∈ E∗

+ . Following the remark preceding Proposition 4.8this holds true if and only if there is a neighborhood V ∈ V such that the function

c �−→ f(c) + inf{λ ≥ 0 | c ∈ λV ↑ }

bounded below on C. Now a reformulation of Proposition 4.8 yields:

Proposition 4.11 Let E be a locally convex ordered topological vector space, let C be a non-empty convexsubset of E and let a ∈ E.

(a) If f is a convex R-valued function on C such that h ≤C f for at least one h ∈ A+ , then

suph∈A+

h≤C f

h(a) = supV ∈V

ε>0

inf {λf(c) | c ∈ C, |λ − 1| ≤ ε, a ∈ λc + V↓ }.

(b) If g is a concave R-valued function on C such that h ≥C g for at least one h ∈ A+ , then

infh∈A+

h≥C g

h(a) = infV ∈V

ε>0

sup {λg(c) | c ∈ C, |λ − 1| ≤ ε, λc ∈ a + V↓ }.



For a particular case, suppose that the convex function f on C satisfies the assumption of Proposition 4.11(a)and is both monotone and lower semicontinuous on C in the following sense: For every point c0 ∈ C we have

f(c0) = supV ∈V

inf{f(c) | c ∈ C, c0 ∈ c + V↓ }.

Then Proposition 4.11 implies that f is the pointwise supremum on C of all functions h ∈ A+ such that h ≤C f.Indeed, let c0 ∈ C. We shall have to distinguish two cases. First assume that f(c0) < +∞. According to theabove condition, given ε > 0 there is V ∈ V such that c0 ∈ c + V↓ for c ∈ C implies that f(c) ≥ f(c0) − ε/3.There is ρ > 0 such that c0 ∈ ρV. We choose U = (1/4)V and δ = min{1/2 , 1/4ρ , ε/2|f(c0)|}. Then for allc ∈ C and λ ∈ R such that |λ − 1| ≤ δ and c0 ∈ λc + U↓ we have

λc0 = c0 + (λ − 1)c0 ∈ (λc + U↓ ) + (4ρ |λ − 1|U) ⊂ λc + (1 + 4δρ)U↓ .

Hence

c0 ∈ c +1 + 4δρ

λU↓ ⊂ c + V↓

since (1 + 4δρ)/λ ≤ 2(1 + 4δρ) ≤ 2(1 + 1) = 4. This implies f(c) ≥ f(c0) − ε/3, hence

λf(c) ≥ λf(c0) −λ

3ε ≥ (f(c0) − δ|f(c0)|) −

(1 + δ)3

ε ≥ f(c0) − ε

since δ|f(c0)| ≤ ε/2 and (1 + δ)/3 ≤(1 + (1/2)

)/3 = 1/2. Thus

inf {λf(c) | c ∈ C, |λ − 1| ≤ ε, c0 ∈ λc + U↓ } ≥ f(c0) − ε.

For the second case, assume that f(c0) = +∞. Then for every α > 0 there is V ∈ V such that c0 ∈ c + V↓ forc ∈ C implies that f(c) ≥ 2α. Thus for ε = 1/2 and λ ∈ R such that |λ − 1| ≤ ε we have λf(c) ≥ α for allsuch c ∈ C, hence

inf {λf(c) | c ∈ C, |λ − 1| ≤ ε, c0 ∈ λc + V↓ } ≥ α.

In both cases we therefore infer that

suph∈A+

h≤C f

h(c0) = supV ∈V

ε>0

inf {λf(c) | c ∈ C, |λ − 1| ≤ ε, c0 ∈ λc + V↓ } = f(c0)

holds, as claimed. A similar argument leads to a corresponding statement for monotone and upper semicontinuousconcave functions.

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