Korovkin theory for vector-valued functions on a locally compact space

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Korovkin theory for vector-valued functions on a locally compact space Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam Received 24 May 2013; received in revised form 31 August 2013; accepted 14 September 2013 Available online 23 September 2013 Communicated by Dany Leviatan

Abstract We consider ordered spaces of continuous vector-valued functions on a locally compact Hausdorff space, endowed with appropriate locally convex topologies. Using suitable sets of such functions as test systems a Korovkin type approximation theorem for equicontinuous nets of positive operators is established. As in the classical theory, the Korovkin closure is characterized both through envelopes of functions and through measure theoretical conditions. c 2013 Elsevier Inc. All rights reserved. ⃝ Keywords: Vector-valued functions; Korovkin type approximation

1. Introduction This paper is based on an earlier publication by the author [6] which contains a classification of spaces of vector-valued functions on a locally compact space and establishes an integral representation for linear functionals. We shall use a special case of this to transfer Korovkin theory to function spaces of this type. The parameters of the theory can be adjusted to suit a wide range of situations. We shall use the following notations: Let (E, V, 6) be a locally convex ordered topological vector space, that is a real vector space E endowed with an order relation 6 and a basis V of balanced convex neighborhoods of the origin. For our purposes we assume that E ∈ V. The order is supposed to be reflexive, transitive and compatible with the algebraic operations, that is a 6 b implies a + c 6 b + c and αa 6 αb E-mail addresses: [email protected], [email protected]. c 2013 Elsevier Inc. All rights reserved. 0021-9045/$ - see front matter ⃝ http://dx.doi.org/10.1016/j.jat.2013.09.003

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for all a, b, c ∈ E and α > 0. Since equality in E is obviously such an order, this concept will apply to locally convex vector spaces in general. E + denotes the subcone of all positive elements of E. A non-empty subset A of E is called decreasing or increasing, respectively, if b ∈ A whenever b 6 a, or a 6 b, for some a ∈ A. The decreasing hull A ↓ and the increasing hull A ↑ of A are defined as A ↓= {b ∈ E | b 6 a for some a ∈ A}

and

A ↑= {b ∈ E | a 6 b for some a ∈ A},

respectively. Note that (−A) ↓= −(A ↑). The decreasing, or increasing, hull of a convex or balanced set is again convex or balanced and the sum of two decreasing, increasing or convex sets is again decreasing, increasing or convex. A subset A of E is called order convex if A = A ↓ ∩A ↑, that is c ∈ A whenever a 6 c 6 b for some a, b ∈ A. As usual, we assume that the neighborhoods in V are also order convex. The topological closure of a subset A of E is denoted by A. The upper and lower topologies of E are defined by the neighborhoods V ↓ (a) and V ↑ (a) for all a ∈ E and V ∈ V, respectively, that is V ↓ (a) = a + V ↓

and

V ↑ (a) = a + V ↑ .

Note that b ∈ V ↓ (a) if and only if a ∈ V ↑ (b). The upper and lower topologies on E are generally non-Hausdorff and their common refinement is the given vector space topology (see Chapter I.2 in [1]). By E ∗ we denote the topological dual of E, that is the space of all continuous real-valued linear functionals on E. A functional µ ∈ E ∗ is called monotone (or positive) if a 6 b implies ∗ will denote the subcone of all µ(a) 6 µ(b), or equivalently if µ(a) > 0 for all a > 0. E + ∗ ∗ positive functionals in E . Every functional µ ∈ E can be expressed as a difference of two ∗ (see Section V.3 in [8]). The polar A◦ positive ones, that is µ = µ1 − µ2 for some µ1 , µ2 ∈ E + of a subset A of E consists of all µ ∈ E ∗ such that µ(a) 6 1 for all a ∈ A. Let Conv(E) denote the cone of all non-empty convex subsets of E, endowed with its canonical addition and multiplication by non-negative scalars and ordered by A6B

for A, B ∈ Conv(E) if A ⊂ B ↓ .

Corresponding to V ∈ V, upper and lower neighborhoods in Conv(E) are defined for A ∈ Conv(E) by V ↓ (A) = {B ∈ Conv(E) | B ⊂ A + V ↓}

and

V ↑ (A) = {B ∈ Conv(E) | A ⊂ B + V ↓}, respectively, giving rise to the upper and lower topologies on Conv(E). These are locally convex cone topologies in the sense of [1]. Their common refinement is called the symmetric topology. Note that B ∈ V ↓ (A) if and only if A ∈ V ↑ (B). Let X be a locally compact Hausdorff space. As usual, for a subset Y of a topological space, the sets Y and Y ◦ denote its topological closure, and interior, respectively. R represents the family of all relatively compact Borel subsets of X. Let F(X, E) be the space of all E-valued functions on X , endowed with the pointwise algebraic operations and order, and let C(X, E) denote the subspace of all continuous functions in F(X, E). Continuity is meant with respect to the given (symmetric) topology of E. The subspace of all functions in C(X, E) with compact support is denoted by CK (X, E). In case that E = R we write F(X ), C(X ) and CK (X ) for short. The subscript + is used to denote the subcone of positive elements in any of these function spaces. For ϕ ∈ F(X ) and f ∈ F(X, E) the function ϕ ⊗ f ∈ F(X, E) is the mapping

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x → ϕ(x) f (x) : X → E. If f is the constant function x → a for some a ∈ E, we write ϕ ⊗ a for the function x → ϕ(x)a. If ϕ ∈ C(X ) and f ∈ C(X, E), then ϕ ⊗ f ∈ C(X, E). The characteristic function χY ∈ F(X ) of a subset Y of X takes the value 1 on Y and 0 else. R = R ∪ {+∞} denotes the extended real number system with the usual order and algebraic operations, in particular a + ∞ = +∞ for all a ∈ R, α · (+∞) = +∞ for all α > 0 and 0 · (+∞) = 0. The multiplication of +∞ with negative numbers is not defined. Unfortunately, the previously introduced upper, lower and symmetric locally convex cone topologies for Conv(E) are too restrictive for the concept of continuity of Conv(E)-valued functions, since for unbounded sets even the scalar multiplication turns out to be discontinuous (see I.4 in [5]). This is remedied by using the coarser (but somewhat cumbersome) upper and lower relative topologies on Conv(E) instead. These topologies are defined by the neighborhoods Vε ↓ (A) = {B ∈ Conv(E) | B ⊂ γ A + εV ↓ for some 1 6 γ 6 1 + ε}

and

Vε ↑ (A) = {B ∈ Conv(E) | A ⊂ γ B + εV ↓ for some 1 6 γ 6 1 + ε}, for A ∈ Conv(E), V ∈ V and ε > 0. Note that B ∈ Vε ↓ (A) if and only if A ∈ Vε ↑ (B). The relative topologies are locally convex but not necessarily locally convex cone topologies in the sense of [1] (for details see I.4 in [5]) since the resulting uniformity need not be convex. They locally coincide on bounded sets with the canonical upper and lower topologies on Conv(E), but render the scalar multiplication (at scalars other than zero) continuous. Their common refinement is called the symmetric relative topology. We shall refer to continuity with respect to the relative topologies as r-upper, r-lower and r-continuity, respectively. For a non-negative Rvalued function ϕ on X and C ∈ Conv(E), we denote the mapping x → ϕ(x)C : X → Conv(E) by ϕ ⊗ C, where (+∞) C is meant to be the full space E. Proposition III.1.8 in [5] states that at a point x ∈ X the function ϕ⊗C is (i) r -lower continuous, provided that ϕ is lower semicontinuous at x in the usual sense, and (ii) r -upper continuous, provided that ϕ is upper semicontinuous at x and that either ϕ(x) > 0 or the set C is bounded in E. With a non-empty convex subset u of C(X, E) we associate the set-valued function Fu : X → Conv(E) such that   Fu (x) = f (x) | f ∈ u , and similarly, with a set-valued function F : X → Conv(E) we associate the convex subset of C(X, E)   u F = f ∈ CV (X, E) | f (x) ∈ F(x) for all x ∈ X . A subset u of C(X, E) is called C(X )-convex if ϕ ⊗ f +(1−ϕ)⊗ g ∈ u whenever f, g ∈ u and 0 6 ϕ 6 1 for ϕ ∈ C(X ). A simple argument shows that sums, multiples, topological closures, decreasing and increasing hulls of C(X )-convex sets are again C(X )-convex. Proposition 3.1 in [7] demonstrates that u = u Fu whenever u is a closed C(X )-convex subset of CV (X, E). Moreover, if u ⊂ C(X, E) is C(X )-convex and decreasing, then Fu (x) is decreasing in E for every x ∈ X ; more precisely: Fu↓ (x) = Fu (x) ↓ whenever u is C(X )-convex. In Section 2 we quote and summarize the relevant terminology and results and provide some examples for spaces of vector-valued functions. Due to technical requirements, our notation of function space neighborhoods will be narrower than in [6] which also includes inductive limit type topologies. Section 3 contains our main results, including a Korovkin-type approximation theorem for equicontinuous nets of positive linear operators on spaces of E-valued functions. We conclude with some special cases and examples.

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2. Spaces of functions that vanish at infinity We proceed to introduce various subspaces of C(X, E) which are generated by suitable systems of neighborhoods. A function space neighborhood v in F(X, E) is defined by an r lower continuous neighborhood-valued mapping Vv : X → V. (It is straightforward to realize (see Lemma 2.6 below) that r -lower continuity coincides with the abridged notions of lower semicontinuity for functions of this type used in [6,7].) We require that for every A ∈ R there is W ∈ V such that W ⊂ Vv (x) for all x ∈ A. The neighborhood v then consists of all functions f ∈ F(X, E) such that f (x) ∈ Vv (x) for all x ∈ X , that is v = uVv . Obviously, v is both C(X )-convex and order convex in CV (X, E). This is a special (reduced) case of the concept of function space neighborhoods defined in [6] which fits our purpose. The functions Fv , Vv↓ and Fv↓ are also r -lower continuous. A straightforward argument using Lemma 2.6 below shows Fv (x) ⊂ Vv (x) ⊂ Fv (x) holds for all x ∈ X. A family V of function space neighborhoods is called a function space neighborhood system for F(X, E) if it is closed for multiplication by positive scalars and downwards directed with respect to set inclusion. The following summary is taken from [6] and included here for the sake of accessibility: a function f ∈ F(X, E) vanishes at infinity with respect to a function space neighborhood system V if for every v ∈ V there is A ∈ R such that χ(X \A) ⊗ f ∈ v. Let CV (X, E) denote the space of all functions in C(X, E) that vanish at infinity with respect to V, and abbreviate CV (X ) if E = R. All neighborhoods v ∈ V are absorbing in CV (X, E), hence for every choice of V, the intersection of CV (X, E) with the sets in V forms a basis for a locally convex vector space topology on CV (X, E). Endowed with the pointwise order, CV (X, E), V is a locally convex ordered topological vector space. Examples 2.1. (a) The topology of uniform convergence on X is generated by the function space neighborhoods vV = { f ∈ F(X, E) | f (x) ∈ V for all x ∈ X }, generated by the neighborhood function x → V , corresponding to all V ∈ V. A function f ∈ C(X, E) vanishes at infinity if for every V ∈ V there is a compact subset K of X such that f (x) ∈ V for all x ∈ X \K . (b) The topology of compact convergence is generated by the lower semicontinuous function space neighborhoods v(K ,V ) = { f ∈ F(X, E) | f (x) ∈ V for all x ∈ K }, generated by the neighborhood function χ K ⊗ V , corresponding to all V ∈ V and compact subsets K of X. We have CV (X, E) = C(X, E) in this case. (c) The topology of pointwise convergence is generated by the lower semicontinuous function space neighborhoods v(Y,V ) = { f ∈ F(X, E) | f (x) ∈ V for all x ∈ Y }, generated by the neighborhood function χY ⊗ V , corresponding to all V ∈ V and finite subsets Y of X. We have CV (X, E) = C(X, E). (d) Parts (a)–(c) are special cases for weighted space topologies which were introduced for real-valued functions by Nachbin and Prolla (see [2,3]). A family Ω of non-negative real-valued

W. Roth / Journal of Approximation Theory 176 (2013) 23–41

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upper semicontinuous functions on X is called a family of weights if for all ω1 , ω2 ∈ Ω there are ω3 ∈ Ω and ρ > 0 such that ω1 6 ρ ω3 and ω2 6 ρ ω3 . For V ∈ V and ω ∈ Ω , v(ω,V ) = { f ∈ F(X, E) | ω(x) f (x) ∈ V for all x ∈ X } is generated by the neighborhood function (1/ω) ⊗ V . (Recall that (+∞) V = E.) Sums of neighborhoods of this type establish a function space neighborhood system V such that CK (X, E) ⊂ CV (X, E) ⊂ C(X, E). (e) The case X = N covers a variety of sequence spaces. We quote Propositions 2.3 and 2.4 from [6]: Proposition 2.2. If E is Hausdorff and if convergence for functions in CV (X, E) implies pointwise convergence, then CV (X, E) is Hausdorff. Proposition 2.3. If the positive cone E + is closed in E and if convergence for functions in CV (X, E) implies pointwise convergence, then the positive cone CV (X, E)+ is closed in CV (X, E). If E is a vector lattice, then the lattice operations transfer pointwise to the functions in F(X, E). proposition 2.6 in [6] states:   Proposition 2.4. Let CV (X, E), V be a function space. If E is a topological vector lattice and if for every v ∈ V there is u ∈ V such that f ∈ v whenever | f | 6 |u| for some u ∈ u, then CV (X, E) is also a topological vector lattice. The main result in [6] states that every continuous linear functional on a function space CV (X, E) can be represented as an integral with respect to an E ∗ -valued measure θ . This measure is defined on the sets in R, which forms a weak σ -ring, but does generally not contain the space X itself. θ is countably additive within R and integrates measurable E-valued functions (see Section 3 in [6]) over sets of the σ -field A = {B ⊂ X | A ∩ B ∈ R for all A ∈ R} which contains all Borel subsets of X and coincides with the latter if X is countably compact. The values of the integrals are in R. A suitable notion of regularity for such measures is introduced. We quote Theorem 4.4 in [6]. The dual of CV (X, E) is denoted by CV (X, E)∗ , endowed with the weak*-topology induced by CV (X, E). Its positive cone is CV (X, E)∗+ . Theorem 2.5. Let V be a function space neighborhood system. For every linear functional µ ∈ CV (X, E)∗ there exists a unique regular E ∗ -valued measure θ on R such that all functions  in CV (X, E) are integrable with respect to θ and µ( f ) = X f dθ holds for all f ∈ CV (X, E). ∗ -valued. The functional µ is positive if and only if θ is E + ∗ -valued measure is therefore called positive. Theorem 2.5 allows to consider continuous An E + linear functionals on CV (X, E) both as elements of its topological dual and as vector-valued measures, thus permitting to employ techniques from either theory. A point evaluation is a linear η functional (measure) with singleton support, that is a functional δx , for x ∈ X and η ∈ E ∗ which evaluates   f → η f (x) : CV (X, E) → R. η

Note that a point evaluation δx is continuous on CV (X, E) if and only if η ∈ Vv (x)◦ for some v ∈ V.

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A neighborhood in CV (X, E) is a (not necessarily balanced) convex subset u of CV (X, E) such that v ⊂ u for some v ∈ V. Obviously, if u ⊂ CV (X, E) is a neighborhood, then for every x ∈ X the set Fu (x) = { f (x) | f ∈ u} is a convex neighborhood (of the origin) in E. The polar u◦ of a decreasing neighborhood in CV (X, E) is contained in the positive cone CV (X, E)∗+ of the ∗. dual, hence for all x ∈ X the polar of Fu (x) is in E + Lemma 2.6. Let u be a C(X )-convex neighborhood in CV (X, E). The corresponding Conv(E)valued function Fu is r -lower (or r -upper) continuous if and only if for every x ∈ X and ε > 0 there is a neighborhood U of x such that Fu (x) ⊂ (1 + ε)Fu (y) ↓ (or Fu (y) ⊂ (1 + ε)Fu (x) ↓) for all y ∈ U. Proof. The sufficiency   of this condition for r -lower continuity is obvious, since it implies that Fu (y) ∈ Vε ↑ Fu (x) , that is Fu (x) ⊂ (1 + ε)Fu (y) ↓⊂ (1 + ε)Fu (y) + εV ↓ holds for all V ∈ V and y ∈ U . Conversely, suppose that Fu is r -lower continuous and let U0 ∈ R be a neighborhood of x ∈ X. Since u is a neighborhood in CV (X, E), there is V ∈ V such that V ⊂ Fu (y) for all y ∈ U0 . Given ε > 0 by the r -lower continuity of the function Fu  there is a neighborhood U1 ⊂ U0 of x such that for all y ∈ U1 we have Fu (y) ∈ Vε ↑ Fu (x) , that is Fu (x) ⊂ γ Fu (y) + εV ↓ for some 1 6 γ 6 1 + ε. Since 0 ∈ Fu (y), the latter implies Fu (x) ⊂ (1 + ε)Fu (y) + εV ↓⊂ (1 + ε)Fu (y) + εFu (y) ↓= (1 + 2ε)Fu (y) ↓, hence our claim. The argument for r -upper continuity is similar.



Corollary 3.2 in [7] states that every extreme point of the polar u◦ of a C(X )-convex η neighborhood u in CV (X, E) is a point evaluation δx such that η is an extreme point of the ◦ ∗ polar Fu (x) of Fu (x) in E . For this reason we shall introduce an additional parameter for our ∗ is said to support the separation property for E if approach. A closed subset S of E + (S1) αη ∈ S whenever η ∈ S and α > 0. (S2) There is a constant β > 1 such that for every V ∈ V     sup ζ (a) | ζ ∈ V ↓◦ 6 β sup η(a) | η ∈ S ∩ V ↓◦ holds for all a ∈ E. Equivalently, (S2) signifies that V ↓◦ is contained in the β-multiple of the closed convex hull ∗ supports the separation property with β = 1 in this sense. For of S ∩ V ↓◦ . Of course S = E + n example, if E = R carries the componentwise order, then we may choose the positive multiples of the unit vectors en for S. If E is an abstract M-space, that is a vector lattice with supremum∗ form a suitable subset S. In particular, stable neighborhoods, the lattice homomorphisms in E + if E is a function space CW (Y ), then the positive multiples of the point evaluations δ y for y ∈ Y , that is a → a(y) for a ∈ E, may constitute S. Condition (S2) holds with β = 1 in this case. We denote ∆S = {δxη ∈ CV (X, E)∗+ | x ∈ X, η ∈ S}

and ∆uS = ∆S ∩ u◦

for a decreasing C(X )-convex neighborhood u in CV (X, E). We proceed to verify that ∆uS is a weak*-compact subset of CV (X, E)∗+ . For the following let ∞ denote the adjoined element in the one-point compactification of the locally compact space X. Lemma 2.7. Let u be a decreasing C(X )-convex neighborhood in CV (X, E) such that the η  corresponding Conv(E)-valued function Fu is r -lower continuous. Every net δxii i∈I in ∆uS

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 η  has a subnet δx jj j∈J such that either x j → ∞ and

η

δx jj → 0,

or

x j → x,

ηj → η

and

η

δx jj → δxη ∈ ∆uS .

 η   η  η Proof. Let δxii i∈I be a net in ∆uS . There is a subnet δx jj j∈J of δxii i∈I such that (x j ) j∈J η

converges to some x ∈ X or to x = ∞. If x = ∞, then δx jj → 0. Indeed, given f ∈ CV (X, E) and ε > 0 there is A ∈ R such  that f (y) ∈ εFu (y)for all y ∈ X \A. There is j0 ∈ J such   η that x j ∈ X \A, and therefore δx jj ( f ) = η j f (x j )  6 ε for all j > j0 , since η j ∈ Fu (x j )◦ . η

η

Thus δx jj ( f ) → 0, hence δx jj → 0 in the weak*-topology of CV (X, E)∗ . If x ∈ X , we argue as follows. According to Lemma 2.6 there is a neighborhood U0 of x such that Fu (x) ⊂ 2Fu (y) ↓= 2Fu (y),  ◦  ◦ and therefore Fu (y) ⊂ 2 Fu (x) holds for all y ∈ U0 . Since Fu (x) is a neighborhood in E, ∗ therefore itsηlpolar   η j  being compact, and S is closed in CV (X, E) , there exists a further subnet ◦ δxl l∈L of δx j j∈J such that ηl → η ∈ S. We continue to argue that η ∈ Fu (x) , and η therefore that δx ∈ ∆uS . Surely, given ε > 0, again using Lemma 2.6 we find l0 ∈ L such that  ◦  ◦ Fu (x) ⊂ (1 + ε)Fu (xl ), hence ηl ∈ Fu (xl ) ⊂ (1 + ε) Fu (x) for all l > l0 . This shows  ◦  ◦ η ∈ (1 + ε) Fu (x) for all ε > 0, and therefore η ∈ Fu (x) . η η All left to verify is that δxll → δx . For this, given f ∈ CV (X, E) and ε > 0 there is a neighborhood U1 ⊂ U0 of x such  that f (y) − f (x) ∈ εFu (x) for all y ∈ U1 , and l0 ∈ K such that both xl ∈ U1 and  ηl − η f (x)  6 ε for all l > l0 . For every such l we have     η    δ l ( f ) − δ η ( f ) = ηl f (xl ) − η f (x)  xl x           6 ηl f (xl ) − ηl f (x)  + ηl f (x) − η f (x)  6 2ε + ε,  ◦ since ηl ∈ Fu (xl ) ⊂ 2 Fu (x) and f (xl ) − f (x) ∈ εFu (x) by the above. This holds for all η η f ∈ CV (X, E) and ε > 0 and therefore yields δxll → δx as claimed.  As an immediate consequence we obtain the following. Proposition 2.8. Let u be a decreasing C(X )-convex neighborhood in CV (X, E) such that the corresponding Conv(E)-valued function Fu is r -lower continuous. The set ∆uS is weak*-compact in CV (X, E)∗+ . We conclude this section with a brief consideration of the special case that E itself is a space CW (Y ) of real-valued functions for a locally compact Hausdorff space Y anda system W  of function space neighborhoods. In this case we can represent the functions in CV X, CW (Y ) in a canonical way as real-valued functions on the locally compact Hausdorff space X × Y setting      f (x, y) = f x (y) for f ∈ CV X, CW (Y ) .   If both X and Y are compact and both C(Y ) and C X, C(Y ) carry their respective norms   of uniform convergence, this establishes a well-known correspondence between C X, C(Y ) and C(X × Y ). In our more general setting involving locally compact spaces and function space topologies the continuity of the functions  f on X × Y is however not guaranteed. In order to obtain a similar representation we introduce an additional requirement for CW (Y ): (R) For every y ∈ Y there is w ∈ W such that Fw is r -continuous at y and Fw (y) is bounded.

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If this condition holds, then the function  f associated with f ∈ CV (X, E) is continuous on X × Y. Indeed, given (x, y) ∈ X × Y and ε > 0 there is w ∈ W such that Fw is r -continuous at y and Fw (y) ⊂ [−ε, +ε]. There are neighborhoods  U  of x in X and V of y in Y such that f (s) − f (x) ∈ w and both Fw (t) ⊂ 2Fw (y) and | f x (t) − f x (y)| 6 ε for all s ∈ U and t ∈ V. Thus     f s (t) − f x (t) ∈ Fw (t) ⊂ 2Fw (y) ⊂ 2[−ε, +ε],     hence | f s (t) − f x (t)| 6 4ε, and therefore     | f (s, t) −  f (x, y)| = | f s (t) − f x (y)|         6 | f s (t) − f x (t)| + | f x (t) − f x (y)| 6 5ε,   validating our claim. Corresponding to function space neighborhoods for F X, C(Y ) we define function space neighborhoods for F(X ×Y ) as follows: for v ∈ V, associated with the W-valued r -lower continuous function Fv on X we consider for every x ∈ X the functions W Fv (x) on Y , that is the r -lower continuous function associated with the neighborhood Fv (x) ∈ W. The values of W Fv (x) are multiples of the interval [−1, +1] in R. This leads to a function Fv˜ on X × Y with the same range, that is Fv˜ (x, y) = W Fv (x) (y)

for (x, y) ∈ X × Y.

The function space neighborhood v˜ for F(X × Y ) is then defined as the set of all functions  f ∈ F(X × Y ) such that  f (x, y) ∈ Fv˜ (x, y) for all (x, y) ∈ X × Y. Hence f ∈ v for f ∈ CV (X, E) if and only if  f ∈ v. This establishes a one-to-one correspondence between  is the function space neighborhood system generated CV (X, E) and CV (X × Y ), where V  by the neighborhoods v. ˜ Moreover, if S consists of all positive multiples of point evaluations δ y ∈ CW (Y )∗+ , then (S2) holds with β = 1 and ∆S consists of all positive multiples of point ∗  y  = {0 6 η ∈ R} and ∆  evaluations δx ∈ CV X, CW (Y ) + . This corresponds to the set S S ∗ consisting of the positive multiples of point evaluations δ(x,y) ∈ CV  (X × Y )+ . We summarize the following. Proposition 2.9. Let X and Y be locally compact spaces, let W be a system of function space neighborhoods for   F(Y ) satisfying (R), and let V be a system of function  space neighborhoods  for F X, CW (Y ) . Then there is a bijective correspondence between CV X, CW (Y ) and CV  (X ×  are defined by the functions Fv˜ (x, y) = W F (x) (y) for Y ), where the neighborhoods  v ∈ V v (x, y) ∈ X × Y and v ∈ V. This correspondence preserves the algebraic operations, the order and the neighborhoods for both spaces. 3. Korovkin theory Some of the concepts and techniques in this section rely on ideas developed by the author in [7] for vector-valued Choquet theory. Let CV (X, E) be a function space. Its topological dual CV (X, E)∗ is endowed with the weak*-topology induced by CV (X, E). We consider the pointwise order for R-valued functions on CV (X, E)∗ . The cone of all lower semicontinuous Rvalued sublinear functionals on CV (X, E)∗ is denoted by P. There is a canonical correspondence between non-empty convex subsets u of CV (X, E) and functionals pu ∈ P defined by pu (µ) = sup{µ( f ) | f ∈ u}

for µ ∈ CV (X, E)∗ .

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Since pu = pu¯ , where u¯ denotes the topological closure of u in CV (X, E), this correspondence is indeed one-to-one and onto if one regards only closed convex subsets of CV (X, E). This follows from the Hahn–Banach theorem. We have f ∈ u¯ for f ∈ CV (X, E) if and only if µ( f ) 6 pu (µ) holds for all µ ∈ CV (X, E)∗ . If u is bounded, then pu is indeed real-valued on CV (X, E)∗ . Note that p(u+w) = pu + pw and p(αu) = αpu for convex subsets u, w of CV (X, E) and α > 0. If u = conv{ui | i ∈ I} denotes the convex hull of the union of the sets ui , then pu = ∨i∈I pui , that is the pointwise supremum of the functionals pui . Every vector-valued function f ∈ CV (X, E) can be considered as a continuous linear functional µ → f (µ) = µ( f ) on CV (X, E)∗ , and every continuous linear functional on CV (X, E)∗ is of this type. Hence CV (X, E) can be considered to be a subspace of P. Proposition 3.1 in [7] states that pu (ν + µ) = pu (µ) + pu (ν) holds for a C(X )-convex subset u whenever the functionals µ, ν ∈ CV (X, E)∗ are represented by mutually disjoint measures. A subset A of a locally convex ordered topological vector space E is called upper precompact if it is precompact with respect to the upper topology of E, that is if for each neighborhood V ∈ V there are points a1 , . . . , an ∈ A such that A⊂

n 

(ai + V ↓).

i=1

Upper precompact sets are bounded in the upper topology, and therefore their relative upper neighborhoods coincide with the canonical ones. Upper precompactness is a far weaker condition than precompactness with respect to the given (symmetric) topology. If E carries an order unit norm, for example, then its unit ball is upper, but in the infinite dimensional case not symmetrically precompact. A non-empty convex subset f of CV (X, E) is called a K-set if the corresponding Conv(E)valued function Ff is r -upper continuous, for every x ∈ X the set Ff (x) is upper precompact, and for every v ∈ V there is A ∈ R such that Ff (x) ⊂ Vv (x) for all x ∈ X \A. Singleton sets { f } for f ∈ CV (X, E) are of course K-sets. It is straightforward to verify that sums and positive multiples of K-sets are again K-sets. With some benefit we shall use K-sets in place of single functions in the forthcoming formulation of our Korovkin-type approximation theorem. ∗ supporting the separation Throughout the following we shall assume that S is a subset of E + property with the constant β > 1 for E. Proposition 3.1. Let u be a decreasing C(X )-convex neighborhood in CV (X, E) such that the corresponding Conv(E)-valued function Fu is r -lower continuous. For every K-set f ⊂ CV (X, E) the functional pf is real-valued and weak*-continuous on ∆uS . Proof. Let f ⊂ CV (X, E) be a K-set and u a decreasing C(X )-convex neighborhood in CV (X, E) η η such that Fu is r -lower continuous. First we argue that pf (δx ) < +∞ for all δx ∈ ∆uS . This follows from the  upper precompactness of Ff (x) with a neighborhood V ∈ V such that η ∈ V ↓◦ . n Indeed, Ff (x) ⊂ i=1 (ai + V ↓) implies that n

pf (δxη ) = sup{η(a) | a ∈ F f (x)} 6 max{η(ai ) + 1}. i=1

 η η η Now let be a net in ∆uS converging to δx ∈ ∆uS . Then pf (δx ) 6 limi∈I pf δxii by the  ηi  η lower semicontinuity of the functional pf . Assume that limi∈I pf δxi > pf (δx ). In this case we 

η  δxii i∈I

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 η   η find ε > 0 and a subnet δx jj j∈J of δxii i∈I such that  η  pf δx jj > pf (δxη ) + ε for all j ∈ J .  η  η  Then there is a further subnet δxll l∈L of δx jj j∈J as constructed in Lemma 2.7 which allows η η for two cases: if xl → ∞, then δx = 0, pf (δx ) = 0 and we argue as follows. There is A ∈ R such that Ff (y) ⊂ Fεv (y) = εFv (y) ⊂ ε Fu (y) for all y ∈ X \A. In turn, there is l0 ∈ L such that xl ∈ X \A for all l > l0 . Thus   η          pf δ l  = sup ηl f (xl ) | f ∈ f  = sup ηl (a) | a ∈ Ff (xl )  6 ε xl  ◦  ◦  η for all l > l0 , since ηl ∈ Fu (xl ) ⊂ ε Ff (xl ) . This contradicts pf δxll > ε for all l ∈ L. In the second case, that is xl → x and ηl → η, we set ε ′ = ε/5 and choose a neighborhood U of x such that both Fu (x) ⊂ 2Fu (y) and Ff (y) ⊂ F f (x) + ε ′ Fu (x) for every y ∈ U. The former uses the r -lower continuity of the function Fu and Lemma 2.6, the latter the r -upper continuity of the function Ff with the decreasing neighborhood Fu (x) in E. Recall that the upper precompactness of the sets Ff (y) allows to consider their canonical rather neighborhoods for  ◦ than  the relative ◦ η continuity. For y ∈ U and δ y ∈ ∆uS , that is η ∈ Fu (y) ⊂ 2 Fu (x) this implies     pf (δ ηy ) = sup η(a) | a ∈ Ff (y) 6 sup η(a) | a ∈ Ff (x) + 2ε′ . Moreover, since the set Ff (x) is supposed to be upper precompact and  since Fu (x) is a decreasing neighborhood in E, we find a1 , . . . , an ∈ Ff (x) such that Ff (x) ⊂ nk=1 ak + ε ′ Fu (x) . Using the convergence of the nets (xl )l∈L and (ηl )l∈L we find l0 ∈ L such that both xl ∈ U and ◦ |ηl (ak ) − η(ak )| 6 ε ′ for all k = 1, . . . , n whenever l > l0 . Once more using that ηl ∈ 2 Fu (x) for all l > l0 , we infer that    n  n  sup ηl (a) | a ∈ Ff (x) 6 max ηl (ak ) + 2ε ′ } 6 max η(ak ) + 3ε ′ 6 pf (δxη ) + 3ε ′ . k=1

k=1

Combining with the above, this yields   pf δxηll 6 pf (δxη ) + 5ε ′ = pf (δxη ) + ε,  ηl  again contradicting  ηi  the assumption that pf δxl > pf (µ) + ε holds for allul ∈ L. We conclude that limi∈I pf δxi = pf (µ), hence the continuity of the functional pf on ∆S .  A continuous linear operator T : CV (X, E) → CV (X, E) is monotone (or positive) if f 6 g implies that T ( f ) 6 T (g) for f, g ∈ CV (X, E), or equivalently if T ( f ) > 0 whenever f > 0. For a subset w of CV (X, E) we denote T (w) = {T ( f ) | f ∈ w}. For a net (Ti )i∈I of positive linear operators on CV (X, E) and K-sets f and g we abbreviate Ti (f) ↓ g for convergence with respect to the upper relative topology in the cone of convex subsets of CV (X, E), that is Ti (f) ↓ g if for every v ∈ V and ε > 0 there is i 0 ∈ I such that for every i > i 0 we have Ti (f) ⊂ γ g + εv with some 1 6 γ 6 1 + ε. This notion of convergence will also be used for functions in CV (X, E), that is singleton K-sets, where it simplifies into convergence with respect to the (locally equivalent) given upper topology, that is Ti ( f ) ↓ g for f, g ∈ CV (X, E) if for every v ∈ V there is i 0 ∈ I such that Ti ( f ) ∈ g + v ↓ for all i > i 0 . We shall use convergence with respect to the lower and symmetric topologies only for functions

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in CV (X, E), writing Ti ( f ) ↑ g for f, g ∈ CV (X, E), if for every v ∈ V there is i 0 ∈ I such that Ti ( f ) ∈ g + v ↑ (or g ∈ Ti ( f )+v ↓) for all i > i 0 . Obviously, Ti ( f ) ↓ g if and only if Ti (− f ) ↑ (−g). Convergence with respect to the symmetric topology, that is Ti ( f ) → g, means that both Ti ( f ) ↓ g and Ti ( f ) ↑ g.   Let M be a collection of K-sets and let MC denote the subcone of Conv CV (X, E) generated by M. The upper Korovkin closure K ↓ (M) of M is defined to be the family of all functions f ∈ CV (X, E) such that for every equicontinuous net (Ti )i∈I of positive linear operators on CV (X, E), Ti (g) ↓ g

for all g ∈ M implies that Ti ( f ) ↓ f.

Similarly, the lower and symmetric Korovkin closures K ↑ (M) and K(M) are defined for subsets M ⊂ CV (X, E) using lower and symmetric convergence in the above definition, respectively. Obviously, MC ⊂ K ↓ (M) ∩ K ↑ (M) ⊂ K(M) whenever M ⊂ CV (X, E). The respective Korovkin closures form subcones of CV (X, E). ∗ supports the Theorem 3.2. Let M be a collection of K-sets and suppose that S ⊂ E + separation property for E. For a function f ∈ CV (X, E) consider the following:      η η (i) µ( f ) 6 η f (x) whenever pg µ 6 pg δx for all g ∈ M, for 0 ̸= δx ∈ ∆S and ∗ µ∈ CV(X, E)+ . η η (ii) η f (x) = supv∈V inf{pg (δx ) | g ∈ MC , f ∈ g + v ↓} for all 0 ̸= δx ∈ ∆S . (iii) f ∈ K ↓ (M).

Conditions (i) and (ii) are equivalent and imply (iii). Proof. The equivalence of (i) and (ii) is a consequence of a corollary to the Range Theorem 5.1 in [4]. Indeed, the cone P of all lower semicontinuous R-valued sublinear functionals on CV (X, E)∗ , restricted to the domain CV (X, E)∗+ and endowed with the pointwise order on CV (X, E)∗+ , becomes a locally convex cone in the sense of [4,5], if we define neighborhoods corresponding to v ∈ V by p1 6 p2 + v

if p1 (µ) 6 p2 (µ) + 1 for all µ ∈ v↓◦

for p1 , p2 ∈ P. The space CV (X, E) is canonically embedded into P, as are the K-sets g ∈ M η which correspond to the functionals pg ∈ P. Corollary 5.3 in [4] if applied to P (with δx in place of µ and MC in place of C) states that for the given function f ∈ CV (X, E) ⊂ P the following two conditions are equivalent: (i′ ) for every Φ ∈ P∗ (the dual cone of P) Φ( f ) 6 δxη ( f )

holds whenever Φ(pg ) 6 δxη (pg ) for all g ∈ M,

and (ii′ ) δxη ( f ) = sup inf{δxη (pg ) | g ∈ MC , f 6 pg + v}. v∈V

η We note that δx operates as a point evaluation on P, η in particular δx (h) = η h(x) for all h ∈ CV (X, E).

η

η

that is δx (p) = p(δx ) for all p ∈ P and Moreover, f 6 pg + v in P means that

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µ( f ) 6 pg (µ) for all µ ∈ v↓◦ , that is by the Hahn–Banach theorem that f is contained in the η topological closure of g + v ↓. Condition (ii′ ) for all δx ∈ ∆S is therefore equivalent to (ii) in η our theorem. Condition (i′ ) for all δx ∈ ∆S , on the other hand appears to be stronger than (i), since CV (X, E)∗+ is only a subset of the dual cone P∗ , consisting only of the point evaluations η for the functions in P. Thus (i′ ) implies (i). But if (i) holds, if Φ ∈ P∗ and if Φ(pg ) 6 δx (pg ) for all g ∈ M (that is of course then for all g ∈ MC ), then we may consider the restriction µ of Φ to the subspace CV (X, E) of P. Then µ ∈ CV (X, E)∗+ , that is u is a point in the domain of the functionals in P, we have h(µ) = µ(h) = Φ(h) for all h ∈ CV (X, E) ⊂ P, and pg (µ) = sup{µ(g) | g ∈ g} = sup{Φ(g) | g ∈ g} 6 Φ(pg ), η

since g 6 pg for every g ∈ g and Φ is monotone on P. Thus pg (µ) 6 δx (pg ) for all g ∈ M and η therefore Φ( f ) = µ( f ) 6 δx ( f ) by (i). Hence (i′ ) and (i) are indeed equivalent. (i) ⇒ (iii): If ∆S = {0}, then Fv↓ (x)◦ = {0}, that is Fv↓ (x) = E, for all x ∈ X and v ∈ V, and therefore  v ↓= v ↑= v = CV (X, E). Hence CV (X, E) carries the trivial topology, every net Ti ( f ) i∈I is convergent in CV (X, E) in any of our topologies. We may therefore assume that ∆S contains a non-zero functional. Suppose that (i) holds for the function f and there is an equicontinuous net (Ti )i∈I of positive linear operators on CV (X, E) such that Ti (g) ↓ g for all g ∈ M, but Ti ( f ) ↓ f fails. Then there is a neighborhood v ∈ V and a subnet (T j ) j∈J of (Ti )i∈I such that T j ( f ) ̸∈ f + v ↓, that is T j ( f ) − f ̸∈ v ↓, for all j ∈ J . Hence  by the Hahn–Banach theorem we find linear functionals µ j ∈ v↓◦ such that µ j T j ( f ) − f > 1. We may assume ζ

that the functionals µ j are indeed point evaluations δx jj ∈ CV (X, E)∗+ , since according to   Corollary 3.2 in [7] all extreme points of v↓◦ are of this type. That is ζ j T j ( f )(x j )− f (x j ) > 1,  ◦ where ζ j ∈ Fv↓ (x j ) . Now, since S supports the separation property for E, according to (S2),   η v↓ there are functionals η j ∈ S ∩ v↓◦ , that is δx jj ∈ ∆S , such that η j T j ( f )(x j ) − f (x j ) > 1/β, where β > 1 denotes the constant from (S2). Hence  1 η  η δx jj T j ( f ) > δx jj ( f ) + . β η

According to Lemma 2.7 we may also assume that (after choosing a further subnet) δx jj → η

v↓

δx ∈ ∆S . (Since v ∈ V, the function Fv↓ is r -lower continuous and therefore satisfies the assumptions of the lemma.) Let T j∗ : CV (X, E)∗ → CV (X, E)∗ denote the adjoint of the operator T j . Rewriting the above we have  η  1 η T j∗ δx jj ( f ) > δx jj ( f ) + . β The equicontinuity of the net (T j ) j∈J implies the existence of a neighborhood u ∈ V such that η T j∗ (v↓◦ ) ⊂ u↓◦ for all j ∈ J . We may therefore assume in addition that T j∗ (δx jj ) → µ ∈ u↓◦ . Thus   1 µ( f ) > η f (x) + . β On the other hand, for every g ∈ M and ε > 0 there is j0 ∈ J such that T j (g) ⊂ γ j g + εv ↓ with some 1 6 γ j 6 1 + ε for all j > j0 . Hence    η  η η η η pg T j∗ (δx jj ) = pT j (g) (δx jj ) 6 γ j pg (δx jj ) + ε 6 pg (δx jj ) + ε |pg (δx jj )| + 1 .

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Now using the lower semicontinuity of the functional pg on CV (X, E)∗ and its continuity on ∆vS , taking the limits on both sides yields      η  pg (µ) 6 lim pg T j∗ (δx jj ) 6 pg δxη + ε |pg (δxη )| + 1 . j∈J

 η η The latter holds for all ε > 0, and therefore pg (µ) 6 pg δx . If δx ̸= 0, this together with the η ζ above inequality for f contradicts (i). If δx = 0, we choose some 0 ̸= δ y ∈ ∆S and observe that pg (µ + δ ζy ) 6 pg (µ) + pg (δ ζy ) 6 pg (0) + pg (δ ζy ) = pg (δ ζy ) holds for all g ∈ M, but      1   1 µ + δ ζy ( f ) > η f (x)) + ζ f (x) + = ζ f (x) + , β β thus contradicting (i) in this case as well.



If M consists of singleton K-sets, that is M ⊂ CV (X, E), then we can obtain similar results for the lower and symmetric Korovkin closures as well. Since Ti ( f ) ↑ f for f ∈ CV (X, E) and a net (Ti )i∈I of operators if and only if Ti (− f ) ↓ (− f ) we have f ∈ K ↑ (M) if and only − f ∈ K ↓ (−M) in this case. Thus Theorem 3.2 yields immediately: ∗ supports the separation property Theorem 3.3. Let M ⊂ CV (X, E) and suppose that S ⊂ E + for E. For a function f ∈ CV (X, E) consider the following:     η (i) µ( f ) > η f (x) whenever µ(g) > η f (x) for all g ∈ M, for 0 ̸= δx ∈ ∆S and µ∈ CV(X, E)∗+ .   η (ii) η f (x) = infv∈V sup{η f (x) | g ∈ MC , g ∈ f + v ↓} for all 0 ̸= δx ∈ ∆S . (iii) f ∈ K ↓ (M).

Conditions (i) and (ii) are equivalent and imply (iii). We obtain a similar result for the symmetric Korovkin closure K(M) of a subset M ⊂ CV (X, E) by inserting its linear hull M L for MC , the cone generated by M, in Theorems 3.2 and 3.3. ∗ supports the separation property Theorem 3.4. Let M ⊂ CV (X, E) and suppose that S ⊂ E + for E. For a function f ∈ CV (X, E) consider the following:     η (i) µ( f ) = η f (x) whenever µ(g) = η f (x) for all g ∈ M, for 0 ̸= δx ∈ ∆S and µ∈ CV(X, E)∗+ .     (ii) η f (x) = supv∈V inf{η f (x) | g ∈ M L , f ∈ g + v ↓} = infv∈V sup{η f (x) | g ∈ η M L , g ∈ f + v ↓} for all 0 ̸= δx ∈ ∆S . (iii) f ∈ K(M).

Conditions (i) and (ii) are equivalent and imply (iii). In special situations we are able to establish the equivalence of all three conditions in Theorem 3.2 (and therefore in Theorems 3.3 and 3.4 with the suitable insertions for M). We consider the case that E is a space CW (Y ) of real-valued functions as in Proposition 2.9. Proposition 3.5. Suppose that the function space CW (Y ) satisfies Condition (R) and S consists of all positive multiples of point evaluations δ y ∈ CW (Y )∗+ . If all K-sets  g ∈ M are  C(X )-convex, then all three conditions in Theorems 3.2–3.4 are equivalent for CV X, CW (Y ) .

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Proof. We shall prove only the case of Theorem 3.2. All left to show is that Condition (iii) implies (i) in case that E = CW (Y ) with the given insertion for S. Condition (S2) holds with β = 1. We shall first consider the case that E = R, that is Y is a singleton set. Then S = {0 6 η ∈ R} and ∆S consists of non-negative multiples of point evaluations δx for x ∈ X. Let us assume that (i) fails for the function f ∈ CV (X ), the point evaluation 0 ̸= ηδx ∈ ∆S and the functional µ ∈ CV (X )∗+ , that is   pg µ 6 pg (ηδx ) holds for all g ∈ M, but µ( f ) > ηf(x). We shall show that (iii) fails as well. Let U0 be a fixed compact neighborhood for x ∈ X and let U be a basis of open neighborhoods of x that are all subsets of U0 . By Urysohn’s Lemma, for all U ∈ U there are functions φU ∈ C(X ) such that 0 6 φU 6 1,

φU (x) = 1,

and

φU (y) = 0 for all y ∈ X \U.

Using these functions we define operators TU on CV (X ) by   1 TU (h) = 1 − φU h + µ(h) φU η for all h ∈ CV (X, R). Clearly TU (h) ∈ CV (X ), and the operators TU are linear and positive. Equicontinuity is easily checked: given v ∈ V, there is ρ > 0 such that [−ρ, +ρ] ⊂ Vv (y) for all y ∈ U0 . We choose u ∈ V such that both u ⊂ (1/2)v and µ ∈ (ηρ/2)u ◦ . Then for every U ∈ U and h ∈ u we have  1 ρ 1 1 µ(h) φU y) 6 for all y ∈ U0 , hence µ(h)φU ∈ v. η 2 η 2 Since h(1 − φU ) ∈ u ⊂ v/2, this yields TU (h) ∈ v, our claim. Ordered by reverse set inclusion, the neighborhood system U serves as the index set of the equicontinuous net (TU )U ∈U of positive linear operators on CV (X ). We proceed to verify that Ti (g) ↓ g for all g ∈ M: let g ∈ M and v ∈ V. There is ρ > 0 such that [−ρ, +ρ] ⊂ Fv . There is g0 ∈ g such that g0 (x) > pg (δx ) − ρ/2. Choose U1 ∈ U such that g0 (y) > g0 (x) − ρ/2 > pg (δx ) − ρ for all y ∈ U1 . Then for any g ∈ g and U ∈ U such that U ⊂ U1 we set gU = (1 − φU )g + φU g0 ∈ g (recall that g is supposed to be C(X )-convex) and calculate     1 1 TU (g) − gU = µ(g) − g0 φU 6 µ(g) − pg (δx ) + ρ φU 6 ρφU . η η The latter follows from µ(g) 6 pg (µ) 6 pg (ηδx ) = ηpg (δx ). Since ρφU ∈ v by the above, this yields TU (g) ∈ g + v ↓ for all g ∈ g, hence TU (g) ⊂ g + v ↓ for all U ⊂ U1 . This yields Ti (g) ↓ g for all g ∈ M as claimed. However Ti ( f ) ↓ f is false, since   TU f (x) − f (x) = (1/η)µ( f ) − f (x) > 0 for all U ∈ U. Indeed, 0 ̸= ηδx ∈ ∆S contradicts the assumption that Vv (x) = R, since 0 ̸= η ∈ Vv (x)◦ . Thus Vv (x) ⊂ [−σ, +σ ] for some σ > 0. If we choose ε > 0 such that εσ < (1/η)µ( f ) − f (x), then TU ( f ) ̸∈ f + εv ↓ for all U ∈ C. Therefore (iii) fails for the function f. In the general case, that is if E is a function space CW (Y ) satisfying (R), we use Proposition 2.9 to represent CV X, CW (Y ) as a space CV functions. Then  (X × Y ) of real-valued   the first part of our argument applies since the elements of ∆S for CV X, CW (Y ) correspond to the multiples of point evaluations on X × Y. 

W. Roth / Journal of Approximation Theory 176 (2013) 23–41

37

A family M of K-sets in CV (X, E) is called an upper, lower or symmetric Korovkin system if its upper, lower or symmetric Korovkin closure is all of CV (X, E). Since CV (X, E) contains the negatives of its elements, we have indeed symmetric convergence Ti ( f ) → f for all f ∈ CV (X, E) and the nets of operators concerned in all of these cases. An application of Proposition 3.5 yields a Stone–Weierstrass type theorem for spaces of vector-valued functions provided that the range space E is a function space CW (Y ) satisfying (R). CW (Y ) is a topological vector lattice and the condition in Proposition 2.4 is satisfied. Indeed, every neighborhood w ∈ W is defined by an r -lower continuous function y → ρ(y)[−1, +1] on Y . Therefore |g| ∈ w whenever g ∈ w for every real-valued function g on Y . For every v ∈ V, in turn, the defining function V Therefore the above implies that | f | ∈ v  v is W-valued.  whenever f ∈ w for every f ∈ CV X, CW (Y ) , and the condition in 2.4 holds  with u =  v. Hence CV X, CW (Y ) is also a topological vector lattice. For a subcone N of CV X,CW (Y ) we denote by N∧ (or N∨ ) the smallest infimum- (or supremum-)stable subcone of CV X, CW (Y ) containing N. The vector sublattice generated by N is denoted by N∧∨ . Corollary 3.6. Supposethat the function space CW (Y ) satisfies Condition (R).  If M is an upper   = {g | g ∈ MC }, Korovkin system for CV X, CW (Y ) , its elements are C(X )-convex and M    ∧ is dense in CV X, CW (Y ) . then M −   Proof. Since CV X, CW (Y ) can be represented as a function space of real-valued functions (see Proposition 2.9), it suffices to consider the case   that CW (Y ) = R. Let M be an upper Korovkin system for CV (X ) and let f ∈ CV X, CW (Y ) − . According to Proposition 3.5 Condition (ii) of Theorem 3.2 holds for f with S = {0 6 η ∈ R} and the positive multiples of point evaluations δx ∈ CV (X ) for ∆S . That is f (x) = sup inf{pg (δx ) | g ∈ MC , f ∈ g + v ↓}

for all x ∈ X such that δx ∈ ∆S .

v∈V

Recall that δx ∈ ∆S for x ∈ X if and only if Fv (x) is bounded in R for some v ∈ V. We shall verify that every f ∈ CV (X )− can be approximated in the topology of CV (X ) by functions  ∧ : let v ∈ V be a function space neighborhood, that is Fv (x) = ρ(x)[−1, +1] for some in M strictly positive R-valued lower semicontinuous function ρ on X. The set Z = {x ∈ X | f (x) 6 −ρ(x)} is therefore closed in X and indeed compact following the requirement that f vanishes at infinity. The latter follows since there is a set A ∈ R such that f (x) ∈ (1/2)ρ(x)[−1, +1] for all x ∈ X \A, hence Z ⊂ A. Let σ > 0 be a lower bound for the function ρ on Z . Since ρ(x) < +∞ for every x ∈ Z , the functional δx is in ∆S . Thus there is gx ∈ MC such that f ∈ gx + v ↓

and pgx (δx ) < f (x) + σ.

 such that f 6 gx + v ↓, that is f 6 gx + ρ, and Thus we find a function gx ∈ gx ⊂ M gx (x) 6 pgx (δx ) < f (x) + σ. The latter inequality holds indeed on an open neighborhood Ux of x, and by the compactness of Z , finitely many of those neighborhoods, say Ux1 , . . . , Uxn , cover all of Z . Now we choose the function g = gx1 ∧ · · · ∧ gxn ∧ 0  ∧ and observe that in ∈ M f 6 g + ρ,

that is f ∈ g + v ↓

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since f 6 0 and g(x) < f (x) + σ 6 f (x) + ρ(x)

for all x ∈ Z .

But for x ∈ X \Z we conclude that g(x) 6 0 6 f (x) + ρ(x) holds as well, since f (x) > −ρ(x) on this set. This yields g ∈ f + v ↓, hence together with the above g ∈ f + v as claimed.  Again, this result can be transferred to lower and symmetric Korovkin systems in an obvious way. A subset M of CV (X, E) is an upper Korovkin system for CV (X, E) if and only if −M is a lower Korovkin system, and M is a symmetric Korovkin system if and only if its linear hull M L is an upper (and therefore also lower) Korovkin system.  Corollary CW (Y) satisfies Condition (R). If M ⊂ CV X,  3.7. Suppose that the function space  CW (Y ) is a lower Korovkin system for CV X, CW (Y ) , then M∨ is dense in CV X, CW (Y ) + .  Corollary (R). If M ⊂ CV  X,  3.8. Suppose that the function space CW (Y ) satisfies  Condition CW (Y ) is a symmetric Korovkin system for CV X, CW (Y ) , then M∧∨ is dense in CV X, CW (Y ) . The following criterion provides a useful tool to identify Korovkin systems in CV (X, E) in a special case. We shall say that an element 0 ̸= e ∈ E + is strictly positive if η(e) > 0 for all ∗ . Such elements exist in many cases. If for example E has an order unit, this is 0 ̸= η ∈ E + of course a strictly positive element. If E is indeed a topological vector lattice and e ∈ E + is a (topological) weak order unit, that is the sequence (a ∧ ne)nıN converges to a for every a ∈ E + , then e is strictly positive. In the spaces l p for 1 6 p < +∞ all sequences with strictly positive terms are strictly positive elements, in l ∞ those sequences where the infimum of all terms is strictly positive. ∗ supports Corollary 3.9. Suppose that e ∈ E is a strictly positive element and that S ⊂ E + the separation property for E. Let M be a family of K-sets in CV (X, E) such that for every η 0 ̸= δx ∈ ∆S η

η

(i) there are g1 , g2 ∈ M such that pg1 (δx ) > 0 and pg2 (δx ) < 0, (ii) for every y ̸= x ∈ X there is a neighborhood U of y, a K-set g ∈ MC and 0 6 g ∈ g such that pg (δxη ) = 0

and

g(z) > e for all z ∈ U,

(iii) for every ζ ∈ S such that η and ζ are linearly independent there is g ∈ MC such that   pg (δxη ) = 0 and ζ g(x) > 0 for some g ∈ g such that g(x) > 0. Then M is an upper Korovkin system for CV (X, E). η

Proof. Let e ∈ E + and S be as stated. Then ∆S consists of all point evaluations δx ∈ CV (X, E)∗+ such that η ∈ S. We shall verify Condition (i) of Theorem 3.2 for all f ∈ CV (X, E) η under the assumptions of this corollary. 0 ̸= δx ∈ ∆S , that is δx◦ ∈ v↓◦ for some v ∈ V, and   Let η ∗ µ ∈ CV (X, E)+ such that pg µ 6 pg δx for all g ∈ M. According to Theorem 2.5 there exists ∗ -valued measure θ representing µ. Let y ̸= x ∈ X and choose the K-set g ∈ M, a regular E + g ∈ g and an open neighborhood U of y as in (ii). Since g > 0 and θ is positive, we can conclude that     µ(g) = g dθ > g dθ > θ U (e) > 0. X

U

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On the other hand we have µ(g) 6 pg (µ) 6 pg (δxη ) 6 0.   ∗ and e is a strictly positive This yields θ U (e) = 0 and indeed θ (U ) = 0 since θ (U ) ∈ E + element of E. Now a regularity argument yields that θ (A) = 0 for every set A ∈ R that does not ξ contain x. Thus θ is indeed a point evaluation δx ∈ CV (X, E)∗+ at x. We use a similar argument to verify that ξ is a positive multiple of η. There is a neighborhood V ∈ V such that V ↓◦ contains both functionals η and ξ. According to (S2) V ↓◦ is contained in the closed, hence compact convex hull C of β(S ∩ V ↓◦ ). Since β(S ∩ V ↓◦ ) is itself compact, it contains the extreme points of its closed convex hull C. This is a consequence of the classical Krein–Milman theorem which also guarantees the existence of a regular positive real-valued Borel measure ω on C, supported by S ∩ C that represents ξ , that is ξ(a) = C a dω =  (S∩C) a dω holds for all a ∈ E, where a is considered to be a function on C. Now a similar ξ

η

argument as before using (iii) yields that ξ is a multiple of η, that is δx = λδx for some λ > 0. η η Finally, using the K-sets g1 and g2 in M from (i) we observe that both λpg1 (δx ) 6  pg1 (δ  x ) and η η η λpg2 (δx ) 6 pg2 (δx ) implies that λ = 1. This yields µ = δx , hence µ( f ) = η f (x) for all f ∈ CV (X, E), our claim.  Again, if M ⊂ CV (X, E) this criterion can be readily transferred to the lower and symmetric cases using −M and M L in place of M, respectively. Examples 3.10. (a) Let X be a locally compact Hausdorff space. Suppose that e ∈ E + is strictly ∗ supports the separation property for E. Let A be a subset of E such positive and that S ⊂ E + that for any two linearly independent functionals η, ζ ∈ S there is a ∈ A such that η(a) = 0 and ζ (a) > 0. (A = E, of course satisfies this requirement.) Let ϕ, ψ ∈ C(X ) such that ϕ is strictly positive and ψ is one-to-one. If CV (X, E) contains the set     M = (ϕψ) ⊗ (−e), (ϕψ 2 ) ⊗ e ∪ ϕ ⊗ a | a ∈ A ∪ {+e, −e} , then M fulfils the criteria of Corollary 3.9: clearly, the functions ϕ ⊗ e, ϕ ⊗ (−e) ∈ M satisfy 3.9(i), and for a fixed x ∈ X the function g ∈ MC , that is  2 g(y) = ϕ(y) ψ(y) − ψ(x) e for all y ∈ X, is positive, vanishes at x, and for any y ̸= x there is a suitable λ > 0 such that λg(z) > e for all z in some neighborhood U of y. Hence (ii) is also satisfied. Using the specified element a ∈ A, the function ϕ ⊗ a then satisfies (iii). Thus according to Corollary 3.9, M is an upper Korovkin system for CV (X, E). (b) For a concrete example let X = [0, +∞) and E = l ∞ with its usual (termwise) order and topology and the order unit e = (1, 1, . . .). For S we choose all positive multiples of the functionals ηk in the dual of l ∞ such that ηk (α1 , α2 , . . .) = αk . Clearly S supports the separation property for l ∞ with β = 1. For A we choose the set of all unit sequences ei ∈ l ∞ , where ei = (0, . . . , 0, 1, 0, . . .). For α > 0 let vα be the function space neighborhood defined by the set-valued function Vvα (x) = eαx B for x ∈ X, where B denotes the unit ball in l ∞ , and let V be the function space neighborhood system generated by these neighborhoods. Then the subset M = { f k | f k (x) = x k e for k = 1, 2} ∪ { f a | f a (x) = a, for a = +e, −e, ei }

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is an upper Korovkin system for CV (X, E): Indeed, in 3.10 insert the functions ϕ(x) = 1 and ψ(x) = x. The functions in M are of course contained in CV (X, E), since they all vanish at infinity with respect to the neighborhoods vα . Let us illustrate this example with an approximation process modeled by a modified version of the classical Bernstein operators. For a function f ∈ CV (X, E) define  2     n  x n 2 −k k  n 2  x k    , for x < n 1− f Tn f (x) = n n n k  k=0 f (n), for x > n. With some straightforward computations one may check the following:   Tn f a (x) = a for all x ∈ [0, +∞) and a ∈ E,   Tn f 1 (x) = x e for all x < n,  2    n −1 2 1 Tn f 2 (x) = x + x e for all x < n. n n2 This shows in particular that Tn ( f ) → f for all f ∈ M in the topology of CV (X, E). Furthermore, one may check that the sequence Tn n∈N is equicontinuous, as for any and vα ∈ V   and f ∈ vα , that is f (x) ∈ eαx B for all x ∈ X , we immediately realize that Tn f (x) ∈ eαx B for all x ∈ X , hence Tn ( f ) ∈ va holds as well for all n ∈ N. Thus Theorem 3.2 applies, and we may conclude that Tn ( f ) → f for all f ∈ CV (X, E). (c) Part (b) can be slightly modified to cover the general case when E contains a strictly ∗ for S and E for A. This yields an upper Korovkin positive element e. Then we choose E + system M = { f k | f k (x) = x k e for k = 1, 2} ∪ { f a | f a (x) = a, for all a ∈ E}. The definition of the Bernstein operators remains unchanged. (d) Some effort has been made to allow the inclusion of K-sets in the test systems M. For an example using non-singleton K-sets let X = [0, 1] and let E = Rn with the Euclidean unit ball B and with the equality as its order. Thus the upper, lower and symmetric topologies coincide. We denote the unit vectors in Rn by e1 , . . . , en and the units in the dual by η1 , . . . , ηn , that is ηi (ek ) = 0 if i ̸= k and ηi (ei ) = 1. For S we choose all √ multiples (positive and negative) of  the functionals ηi . Requirement (S2) then holds with β = n. For CV [0, 1], Rn we consider the topology of uniform convergence, that is V consists of the strictly positive multiples of the neighborhood v defined by the set-valued function x → B for all x ∈ [0, 1]. Now for a fixed x ∈ [0, 1] let ρx : [0, 1] → R be the function ρx (y) = (y − x)2 for y ∈ [0, 1] and let gx be the K-set defined by the set-valued function ρx ⊗ B, that is     gx = g ∈ CV [0, 1], Rn | ∥g(y)∥ 6 (y − x)2 for all y ∈ [0, 1] . We claim that the family of K-sets M = {gx | x ∈ [0, 1]} ∪ {ga | ga (x) = a, for a = +ei , −ei }      η  is a Korovkin system for CV [0, 1], Rn . Indeed, suppose that pg µ 6 pg δx k for all g ∈ M, ∗  holds for x ∈ [0, 1], ηk ∈ S and µ ∈ CV [0, 1], Rn . Since µ is an Rn -valued regular Borel measure on [0, 1], a similar argument as above using the K-set gx yields that µ is a point ξ evaluation δx at x. Now using the constant functions ga ∈ M we shall argue that ξ = ηk . Indeed,

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we have ξ = α1 η1 + · · · + αn ηn . For a = +ei , −ei this leads to αi = µ(ga ) = δx k (ga ) = ηk (a), η that is αi = 0 for i ̸= k and αk = 1. Thus ξ = η and µ = δx k , as claimed. Since equality servers as the order in this  example,  the conclusions of Theorem 3.2 apply to equicontinuous nets of operators on CV [0, 1], Rn without further qualifications. References [1] K. Keimel, W. Roth, Ordered Cones and Approximation, in: Lecture Notes in Mathematics, vol. 1517, Springer Verlag, Heidelberg–Berlin–New York, 1992. [2] L. Nachbin, Elements of Approximation Theory, D. Van Nostrand Co., Princeton, 1967. [3] J.B. Prolla, Approximation of Vector-Valued Functions, in: Mathematical Studies, vol. 25, North Holland, 1977. [4] W. Roth, Hahn–Banach type theorems for locally convex cones, Journal of the Australian Mathematical Society (Series A) 68 (1) (2000) 104–125. [5] W. Roth, Operator-Valued Measures and Integrals for Cone-Valued Functions, in: Lecture Notes in Mathematics, vol. 1964, Springer Verlag, Heidelberg–Berlin–New York, 2009. [6] W. Roth, Spaces of vector-valued functions and their duals, Mathematische Nachrichten 285 (8–9) (2012) 1063–1081. [7] W. Roth, Choquet theory for vector-valued functions on a locally compact space (in press). [8] H.H. Sch¨afer, Topological Vector Spaces, Springer Verlag, Heidelberg–Berlin–New York, 1980.

Further reading [1] F. Altomare, M. Campiti, Korovkin Type Approximation Theory and Its Applications, in: Gruyter Studies in Mathematics, vol. 17, Walter de Gruyter, Berlin–New York, 1994. [2] H. Bauer, K. Donner, Korovkin approximation in C0 (X ), Mathematische Annalen 236 (1978) 225–237. [3] M. Campiti, Korovkin theorems for vector-valued continuous functions, Approximation Theory, Spline Functions and Applications 356 (1992) 293–302. [4] T. Nishishiraho, Bernstein-type approximation processes for vector-valued functions, Acta Mathematica Hungarica 84 (1–2) (1999) 149–158.