Contents

1. Introduction 2 1.1. Abstract 2 1.2. Basic definitions, notations and results 4 2. The Bishop Theorem 7 2.1. Antisymmetric sets 7 2.2. Orthogonal measures 8 2.3. Machado’s proof 10 2.4. Ransford’s proof 14 3. The strict topology 17 3.1. Bounded functions on locally compact spaces 17 3.2. The strict topology 17 3.3. The Stone-Weierstrass Theorem for C_{b}(X) 19 3.4. The Riesz Representation Theorem for (C_{b}(X; E), β) 21 3.5. A Stone-Weierstrass Theorem for C_{b}(X; E) 26 4. The Korovkin closure 28 4.1. Motivations of the Korovkin-type theorems 28 4.2. The Korovkin shadow 29 4.3. Separating a compact set from a single point 32 4.4. Approximation at one point 33 4.5. The Korovkin closure in some special case 34 References 36 GENERALIZATIONS OF THE STONE-WEIERSTASS

THEOREM

BALÁZS SZENTES

1. Introduction

1.1. Abstract. (In this part we are not going to give pecise definitions and notations.) The main goal of this work is to show several ways, how the standard Stone-Weierstrass theorem can be generalized. The original theorem is known as follows

Theorem 1.1 (Stone-Weierstrass). Let f be a continous function on the compact interval [a, b] ⊂ ℝ. Then for all ϵ > 0 there exists a polynomial p such that

sup_{x∈[a,b]} |f(x) − p(x)| < ϵ .

Although this theorem has become very important especially in the field of Applied Approximation Theory, we don’t deal with its proof at the beginning of this paper but later we provide a constructive proof due to H.Bauer (see [2]). There is a much more abstract version of this theorem that we consider as a basic theorem which we would like to generalize.

Theorem 1.2. Let X be a compact topological space, and A be a subalgebra of C_{ℝ}(X), containing the neutral element. If A even separates X (i.e. for each pair x, y in X there exists an a ∈ A with a(x) ≠ a(y) ), then the closure of A (with respect to the supremum norm) is C_{ℝ}(X).

If we consider C_{ℂ}(X) instead of C_{ℝ}(X) the statemant of the theorem remain valid with only one more assumption; A has to be closed under the complex conjugation (i.e. if a ∈ A then a ∈ A too).

We try to weaken every condition and assumption of this theorem. In the first part of this paper we assume about A only, that it is

closed and contains the constant function, and we try to give simple condition, due to E.Bishop (see [14]), how one can check whether a function belongs to A or not.

Theorem 1.3 (Bishop). If f ∈ C(X) and f|_{S} ∈ A|_{S} for all antisymmetric set S, then f ∈ A.

We try to investigate not only functions, mapping from X into a field, but also ones into a Banach-space (or sometimes a complete, locally convex space). In this more general case S.Machado could develope Bishop’s idea, and he could say something about the distance between the subalgebra and any function.

Theorem 1.4 (Machado). Let G be a vector subspace of C(X; E) which is a module over a unital subalgebra of C_{ℂ}(X). Then for every f ∈ C(X; E) there exists an antisymmetric set S_{f} such that

inf{∥f − g∥_{E} : g ∈ G} = inf{∥f|_{S_{f}} − g|_{S_{f}} ∥_{E} : g ∈ G}

If we deal with locally compact spaces, A can be a subalgebra vanishing at infinity. In this case the assumption about the neutral element has to be replaced by an assumption of the existence of a function vanishing nowhere, or for all x ∈ X we need a function f ∈ A with f(x) ≠ 0. This theorem was proved for instance by La Brange (see [13]). In the last section of the first part we provide a short, elementary proof for Machado’s theorem by Ransford (see [17]).

In the second part we deal with locally compact spaces, but we try to approximate all bounded functions. For this purpose we introduce a new topology (called strict topology) and we establish the Riesz Representation Theorem for this case. This new topology was introduced by Buck (see [5]) and was investigated in [8] [10] [11]. The main theorem of this part is the following

Theorem 1.5. Let X be a locally compact space, and E a locally convex, complete topological space. Let A be a closed subspace of C_{b}(X; E) with respect to the strict topology that satisfies the following conditions (i) A(x) = {f(x) : f ∈ A} = E for every x ∈ X and (ii) If ϕ ∈ C(X) has values in [0, 1] and f ∈ A, then ϕf ∈ A.

Then A = C_{b}(X; E).

However this theorem is strong and generalized, we stress that perhaps the tecnique is more important what we established at its proof. The generalized Riesz Representation Theorem, where X is locally compact, Hausdorff is the following

Theorem 1.6. If L is a strictly continous linear functional on C_{b}(X; E), where E is a locally convex, complete space, then there exists a μ in M(X; E^{⋆}) such that

L(f) = ∫_{X} f dμ

for each f ∈ C_{b}(X; E). Conversely for every μ ∈ M(X; E^{⋆} the formula ∫_{X} f dμ defines a strictly continous linear functional on C_{b}(X; E).

4 BALÁZS SZENTES

In the last part we deal with Korovkin-type theorems, that can be considered as a related application motivated by the Stone-Weierstrass Theorem, in this context. We try to answer the question, in some cases, how it can be proved about a sequence corresponding to a function, that it converges to the function. For instance, we show a proof about the convergence of the Bernstein polynomials. At the end of this section we will use the tools developed in the Korovkin-type theorems to prove the Stone-Weierstrass Theorem for locally compact spaces and functions vanishing at infinity.

1.2. Basic definitions, notations and results. As we mentioned, we deal with usually approximations of continous functions defined on a Hausdorff topological space X, and its values taken from another topological space Y. Then we denote the set of all such continous function by C(X; Y). If Y = ℝ or ℂ we use the simple notations C_{ℝ}(X) and C_{ℂ}(X). (When it doesn’t make any confusion we ignore to sign the the field in the lower index.) Denote C_{0}(X) the subset of C(X) consisting of all function f which vanishs at infinity i.e. for every ϵ > 0 there exists a compact set K ⊂ X such that |f| < ϵ outside of K. Define C_{00}(X) ⊂ C_{0}(X) as the set of all continous function which is zero except on a compact set. C_{b}(X) denotes the set of all bounded function in C(X).

Moreover we need to recall some topological definitions and results.

Definition 1.1. A topological space (X, τ) is said to be σ-compact if X = ∪_{n=1}^{∞}K_{n}, where K_{n} is compact set for all n ∈ ℕ.

Definition 1.2. A topological space (X, τ) is regularly σ-compact if it is σ-compact and K_{n} ⊂ intK_{n+1}.

Definition 1.3. A K ⊂ X is said to be relatively compact if its closure is compact.

Definition 1.4. A topological space (X, τ) is said to be normal if every two disjoint closed sets in X can be separated by open sets.

Theorem 1.7 (Urüsszon’s Lemma). If (X, τ) is normal then every two disjoint, closed sets can be separated by functions.

(For proof see [26] pg 92.)

Definition 1.5. A topological space (X, τ) is said to be completely regular if for every x ∈ X and its every neigbourhood U, x and U can be separated by function.

Moreover wee need some staandard theorem from Functional Analysis. (For proof see [24] 3.15, 3.2, 3.21 .)

Theorem 1.8 (Banach-Alaoglu). Let X be a Banach space and denote X^{⋆} the set of 𝕂- valued continous functions defined on X. (Where 𝕂 = ℝ or ℂ.) Then the unit ball of X^{⋆} is weak^{⋆}- compact.

Theorem 1.9 (Hahn-Banach). Let X be a real linear space and p be a homogen, subadditive function (i.e. p(tx) = tp(x) for all x ∈ X and t ∈ ℝ_{+}, and p(x + y) ≤ p(x) + p(y)). Let Y be a linear subspace of X, and let f_{0} be a linear functional defined on Y with f_{0}(y) ≤ p(y) (for all y ∈ Y).

Then there is a linear extention of f_{0} (denoted by f) into X such that f(x) ≤ p(x) for all x ∈ X.

The Hahn-Banach Theorem can be generalized for complex, linear spaces. In this case we need only a simple modification, that is, a seminorm p majorizes the absolute value of f_{0} and later the absolute value of f.

Theorem 1.10 (Krein-Milman). Let (X, τ) be a locally convex topological space, on which X^{⋆} separates points, and K be a conpact, convex subset of X. Then the closure of the convex hull of all extreme points of K is K itself.

Theorem 1.11 (Riesz Representation Theorem). Let X be a locally compact, Hausdorff space. Then for every positive linear functional Λ on C_{00}(X), there exists a Borel measure μ, such that, for every f ∈ C_{00}(X)

Λ(f) = ∫_{X} f dμ .

However the proof of this theorem is well-known (see [27] ) here we provide a short one due to J. Kindler (see [23] ).

Let X be a nonvoid set and F a linear function space defined on X such that f, g ∈ F implies that f ∧ g and f ∧ 1 belong to F as well. Let B_{F} denote the σ-ring generated by the sets {f > 1} = {x ∈ X : f(x) > 1} (f ∈ F). Let Λ : F → ℝ be a positive, linear functional such that f ↘ 0 (i.e. f_{i} ≥ f_{i+1} for all i ∈ ℕ and lim_{n→∞} f_{n} = 0) implies lim_{n→∞} Λ(f_{n}) = 0.

Here we prove the Daniell-Stone Theorem and remark that by Dini’s Theorem the standard Riesz Representation Theorem follows.

Theorem 1.12 (Daniell-Stone). There exists a measure μ on B_{F} such that Λ(f) = ∫_{X} f dμ for every f ∈ F.

We use the notation ∑_{1}^{∞} A_{i} for the union of disjoint sets A_{1}, A_{2}, .... We need some lemmata for the proof.

Lemma 1.13. For all sequences a_{n}, b_{n} ∈ ℝ with b_{n} ≥ a_{n}, n ∈ ℕ and [a_{0}, b_{0}) = ∑_{1}^{∞}[a_{n}, b_{n}) we have b_{0} − a_{0} = ∑_{1}^{∞}(b_{n} − a_{n}).

For proof see [25] 4.d..

Lemma 1.14. Every σ-additive non-negative finite set function ϑ on a semiring S can be extended to a measure ρ on the σ-ring generated by S.

For proof see [25] 4.12..

Lemma 1.15. The system S of intervals [f, g) = {(x, t) ∈ X × ℝ : f(x) ≤ t < g(x)} where g, f ∈ F and g ≥ f is a semiring, and ϑ : S → ℝ_{+} , ϑ([f, g)) = Λ(g−f) is a well defined σ-additive function.

Proof. [f, g) = ∑_{1}^{∞}[f_{n}, g_{n}) implies [f(x), g(x)) = ∑_{1}^{∞}[f_{n}(x), g_{n}(x)) for every x ∈ X. From Lemma 1.12 we obtain that g − f = ∑_{1}^{∞}(g_{n} − f_{n}). Let h_{n} = g − f − ∑_{m=1}^{n}(g_{m} − f_{m}), then h_{n} ↘ 0 hence Λ(h_{n}) → 0. (That is, ϑ([f, g)) = ∑_{1}^{∞} ϑ([f_{n}, g_{n})).)

Proof of Daniell-Stone Theorem. For ϑ as defined in Lemma 1.14 choose ρ according to Lemma 1.13 and let 0 ≤ f ∈ F. For f_{n} = (n[f −f ∧1])∧1 we have

[0, βf_{n}) ↗ {f > 1} × [0, β)

for every β > 0. Hence μ(A) = ρ(A×[0, 1)), (A ∈ B_{F}), is a well defined measure with the property ρ({f > 1} × [0, β)) = βμ({f > 1}), f ∈ F, β > 0. Let B^{n} = ∑_{i=1}^{n2^{n}}(C_{i}^{n} − c_{i+1}^{n}) × [0, i2^{−n}) where c_{i}^{n} = {i^{−1}2^{n}f > 1} for 1 ≤ i ≤ n2^{n} and c_{i}^{n} = ∅ for i = 2^{n}n + 1. Then B^{n} ↗ [0, f) and we obtain

Λ(f) = ρ([0, f)) = lim_{n→∞} ρ(B^{n}) = ∫_{X} f dμ .

2. The Bishop Theorem

2.1. Antisymmetric sets.

Definition 2.1. Let F be a subset of C(X). An S ⊂ X is said to be F-antisymmetric set if the restriction of any function of F is real-valued on S, implies that it is constant on S.

Definition 2.2. Let F be a subset of C(X). An S ⊂ X is said to be F-peak if there is a function f ∈ F such that f(S) = 1 but |f(x)| < 1 for every x ∈ X \ S.

Definition 2.3. Let F be a function family defined on X. F is said to be conjugate-closed if f ∈ F implies f ∈ F.

Lemma 2.1. Let S and T be antisymmetric sets with S ∩T ≠ ∅. Then S ∪ T is also antisymmetric.

Proof. Let x ∈ S ∩ T be an arbitrary point of X and suppose that f(S ∪ T) ⊂ ℝ. Then f(S) = f({x}) = f(T) which means that f is constant on S ∪ T.

Corollary 2.2. Let S be contained in an F-antisymmetric sets A_{i} (i ∈ I). Then ∪_{i∈I}A_{i} is also an F-antisymmetric set.

Obviously every single point of X is an F-antisymmetric set for any function family F. By the previous corollary there exists a maximal antisymmetric set which contains x. If F is assumed to be a continous function family, then the closure of an antisymmetric set is antisymmetric as well. It means that any maximal antisymmetric set is closed. This observation with Lemma 2.1. implies that X has a disjoint partition into closed antisymmetric sets. Let F(X) denote the set of all elements of this partition.

Let us try to describe the F-peak sets. Let F ⊂ C(X)_{ℝ} an algebra containing the neutral element. We claim that the finite intersection of nonempty F-peak sets remains F-peak. Indeed, if f_{i} ∈ F corresponding to S_{i} according to the definition 2.2. then ∑_{i=1}^{n} f_{i} ; n is peak on the intersection. If F is even closed then finite can be replaced by countable many, because in this case ∑_{i=1}^{∞} 2^{−i}f_{i} is peak on the intersection and belongs to F.

Lemma 2.3. Let X be a compact set and H ∈ F(X). Then H is an intersection of F peak sets.

Proof. Let y ∈ H be fixed, then H = {x ∈ X : f(x) = f(y) ∀f ∈ F} by the maximality. Thus, setting H_{f} = {x ∈ X : f(x) = f(y)}, we

8 BALÁZS SZENTES

have H = ∪_{f∈F}H_{f}. For any f the set H_{f} is clearly F-peak, because the function f̂ = 1 − ϵ(f(x) − f(y))^{2} is a peak on H_{f} if ϵ is small enough.

2.2. Orthogonal measures. Let M(X) denote the set of all regular complex-valued Borel measures on X, and let M_{ℝ}(X) ⊂ M(X) containing every measures with real values. If f ∈ C(X) and μ ∈ M(X) we can consider fμ as a linear functional such that fμ(g) = ∫_{X} fg dμ. A^{⊥} will denote the linear subspace of M(X) consisting of all the measures which are orthogonal to A. ballA^{⊥} is the subset of A^{⊥} such that every μ ∈ ballA^{⊥} has a norm less or equal than 1. (The norm of a measure is its total variation.) Denote (ballA^{⊥})^{extr} the set of all extreme points of ballA^{⊥}.

This proof of the Bishop’s Theorem, derived from L. De Branges and further developed by Bishop and Glickbers (see [13], [14], [15]), based on the following

Lemma 2.4. If μ ∈ (ballA^{⊥})^{extr} then suppμ is antisymmetric.

Proof. Suppose that f ∈ A is real-valued, we should prove that f is constant on the set suppμ = K. Without loss of generality 0 < f < 1 can be assumed since 1_{X} ∈ A. Therefore neither fμ nor (1 − f)μ are zero measures, and the following equalities hold

(2.1) ∥fμ∥ + ∥(1 − f)μ∥ = ∫_{X} |f| d|μ| + ∫_{X} |1 − f| d|μ|

= ∫_{X} f d|μ| + ∫_{X} 1 − f d|μ| = ∫_{X} 1_{X} d|μ| = 1

hence μ can be considered as

μ = ∥fμ∥ fμ ; ∥fμ∥ + ∥(1_{X} − f)μ∥ (1 − f)μ ; ∥(1_{X} − fμ)∥ , (2.2)

which is clearly a convex combination with (2.1). Recall that g ∈ A implies that fg ∈ A as well, hence ∫_{X} fg dμ = 0 since μ ∈ A^{⊥}, that shows g orthogonal to fμ. Since g was an arbitrary element of A we obtain fμ ∈ A^{⊥} and in a similar way that (1_{X} − f)μ ∈ A^{⊥}. This fact combining with (2.2) shows that μ is a sum of two measures belonging to ballA^{⊥}. Since μ is an extreme point it must be fμ ; ∥fμ∥, therefore f is constant on suppμ.

Let A be a subalgebra of C_{ℂ}(X) containing the constant function.

Theorem 2.5 (Bishop).

(1) If f ∈ C(X) and f|_{S} ∈ A|_{S} for all S ∈ A(X) then f ∈ A.

(2) A|_{S} is uniformly closed in C(S) for all S ∈ A(X).

Before we begin to prove this theorem we need a lemma for (2).

Lemma 2.6. Let E be an intersection of A-peak sets. Then A|_{E} uniformly closed in C(E).

Proof. We show that A|_{E} is isometrically isomorphic with the quotient algebra A/kE where kE = {f ∈ A : f(E) = 0}. Obviously for each x ∈ E the inequality |f(x)| ≤ ∥f + kE∥ holds, so ∥f|_{E}∥ ≤ ∥f + kE∥. Let V_{ϵ} is a neighbourhood of E such that |f| ≤ ∥f|_{E}∥ + ϵ. Then V_{ϵ} contains finitely many A-peak sets containing E. By the comment after Corollary 2.2., this intersection is an A-peak set as well, we will denote it by P. Therefore there exists a g ∈ A with g(P) = 1 but |g(x)| < 1 for every x ∉ P. Since on E g^{n}f = f clearly g^{n}f ∈ f + kE. We claim that ∥f +kE∥ ≤ ∥f|_{E}∥. Indeed lim sup ∥g^{n}f∥ ≤ sup |f(V_{ϵ})| ≤ ∥f|_{E}∥+ϵ and ϵ converging to zero, shows the desired inequality.

Proof of Theorem 2.5. (1) Let μ ∈ (ballA^{⊥})^{extr} an arbitrary measure. Then by Lemma 2.4. suppμ ⊂ S where S is in A(X). Since f|_{S} ∈ A|_{S}, there exists a g_{f} ∈ A such that μ(f) = ∫_{X} f dμ = ∫_{S} f dμ = ∫_{S} g_{f} dμ = μ(g_{f}) = 0 because μ orthogonal to A. Applying the Krein-Milman Theorem we claim that for each μ ∈ A^{⊥} is orthogonal to f i.e. μ(f) = 0. If f ∉ A the Hahn-Banach Theorem provides a linear functional which has an integral representation by the Riesz Representation Theorem such that μ(f) ≠ 0 but μ ∈ A^{⊥} which is a contradiction.

(2) Considering the fact that all maximal antisymmetric sets are an intersection of A-peak sets the previous lemma can be applied.

The next purpose of this capter to ignore the neutral element of A. Unfortunately in this case Bishop’s Theorem can not be proved but using orthogonal measures the Stone-Weierstrass Theorem might be proved. For it we need a lemma that is quite similar to Lemma 2.4. .

Lemma 2.7. Let μ ∈ (ballA^{⊥}) with μ ≠ 0, and g is a complexvalued Borel-measurable function that is, in addition, μ-bounded . If ∫_{X} fg dμ = 0 for all f ∈ A then g is constant μ-almost everywhere.

Proof. Since μ is an extreme point of (ballA^{⊥}), ∥μ∥ = 1. Without loss of generality g can be assumed real-valued (because the propereties of g is satisfied by both reg and img). Moreover if the theorem is true for g it is true for c + g too, where c ∈ ℝ. Therefore ∫_{X} g dμ = 1 can be assumed. Since g is μ-bounded, there is a λ > 0 with g < λ^{−1} μ-almost everywhere.

(a) If λ ≥ 1 then g < 1. Therefore the following equalities hold

∫_{X} |1_{X} − g| |dμ| = ∫_{X} |dμ| − ∫_{X} g |dμ| = 1 − 1 ≡ 0 .

Hence g = 1 on X μ-almost everywhere.

(b) If 0 < λ < 1 let us define the measures μ_{1} and μ_{2} as follows

μ_{1}(A) = ∫_{A} g dμ

μ_{2}(A) = ∫_{A} 1 − λg ; 1 − λ dμ .

Since ∫_{X} g |dμ| = 1, μ_{1} and μ_{2} are elements of ballA^{⊥}. The following equality is true

μ = λμ_{1} + (1 − λ)μ_{2} ,

that shows that μ can be considered as a convex combination of two elements of ballA^{⊥}. Since μ is an extreme point, μ = μ_{1} = μ_{2}, that implies that g is constant μ-almost everywhere.

Now we can prove a theorem which generalises the Stone-Weierstrass

Theorem 2.8. Suppose that A even satifies the following properties (i) For every x ∈ X there can be found an f ∈ A with f(x) ≠ 0. (ii) For all x, y ∈ X with x ≠ y, there is an f ∈ A such that f(x) ≠ f(y). (iii) A is closed under the multiplication.

Then A is dense in C(X).

Proof of L. De Branges. Suppose by contrary that the closure of A is not C(X) although the conditions (i)-(iii) are satisfied. Then there is a μ ∈ (ballA^{⊥})^{extr}. Let g be any element of A. By (iii) for every f ∈ A, fg ∈ A, hence ∫_{X} fg dμ = 0. By the Lemma 2.7. g is constant with respect to μ on the support of μ. Since for a nonzero extreme point μ, ∫_{X} |μ| = 1, we have the formula g(x) = ∫_{X} g |dμ| for every g in A and every x ∈ X. Now (ii) implies that μ is supported at a single point x. Since for every g ∈ A ∫_{X} g dμ = 0 and |μ({x})| = 1 we have that for each g ∈ A g(x) = 0. This contradicts (i).

2.3. Machado’s proof. The proof is based on the following lemma which was proved by Burckell (see [18] and compare with [16]).

Lemma 2.9. Let X be a compact Hausdorff space, B be a unital subalgebra of C_{ℝ}(X) . To each Y in B(X) let there be associated a compact subset D_{Y} of X \ Y. Then there exist elements Y_{1}, ..., Y_{n} of B(X) with the following property: given δ > 0, nonnegative b_{1}, ..., b_{n} can be found in B which satisfy (1) ∑_{i=1}^{n} b_{i} = 1

(2) b_{j} < 2δ on D_{Y_{j}} (j = 1, ..., n).

Proof. By Lemma 2.3. for each Y ∈ B(X) one can find a function c_{Y} such that 0 < c_{Y} ≤ 1, c_{Y} (Y) = {1}, and c_{Y} < 1 2 in D_{Y}. Therefore u_{Y} = sup_{x∈D_{Y}} c_{Y} (x) < 1 2, hence we can choose v_{Y} ∈ ℝ with 0 < u_{Y} < 1 − u_{Y} < v_{Y} < 1. The set Y lies in the open set U_{Y} = {x ∈ X : c_{Y} (x) > v_{Y} (x)} and by the compactness property there can be found finitely many U_{Y_{i}} (i ∈ {1, ..., n}) such that

X = ⋃_{i=1}^{n} U_{Y_{i}} .

Applying the standard Stone-Weierstrass Theorem for C[a, b] there are polinomyals q_{j} (j = 1, ..., n) with 0 ≤ q_{j} < 1 on [0, 1] such that 0 ≤ q_{j} < δ and on [0, u_{Y_{j}}], and 1 − δ ≤ 1 on [v_{Y_{j}}, 1].

Define functions g_{j} (j = 1, ..., n) to be q_{j} ∘ c_{Y_{j}}. (This composition belongs to B since it is an algebra, and its values are taken from the intarval [0, 1].) Define b̂_{1} = g_{1} and b̂_{i} = g_{i} ∏_{j=1}^{i−1}(1 − g_{j}) (i = 1, ..., n), hence the values of b̂_{j} lie in [0, 1] as well. The equation

∑_{i=1}^{n} b̂_{i} = 1 − ∏_{i=1}^{n}(1 − g_{i})

can be easily verified by induction. On D_{Y_{j}} we have c_{Y_{j}} ≤ u_{Y_{j}} so g_{j} < δ by the construction of g_{j}. Hence 0 ≤ b̂_{j} ≤ g_{j} < δ on D_{Y_{j}}. Now for each x ∈ X there is a k ∈ {1, ..., n} such that x ∈ U_{Y_{k}}, then 1 ≥ ∑_{i=1}^{n} b̂_{j}(x) = 1 − (1 − g_{k}(x)) ∏_{j≠k}(1 − g_{j}(x)) ≥ 1 − δ. Finally , set b_{i} = b̂_{i} for i = 1, ..., n−1 and b_{n} = 1−∑_{i=1}^{n} b̂_{i}. Then b_{n} ≤ b̂_{n}+δ < 2δ on D_{Y_{n}} and b_{n} ≥ b̂_{n} ≥ 0 on X. {b_{i}}_{i=1}^{n} satisfies the desired conditions.

Remark 2.1. We could have avoided to use the Stone-Weierstass Theorem by the lemma in [20].

Machado proved a more general theorem which implies the Bishop’s Theorem (see [16] and [18]). Now E denotes a complex Banach space.

Theorem 2.10 (Machado). Let G be a vector subspace of C(X; E), which is a module over a unital subalgebra B of C_{ℝ}(X).

Then for every f ∈ C(X; E) the following equality holds

(2.3) inf{∥f − g∥_{E} : g ∈ G} = sup_{Y ∈B(X)} inf{∥f|_{Y} − g|_{Y} ∥_{E} : g ∈ G} .

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Proof. For every Y ∈ B(X) define d(Y) to be inf{∥f|_{Y} −g|_{Y} ∥ : g ∈ G} and set c = sup_{Y ∈B(Y)}{d(Y)} and d = inf{∥f −g∥ : g ∈ G}. Obviously the inequality d ≥ d(Y) holds, hence we have to verify the oposite inequality. The main idea of the proof as follows: try to construct an approximating function from that functions which approximate f on Y weighted by b_{i} (using the notations of the previous lemma).

For this, find functions g_{Y} for all Y ∈ B(X) with d(Y) + ϵ ; 3 > ∥(f − g)|_{Y} ∥ and consider the sets D_{Y} = {x ∈ X : ∥f(x)−g(x)∥ > d(Y)+ ϵ ; 3}, so D_{Y} ∩ Y = ∅. Therefore the Lemma 2.9. can be applied and there can be found n, b_{Y_{1}}, ..., b_{Y_{n}} such that (1) and (2) are satisfied in Lemma 2.9.. Let K ∈ ℝ greater than ∥f∥ + ∑_{i=1}^{n} ∥g_{Y_{i}}∥ and δ < (nK^{−1}) ϵ ; 3. Let us define g as follows

g = ∑_{i=1}^{n} b_{Y_{i}}g_{Y_{i}} .

We shall show that g approximates f with error less than ϵ ; 3. An upper estimation might be given for the error by

|f(x) − ∑_{i=1}^{n} b_{Y_{i}}(x)f(x)| + | ∑_{i=1}^{n} b_{Y_{i}}(x)(f(x) − g(x))| .

Denote the first part of the sum by e_{1} and the second one by e_{2}. Clearly

e_{1}(x) ≤ ∥f∥ · |1 − ∑_{i=1}^{n} b_{Y_{i}}(x)| < δ∥f∥ < ϵ ; 3. Now try to estimate e_{2}(x).

If x ∉ D_{Y_{i}} then

∥b_{Y_{i}}(x)(f(x) − g_{Y_{i}}(x))∥ ≤ b_{Y_{i}}(x)(d(Y_{i}) + ϵ ; 3) .

If x ∈ D_{Y_{i}} then

∥b_{Y_{i}}(f(x) − g_{Y_{i}}(x))∥ ≤ δ(∥f∥ + ∥g_{i}∥) < δK .

Hence e_{2}(x) < nKδ+∑_{i=1}^{n}(d(Y_{i})+ ϵ ; 3) ≤ c+ 2ϵ ; 3, and the altogether error is less than c+ϵ. If ϵ converges to zero we get the desired inequality.

Remark 2.2. We can replace the supremum by maximum in the previous theorem i.e. for every f in C(X; E) there exists an antisymmetric set Y_{f} ∈ B(X) such that

inf{f − g∥_{E} : g ∈ G} = inf{∥f|_{Y} − g|_{Y} ∥_{E} : g ∈ G} (2.4)

Proof. Using the notations of the previous theorem let us find a sequence {Y_{n}}_{n∈ℕ} in B(X) with d(Y_{n}) > d − 1 ; n. Choose an arbitrary x_{n} from Y_{n} for all n ∈ ℕ by the Axiom of Choice. Let x_{0} be a cluster point of the sequence {x_{n}}_{n∈ℕ}. By the observation taken after Corollary 2.2. x_{0} is contained by a Y_{0} ∈ B(X). We show that Y_{0} is the proper Y_{f}. We have to prove that d(Y_{0}) ≥ d. Let r ∈ ℝ with r > d(Y_{0}) it would be sufficient to verify that r ≥ d holds. If r > d(Y_{0}) there is a function g ∈ G with ∥f|_{Y_{0}} − g|_{Y_{0}}∥_{E} < r, hence Y_{0} is contained by the open set

U = {x ∈ X : ∥f(x) − g(x)∥_{E} < r} . (2.5)

Then for all x ∈ X\U there exists b_{x} ∈ B with b_{x}(x) = 1 but b_{x}(Y_{0}) = 0. Obviously b_{x}^{−1}(( 1 2, ∞)) is a neighbourhood of x, so by the compactness argument X \ U ⊂ ⋃_{i=1}^{n} b_{y_{j}}^{−1}(( 1 2, ∞)). Let us define b to be ∑_{i=1}^{n} b_{y_{i}}^{2} so b(Y_{0}) = 0 but b > 1 4 on X \ U. Recall that x_{0} was a cluster point therefore there are infinetely many n ∈ ℕ such that b(x_{n}) < 1 4. For these n, U_{n} ∩(X \U) ≠ ∅ therefore Y_{n} ⊂ U. If x lies in such an Y_{n} then ∥f(x) − g(x)∥_{E} < r by (2.5) hence d ≤ r by the choice of {U_{n}}_{n∈ℕ}.

Let us see the generalization of the theorem above for the complex case. For this we shall use the Axiom of Choice as Zorn’s Lemma. Basically the proof has two main part. First we show that the Zorn’s Lemma can be applied, secondly we reduce the theorem to the real case.

Theorem 2.11. Let G be a vector subspace of C(X; E) which is a module over a unital subalgebra of C_{ℂ}(X). Then for each f ∈ C(X; E) there exists an A-antisymmetric set S_{f} such that

inf{∥f − g∥_{E} : g ∈ G} = inf{∥f|_{S_{f}} − g|_{S_{f}} ∥_{E}} . (2.6)

Proof. Let f ∈ C(X; E) and d^{⋆} = inf{∥f −g∥_{E} : g ∈ G}. If d^{⋆} = 0 the equality (2.6) holds for all S ∈ A(X). Hence d^{⋆} ≠ 0 can be assumed. Define pairs (R, S) as follows

(i) Let R be a dijoint, closed partition of X consisting of nonempty sets and (ii) S ∈ R with d^{⋆} = inf{∥f|_{S} − g|_{S}∥_{E} : g ∈ G} . Define D as the following set

D = {(R, S) : (R, S) satisfies (i) and (ii) conditions } .

Since ({X}, X) is obviously contained by D, D ≠ ∅. Now let us define a partialy order on D denoted by ”≺” as follows (R, S) ≺ (L, T) if and only if L is a refinement of R and T ⊂ S. We want to show that any chain C in D has an upper bound in D in order to apply the Zorn’s Lemma. To all x ∈ X and to every (R, S) ∈ C assign a set R(x) ∈ R with x ∈ R(x). Let us consider the set C(x) = ⋂_{(R,S)∈C} R(x) hence {C(x) : x ∈ X} obviously a closed partition of X. We should show that this is a disjoint partition. Indeed if C(x)∩C(y) ≠ ∅ then R(x) = R(y) for every R therefore C(x) = C(y). We have to find a suitable set T which satisfies the condition (ii). Let T = ⋂_{(R,S)∈C} S. Now for all

14 BALÁZS SZENTES

g ∈ G correspond two sets as follows

K(g) = {x ∈ T : ∥f(x) − g(x)∥_{E} ≥ d^{⋆}}
K(g, R, S) = {x ∈ S : ∥f(x) − g(x)∥_{E} ≥ d^{⋆}} .

These sets are clearly compact, moreover

K(g) = ⋂_{(R,S)∈C} K(g, R, S)

holds. Each K(g, R, S) is nonempty because d^{⋆} = inf{∥f|_{S} − g|_{S}∥_{E} : h ∈ G}. Obviously if {(R_{i}, S_{i})}_{i=1}^{n} ⊂ C then ⋂_{i=1}^{n} S_{i} = S_{j} for an j ∈ {1, ..., n}. This fact combining with the compactness property shows that K(g) ≠ ∅. Since K(g) ⊂ T for each g ∈ G, we obtain

inf{∥f|_{T} − g|_{T} ∥_{E} : g ∈ G} ≥ inf{∥f|_{K(g)} − g|_{K(g)}∥_{E} : g ∈ G} ≥ d^{⋆}

by definition of K(g). Of course this infimum is not greater than d^{⋆} by its definition, so the desired equality holds.

Zorn provides a maximal element (R, S) ∈ D. We shall prove that S is an antisymmetric set, considering (ii) it would verify the theorem. To do this introduce the set A_{S} = {f ∈ A : f(S) ⊂ ℝ} and B = A|_{S}. B is obviously a unital subalgebra of C_{ℝ}(X). If S is not antisymmetric set then B(S) is not trivial. Now G|_{S} is a B-module and hence the last theorem provides T ∈ B(S) with

d^{⋆} = inf{∥f|_{S} − g|_{S}∥_{E} : g ∈ G} = inf{∥f|_{T} − g|_{T} ∥_{E} : g ∈ G} .

Consider the partition R′ of X consisting of the elements of R different from it only S together with all the sets in B(S), this contradicts the maximality of (R, S).

2.4. Ransford’s proof. In the previous two sections we provided two different proofs for Bishop’s theorem. One of these was quite short, and elegant however it used several tools from functional analisys. The other (due to Machado) is elementary but very long, although its result is stronger than Bishop’s original theorem. The main purpose of this section is to establish a proof (due to Ransford, see [17] ) which is as elementary as short.

Before we prove again the Machado’s theorem we introduce two notations. If F is a non-empty, closed subset of X, ∥f∥_{F} denotes sup{|f(x)| : x ∈ F }, and d_{F} (f) denotes inf{∥f − g∥_{F} : g ∈ A}. Now we are ready to prove

Theorem 2.12 (Machado). Let A be a unital subalgebra of C(X) and f be in C(X). Then there exists a closed A-antisymmetric set S such that d_{S}(f) = d_{X}(f).

Ransford’s proof. Let F denotes the collection of all non-empty closed subsets F of X which satisfies the equality d_{F} (f) = d_{X}(f). Since X ∈ F obviously F ≠ ∅. In order to use the Zorn’s Lemma we try to verify that if C is a chain (ordered by inclusion), then

E = ∩{F : F ∈ C}

belongs to F as well. To do this let us observe that {x ∈ F : |f(x) − g(x)| ≥ d_{X}(f)} is compact and non-empty for each F ∈ C and g ∈ A, since d_{X}(f) = d_{F} (f). Therefore the intersection of the chain of all such sets, namely

{x ∈ E : |f(x) − g(x)| ≥ d_{X}(f)} ,

is also non-empty. Since this property holds for all g ∈ A, it follows that d_{E}(f) = d_{X}(f). So, we can apply the Zorn’s Lemma, which provides a minimal element S in F. We have to show that S is an antisymmetric set.

For contrary we assume that there exists h ∈ A such that h is real-valued but not constant on S. Obviously we might assume that min_{x∈S} h(x) = 0 and max_{x∈S} h(x) = 1. Define two sets as follows

H_{1} = {x ∈ S : 0 ≤ h(x) ≤ 2 3}
H_{2} = {x ∈ S : 1 3 ≤ h(x) ≤ 1} .

Since S is minimal in F and H_{1} and H_{2} are non-empty, closed sets, there exist two functions in A denoted by g_{1} and g_{2} such that:

∥f − g_{1}∥ < d_{X}(f) (2.7)

∥f − g_{2}∥ < d_{X}(f) .

For n ≥ 1 set h_{n} = (1 − h^{n})^{2^{n}} and g_{n} = h_{n}g_{1} + (1 − h_{n})g_{2}. Clearly h_{n}, g_{n} belong to A and 0 ≤ h_{n} ≤ 1 on S. From (2.7) it follows that ∥f − g∥_{H_{1}∩H_{2}} < d_{X}(f), because g_{n} is a convex combination of g_{1} and g_{2} at each point of S. On H_{1} \ H_{2}:

h_{n} = (1 − h^{n})^{2^{n}} ≥ 1 − 2^{n}h^{n} ≥ 1 − 2 3^{n}

while on H_{2} \ H_{1} :

h_{n} = (1 − h^{n})^{2^{n}} ≤ (1 + h^{n})^{−2^{n}} ≤ (2^{n}h^{n})^{−1} ≤ 3 4^{n} .

Thus g_{n} converges uniformly to g_{1} on H_{2} \ H_{2} and to g_{2} on H_{2} \ H_{1}. Hence we obtained that if n is large enough, then ∥f − g_{n}∥ < d_{X}(f) contradicting the fact that S belongs to F.

Remark 2.3. With only obvious modification the same proof yields Machado’s theorem for vector valued functions.

Bishop’s idea was further developed by J.B Porolla for locally compact spaces and for vector valued functions, where the topology is a weighted topology, and the weights are positive continous functions vanishing at infinity (see [21] Theorem 3.2., 4.5.).

3. The strict topology

3.1. Bounded functions on locally compact spaces. In this part we try to generalise the classical Stone-Weierstrass theorem such that the compactness property is repleaced by the locally compactness property. In addition in this part we try to approximate not only the continous functions vanishing at infinity, but all bounded continous functions. Of course, it would be impossible without considering a new weaker topology.

This new topology was introduced in [5] and called strict topology. In this part we follow [6] because in that paper this topology was defined for more general cases.

3.2. The strict topology. Let X be a locally compact topological space and E be a real linear space, with a complete locally convex topology described by an indexed family of semi-norms ∥.∥_{ϑ}. C(X; E) is the real linear space of all continous mappings from X into E. Let C_{b}(X; E) be the linear subspace of C(X; E) containing the all bounded functions. Although the space C_{b}(X; E) is not an algebra, since the product of mappings is not defined, it is naturally a C(X) module. (Where C(X) and C_{b}(X) denotes C(X; ℝ) and C_{b}(X; ℝ).)

The method which we use to obtain the topology β on C_{b}(X; E) is a special case of a general procedure. Let V be a linear space with locally convex topology, denoted by τ, and let H be a collection of all τ-continous linear transformations on V. Then the weak H-topology, w(H), on V is defined as the smallest topology τ′ on V such that each T ∈ H linear operator remains continous as a map: (V, τ′) → (V, τ). (Using the previous notation the ordinary weak topology on V is the topology w(H), where H is the set of all transformations with finite dimensional range.) With these notation we are ready to introduce the following

Definition 3.1. The strict topology β on C_{b}(X; E) is the weak topology w(H) obtained by taking C_{b}(X; E) as the space V, the uniform σ topology as τ, and H as the family of transformations f → ϕf, defined by function ϕ ∈ C_{0}(X). (Where the uniform topology is defined by the semi-norms ∥f∥_{ϑ} = sup_{x∈X} ∥f(x)∥_{ϑ}.)

Later we will use another topology, therefore let us introduce one more

Definition 3.2. The compact open topology κ can be induced by the same way, just replacing C_{0}(x) by C_{00}(X).

Remark 3.1. If X is compact then σ ≡ κ.

18 BALÁZS SZENTES

Remark 3.2. Topology β could have been defined as a locally convex topology generated by the following seminorms:

∥f∥_{ϕ,ϑ} = ∥ϕf∥_{ϑ} = sup_{x∈X} ∥ϕ(x)f(x)∥_{ϑ} ,

where ϑ ranges over the indexed topology on E, and ϕ ranges over C_{0}(X). In the similar way κ could have been introduced by the seminorms:

∥f∥_{κ,ϑ} = max_{x∈K} ∥f(x)∥_{ϑ} .

Remark 3.3. Considering the method of constructing these topologies the following statement can be concluded:

κ ≤ β ≤ σ .

Obviously the following inclusions explain it:

C_{00}(X) ⊂ C_{0}(X) ⊂ C_{b}(X) .

Let us mention a few very important properties about the completness of the topologies above. C_{b}(X) is well-known complete normed space and the closure of C_{00}(X) is C_{0}(X) in topology σ. In the compact open topology C_{00}(X) is dense in the space C(X). The completion of (C_{b}(X), κ) is (C(X), κ). The main result of this chapter is the following

Theorem 3.1. Let X be a locally compact space, E locally convex and complete, and let β be the strict topology on C_{b}(X; E).

Then the space C_{b}(X; E) is topologically complete.

Before proving this theorem we will prove two lemmata which are necessery for it.

Lemma 3.2. If {x_{n}}_{n∈ℕ} is a discrete sequence of points of X and {c_{n}}_{n∈ℕ} is a sequence in ℝ converging to 0, then there is a function ϕ ∈ C_{0}(X) with ϕ(x_{n}) = c_{n} for every n in ℕ.

Proof. By local compactness, a sequence of compact sets {K_{n}}_{n∈ℕ} can be choosen with x_{n} ∈ intK_{n}, but with no two sets overlapping. Take ϕ_{n} with range in [0, 1] vanishing off K_{n} and with ϕ_{n}(x_{n}) = 1. If we define ϕ(x) as ∑_{n∈ℕ} c_{n}ϕ_{n}(x_{n}) it would be clearly the desired function.

Lemma 3.3. Let F ⊂ C(X; E) and suppose that ϕF ∈ C_{0}(X; E) for every ϕ in C_{0}(X). Then F lies in C_{b}(X; E).

Proof. By contrary assume that F is unbounded. Then there exists a sequence {x_{n}} and an index ϑ such that

∥F (x_{n})∥_{ϑ} ≥ n for every n ∈ ℕ .

Since F is continous, {x_{n}}_{n∈ℕ} is discrete, hence the previous lemma provides a ϕ ∈ C_{0}(X) such that

ϕ(x_{n})F (x_{n}) = 1 for all n ∈ ℕ .

It implies that ϕF does not belong to C_{0}(X; E), that is a contradiction.

Now we have all tools to prove the theorem.

Proof of the Theorem 3.1. Let {f_{α}}_{α∈I} be a β-Cauchy net. Since κ ≤ β, {f_{α}}_{α∈I} is κ-Cauchy as well, therefore converges in the topology κ to a map F in C(X; E). For any ϕ ∈ C_{0}(X), ϕf_{α} → ϕF in topology κ. Since {f_{α}}_{α∈I} is β-Cauchy, {ϕf_{α}}_{α∈I} is also σ-Cauchy, hence it converges to H ∈ C_{b}(X; E) in topology σ. H is obviously ϕF. ϕf_{α} lies in C_{0}(X; E), which is σ-closed, so we can conclude that ϕF ∈ C_{0}(X; E), and this holds for every ϕ ∈ C_{0}(X), so the previous lemma can be applied and it says that F ∈ C_{b}(X; E). Since {ϕf_{α}}_{α∈I} σ-converges to ϕF for each ϕ ∈ C_{0}(X), we have {f_{α}}_{α∈I} β-converges to F and β is a complete topology.

3.3. The Stone-Weierstrass Theorem for C_{b}(X). If X is not compact there can be found a σ-closed maximal ideal in C_{b}(X) which is non trivial but satisfies the properties of the standard Stone-Weierstrass Theorem (see [22]). Nevertheless in topology β the theorem above can be generalised for locally compact spaces and for C_{b}(X).

Theorem 3.4. Let X be a locally compact space and A be a β-closed subalgebra of C_{b}(X) with the following properties:

(i) X is separated by A and (ii) A contains a function g vanishing nowhere.

Then A = C_{b}(X).

Proof. The function g in (ii) obviously can be assumed to be a positive function with ∥g∥ > 1. First of all we try to show that 1_{X} ∈ A. Taking the square-root provided by Stone (see [22]) by the usual series argument g^{r} ∈ A for every r > 0. Let us define h_{n} for all n ∈ ℕ as follows: h_{n} = g^{1 ; n}. Therefore ∥h_{n}∥ ≤ ∥g∥ so the sequence {h_{n}}_{n∈ℕ} is uniformly bounded in X and converges to 1 uniformly on every compact sets. We claim that it means 1_{X} ∈ A. It would be sufficient to show that for every neigbourhood of 1_{X} there is an n in ℕ such that h_{n} is contained by this neigbourhood. Let ϕ ∈ C_{0}(X) arbitrary function with ϕ < ϵ ; ∥g∥+1 outside of K where K is a compact set. On K if

n > n_{0} ∈ ℕ then ∥h_{n} − 1∥ < ϵ ; ∥ϕ∥+1 hence ∥h_{n}ϕ − 1_{X}ϕ∥ < ϵ on K.

Outside of K

∥h_{n}ϕ − 1_{X}ϕ∥ ≤ ∥h_{n} − 1_{x}∥∥ϕ∥ < ϵ(∥g∥ + 1) ; ∥g∥ + 1 = 0 .

Therefore 1_{X} is contained by A. Since |f| = √f^{2} , |f| is also in A hence A is a lattice. Let F be an arbitrary function in C_{b}(X) with ∥F ∥ = m. Given any ϕ ∈ C_{0}(X) such that ∥ϕ∥ ≤ 1 and |ϕ(x)| < ϵ ; 2m+1 for every x not in K. (K is a compact subset of X again.) Using the standard Stone-Weierstrass Theorem for K, A_{K} is dense in C(K) where A_{K} contains all functions in A restricted to K. Hence there exists a function f in A such that: |f(x) − F (x)| < ϵ ; ∥ϕ∥+1 for all x ∈ K. Define f_{0} as (f ∧ m) ∨ −m. f_{0} obviously lies in A since A is a lattice, and |f_{0}(x) − F (x)| < ϵ ; ∥ϕ∥+1 holds for all x ∈ K, so for each x ∈ K

|(f_{0}(x) − F (x))ϕ| ≤ sup_{x∈K} |f_{0}(x) − F (x)||ϕ(x)| < ϵ .

If x ∉ K : we obtain the same result using the estimation of the norm of ϕ. Thus any β-neighbourhood of F contains a function in A. Since A is β-closed F lies in A and A = C_{b}(X).

Remark 3.4. If X is locally compact and it is even σ-compact and A ⊂ C_{b}(X) which satisfies the following conditions

(i) separates X, (ii) for every x in X there exists an f in A with f(x) ≠ 0,

then the algebric closure of A in C_{b}(X) is β-dense.

Proof. We shall show that A contains a function that is strictly positive. Let X = ⋃_{n∈ℕ} K_{n} where K_{n} is a compact set in X. A_{K_{n}} is uniformly dense in C(K_{n}). (Where A_{K_{n}} is the set of all function from A restricting into K_{n}.) By (ii) choose f_{n} ∈ A with f_{n}(x) ≥ 1 for each x ∈ K_{n}. Let g_{n} be (n∥f_{n}∥)^{−2}, so ∥g_{n}∥ ≤ 1 ; n^{2} . Set g = ∑_{n∈ℕ} g_{n}. This expression is uniformly convergent so g ∈ A. Clearly g > 0 on X.

Using the previous remark we can prove a theorem without assuming the existence of a function g which is strictly positive. This theorem was proved by C. Todd (see [12]).

Theorem 3.5. Let X be a locally compact Hausdorff space and let A be a strictly closed subalgebra of C_{b}(X) with the following properties:

(i) A separates X and (ii) for each x ∈ X there is a function g contained by A with g(x) ≠ 0.

Then A = C_{b}(X).

Proof. Let f be an arbitrary function in C_{b}(X) which we try to approximate. Let ϕ ∈ C_{0}(X) and ϵ be given. Let H_{1} be the σ-compact set outside of which ϕ vanishes identically. We can replace H_{1} by a regularly σ-compact set denoted by H. H is obviously a locally compact σ-compact Hausdorff space in its relative topology. A restricted to H is β-dense in C_{b}(H) by the previous remark, consequently there is a function g ∈ A such that |ϕ(x)f(x) − ϕ(x)g(x)| < ϵ for every x ∈ H. Since ϕf and ϕg vanish identically outside of H, we have ∥g − f∥_{ϕ} < ϵ. Thus for every neighborhood of f we can find a function in A lying in this neigbourhood.

Remark 3.5. Although at this stage we could have proved a more general theorem in which the C_{b}(X) could have been replaced by C_{b}(X; E) we didn’t do it, because later we can prove a much more general theorem which has almost the same properties and which implies the theorem mentioned above.

To do this we need a Riesz Representation Theorem for (C_{b}(X; E), β).

3.4. The Riesz Representation Theorem for (C_{b}(X; E), β). (In this section our discussion is basically based on [11],[8], [12] and [6].) If X is compact space then the dual space of the (C_{b}(X), σ) is M(X), the space of all complex Borel-measures on X. If X is only locally compact and we consider C_{0}(X) then its dual space is also M(X) provided by the one-point compactification. (Thus it is true for C_{00}(X) as well.) In this chapter we try to show that if we replace C_{0}(X) by the larger space C_{b}(X) the dual space remains M(X).

Before proving this theorem we try to describe the support of these measures.

Lemma 3.6. Each measure μ in M(X) has a σ-compact support.

Proof. Without loss of generality the positivity of μ might be assumed. Let L be the corresponding linear functional defined on C_{00}(X) by

L(f) = ∫_{X} f dμ .

Since μ is bounded L is σ-continous, and there exists a number M = ∥L∥ such |L(f)| ≤ M∥f∥ for all f ∈ C_{00}(X). Choose f_{n} ∈ C_{00}(X) with ∥f_{n}∥ = 1 and lim_{n→∞} L(f_{n}) = M. Let K_{n} be the support of f_{n}, and let S_{μ} = ⋃_{n∈ℕ} K_{n}. We claim that S_{μ} ⊃ suppμ. Let g ∈ C_{00}(X) with ∥g∥ ≤ 1 2 and g(x) = 0 for all x ∈ S_{μ}. Therefore ∥f_{n} +g∥ = ∥f_{n} −g∥ = 1 for each n ∈ ℕ, so

∥L(f_{n}) + L(g)∥ = ∥L(f_{n}) − L(g)∥ ≤ M

which implies that L(g) = 0, by other words, suppμ is contained by S_{μ} which is σ-compact.

Theorem 3.7. Every β-continous linear functional on C_{b}(X) has an integral representation, hence the dual space of (C_{b}(X), β) is M(X).

Proof. Let L be any positive β-continous linear functional on C_{b}(X). L is obviously σ-continous on the space C_{00}(X). Since the uniformal dual space of C_{00}(X) is M(X), L has an integral representation with μ. By the previous lemma suppμ ⊂ ⋃_{n∈ℕ} K_{n} where K_{n} is compact set in X, moreover K_{n} ⊂ intK_{n+1} can be assumed. Put a_{n} = μ(K_{n+1} \ K_{n}), so that a_{n} ≥ 0 and ∑_{n=1}^{∞} a_{n} = M using the notations of the previous lemma. Choose a sequence {b_{n}}_{n∈ℕ} converging to zero with

∑_{n=1}^{∞} a_{n} ; b_{n} < 2M .

Let us define c_{n} as b_{n} − b_{n+1} so that b_{n} = ∑_{j=n}^{∞} c_{j}. Now construct a function ψ_{n} ∈ C_{00}(X) for all n ∈ ℕ taking values in [0, 1] such that ψ_{n}(x) = 1 for all x ∈ K_{n} while ψ_{n}(x) = 0 if x ∉ K_{n+1}. Construct a function ψ by seting

ψ(x) = ∑_{n∈ℕ}c_{n}ψ_{n}(x) .

ψ lies in C_{0}(X) because b_{n} → 0, and ψ is zero exept ⋃_{n∈ℕ} K_{n}, and ψ(x) ≥ b_{n} for each K_{n} \ K_{n−1}. The function 1 ; ψ is continous on suppμ hence it is μ-integrable with ∫_{X} 1 ; ψdμ < 2M. If f ∈ C_{b}(X) with ∥f∥_{ψ} <

ϵ ; 2M then

∫_{X} f dμ ≤ ϵ ; 2M ∫_{X} 1 ; ψ dμ < ϵ

which proves that μ is β-continous.

Remark 3.6. Using the conclusion of the previous theorem and its notations there exists a measure in M(X) with

∫_{X} f ; ψ dμ = ∫_{X} f dϑ

for every f ∈ C_{b}(X). It means that for all μ ∈ M(X) one can find a ψ ∈ C_{0}(X) and ϑ ∈ M(X) such that

μ(H) = ∫_{H} ψ dϑ

for each Borel-set H. The positivity of ψ can be assumed.

Now we shall make some effort to generalise the Riesz Representation Theorem for the topological space (C_{b}(X; E), β). For this we have to give a precise definition to the suitable measure space. (See [11].)

Definition 3.3. M(X; E^{⋆}) denotes the set of all measure μ on X whose values lie in E^{⋆} (where E^{⋆} the a dual space of E) and which satisfies the following two conditions

(i) μ(.)s ∈ M(X) for every s ∈ E and (ii) there is a p seminorm from the family which induces the topology on E and a constant k such that sup{∑_{i=1}^{n} μ(A_{i})s_{i}} ≤ k, where the supremum is taken over all partitions of X into a finite number of dijoint Borel sets {A_{i}}_{1=1}^{n} and {s_{i}}_{i=1}^{n}in E such that p(s_{i}) ≤ 1.

Definition 3.4. For every seminorm p which satisfies (ii) in the previous definition the p-variation |μ|_{p} of μ is defined by

|μ|_{p}(A) = sup | ∑_{i=1}^{n} μ(A_{i})s_{i}|

where A is a Borel set and the supremum is exactly as in the previous definition just X is replaced by A.

It is easy to verify that |μ|_{p} belongs to M(X). We set

∥μ∥_{p} = |μ|(X) .

Now we shall prove an analogous lemma with Remark 3.6. .

Lemma 3.8. Let ϕ in C_{b}(X) and V = {f ∈ C_{b}(X) : ∥ϕf∥ < 1}. Then for all ϑ ∈ M(X) with |ϑ(f)| < 1 if f ∈ V, there exists a γ in M(X) with ∥γ∥ ≤ 1 such that γϕ = ϑ i.e.

∫_{X} fϕ dγ = ∫_{X} f dϑ .

Proof. |ϑ(f)| ≤ ∥ϕf∥ because f ; ∥ϕf∥+ϵ ∈ V for all ϵ > 0, therefore the mapping T defined by T (ϕf) = ∫_{X} f dϑ is a σ- continous functional on the subspace ϕC_{b}(X) ⊂ C_{0}(X), moreover ∥T ∥ ≤ 1. Using the Hahn-Banach Theorem T has a norm-preserving extension T′ to C_{0}(X). Applying the Riesz Representation Theorem for M(X) (see Theorem 3.7.). there exists a γ in M(X) with ∥γ∥ ≤ 1 such that

T′(f) = ∫_{X} f dγ for all f ∈ C_{0}(X).

Combining this with definition of T we see that

∫_{X} fϕ dγ = ∫_{X} f dϑ

for every f ∈ C_{b}(X), hence ϑ = ϕγ.

Lemma 3.9. If μ ∈ M(X; E^{⋆}) then there exist a ϑ ∈ M(X; E^{⋆}) and a ϕ ∈ C_{0}(X) such that μ = ϕϑ i.e.

μ(A)s = ∫_{A} ϕ dϑs

for each s ∈ E and each Borel set A ⊂ X.

Proof. Since |μ|_{p} belongs to M(X) and therefore determines a strictly continous functional on C(X), (where p is the seminorm corresponding to μ), there can be found a ϕ ∈ C_{0}(X) with | ∫_{X} f d|μ|_{p}| < 1 if f ∈ V using the notations of the previous lemma. By Lemma 3.8. there is a measure ϑ ∈ M(X) with |μ|_{p} = ϕϑ_{p}. (We ensure uniquness by requiring ϑ_{p} to vanish on the set S_{ϕ} = {x ∈ X : ϕ(x) = 0}.) ∥ϑ_{p}∥ ≤ 1 and ϕ ≥ 0 can be assumed. The same argument shows that for each s ∈ E there is a unique measure ϑs in M(X) vanishing on S_{ϕ} which satisfies the condition ∥ϑs∥ ≤ p(s) and μs = ϕϑs. Obviously |μs| is majorized by p(s)|μ|_{p} considering the definition of |μ|_{p} and the choice of p. Therefore the following inequalities hold

(3.1) | ∫_{X} f dμs| ≤ ∫_{X} |f| d|μs| ≤ p(s) ∫_{X} |f|d|μ|_{p}

= p(s) ∫_{X} |fϕ| dϑ_{p} ≤ p(s)∥ϕf∥∥ϑ_{p}∥

for all f ∈ C_{b}(X; E). Let us define ϑ on every Borel set A as a function on E whose value at the point s is ϑs (ie. the value of the measure ϑs at A). The idendities μ(s + t) = μ(s) + μ(t) and μs = ϕϑs show that

ϑ(A)(s + t) = ϑ(A)s + ϑ(A)t .

The relation |ϑ(A)s| ≤ p(s) follows from the inequality ∥ϑs∥ ≤ p(s). Hence ϑ(A) ∈ E^{⋆} and ϑ satisfies the condition (i) in the definition 3.3.. We try to show that it satisfies condition (ii) as well. For this let p(s_{i}) ≤ 1 for every i ∈ ℕ and {a_{i}}_{i=1}^{n} a finite partition of X into disjoint Borel sets. The following inequalities might be easily verified

(3.2) | ∑_{i=1}^{n} ϑ(A_{i})s_{i}| = | ∑_{i=1}^{n} ∫_{A_{i}} ϕ^{−1} dμs_{i}| ≤ ∑_{i=1}^{n} ∫_{A_{i}} ϕ^{−1} d|μs_{i}| ≤

∑_{i=1}^{n} ∫_{A_{i}} ϕ^{−1} d|μ|_{p} ≤ ∫_{X} ϕ^{−1} d|μ_{p}| = ∫_{X} dϑ_{p} ≤ 1

which means that the desired equality holds.

Now we shall describe the dual space of C_{0}(X; E_{p}) where E_{p} denotes E with the topology induced by the single seminorm p.

Lemma 3.10. Let L be a linear σ-continous functional on C_{0}(X; E_{p}). Then there exists a μ ∈ M(X; E_{p}^{⋆}) with ∥L∥ = ∥μ∥_{p} such that

L(f) = ∫_{X} f dμ (f ∈ C_{0}(X)) .

Proof. For all s ∈ E_{p} we can define a functional L_{s} on C_{0}(X) by L_{s}(f) = L(fs) where fs: x → f(x)s represents a member of C_{0}(X; E_{p}). L_{s} is clearly linear functional with

|L_{s}(f)| = |L(fs)| ≤ ∥L∥∥fs∥_{p} ≤ ∥L∥p(s)∥f∥ (3.3)

that shows that ∥L_{s}∥ ≤ ∥L∥p(s). Applying the Riesz Representation Theorem for L_{s}, there exists a unique measure μs ∈ M(X) with ∥μs∥ = ∥L_{s}∥ such that

L_{s}(f) = ∫_{X} f dμs .

The relation μ(as+bt) = aμs+bμt (a, b ∈ ℝ s, t ∈ E_{p}) comes from the fact that L is bilinear map on C_{0}(X) × E_{p}. The inequality (3.3) shows that the map s → μ(A)s is continous. We heve to show that μ satisfies the condition (ii) in definition 3.3.. Let {A_{i}}_{i=1}^{n} be a disjoint partition of X into Borel sets. Using the regularity of μs_{i} there exist compact sets F_{i} ⊂ A_{i} such that |μs_{i}|(A_{i} \ F_{i}) < ϵ ; 2n, and open sets S_{i} ⊃ F_{i} such

that |μs_{i}|(S_{i} \ F_{i}) < ϵ ; 2n, for each n ∈ {1, ..., n} and S_{i} ∩ S_{j} = ∅, where

{s_{i}}_{i=1}^{n} is a sequence in E_{p} and ϵ > 0. Choose functions g_{i} ∈ C_{0}(X) for each i ∈ {1, ..., n} taking value from [0, 1] such that g_{i}(S_{j}) = 0 if i ≠ j but g_{i}(F_{i}) = 1. The function

h = ∑_{i=1}^{n} g_{i}s_{i}

lies in C_{0}(X; E_{b}) with ∥h∥_{p} ≤ max_{i∈{i,...,n}} p(s_{i}) and

L(h) = ∑_{i=1}^{n} L(g_{i}s_{i}) = ∑_{i=1}^{n} ∫_{X} g_{i} dμs_{i} .

This information combined with |μ(A_{i}) − ∫_{A_{i}} g_{i} dμs_{i}| < ϵ ; n yields

| ∑_{i=1}^{n} μ(A_{i})s_{i}| < ϵ + ∥L∥ max_{i∈{1,...,n}} p(s_{i})

which is the desired inequality considering that the p-norm of s_{i} involving in the condition (ii) of the definition 3.3. is always less or equal than 1. Simple estimations on approximating sums show that ∫_{X} g dμs = ∫_{X} gs dμ for every g ∈ C_{0}(X) and s ∈ E_{p}. Thus L(h) = ∫_{X} h dμ. Since functions in the form of h with g_{i} ∈ C_{0}(X) (s_{i} ∈ E_{p}) are σ-dense in

26 BALÁZS SZENTES

C_{0}(X; E_{p}), it follows that μ represents L on function f in C_{0}(X; E_{p}). ∥L∥ = ∥μ∥_{p} is obviously true.

Now we can prove the main theorem of this section.

Theorem 3.11. If L is a β-continous linear functional on C_{b}(X; E) then there exists a μ in M(X; E^{⋆}) such that

L(f) = ∫_{X} f dμ (3.4)

for each f ∈ C_{b}(X; E). Conversely for every μ ∈ M(X; E^{⋆}) the formula (3.4) defines a strictly continous linear function on C_{b}(X; E).

Proof. By the β-continuity of L there is a seminorm p and a ϕ in C_{0}(X) such that L(f) < 1 whenever f is in the neigbourhood V defined by the following formula

V = {f ∈ C_{b}(X; E) : ∥ϕf∥_{p} < 1} .

For any f ∈ C_{b}(X; E) and for any ϵ > 0 the function f ; ∥ϕf∥_{p}+ϵ ∈ V, therefore |L(f)| < ∥ϕf∥_{p} + ϵ, and letting ϵ tends to zero shows that |L(f)| ≤ ∥ϕf∥_{p}. It means that T defined by T (ϕf) = L(f) is a σ -continous functional on ϕC_{b}(X; E) whose norm does exceed 1. The Hahn-Banach Theorem provides an extention T′ of T to C_{0}(X; E) such that

|T′(f)| ≤ ∥f∥_{p} (f ∈ C_{0}(X; E_{p})) .

Using the previous lemma there exists a measure ϑ ∈ M(X; E_{p}) such that

T′(f) = ∫_{X} f dϑ (f ∈ C_{0}(X; E_{p})) . (3.5)

By the definition of T′ (3.5) implies that L(f) = ∫_{X} fϕ dϑ. Let us define μ as ϕϑ. Hence the integral on the right side can be written in the form ∫_{X} f dμ, which is the required representation.

We shall show that every μ ∈ M(X; E^{⋆}) defines a β-continous functional on C_{b}(X; E). By Lemma 3.9. μ = ϕϑ for some ϕ ∈ C_{0}(X) and ϑ ∈ M(X; E^{⋆}). Obviously ϕϑ is a β-continous functional. (Indeed, let p the seminorm corresponding to ϑ such that ∥ϑ∥_{p} < 1. Hence if g ∈ V_{1} = {g ∈ C_{b}(X; E) : ∥fϕ − gϕ∥_{p} < 1} then | ∫_{X} fϕ − gϕ dϑ| < 1.)

3.5. A Stone-Weierstrass Theorem for C_{b}(X; E).

Theorem 3.12. Let X be a locally compact space, and E a complete, locally convex topological space whose topology is given by a family, indexed by I. Let A be a β-closed subspace of C_{b}(X; E) that satisfies the following conditions (i) A(x) = {f(x) : f ∈ A} = E for every x ∈ X (ii) If ϕ ∈ C(X) has values in [0, 1] and f ∈ A, than ϕf ∈ A.

Then A = C_{b}(X; E).

Proof. Suppose the conclusion is false. Then using the same argument like in Bishop’s proof, there exists a μ ∈ M(X; E^{⋆}) with μ⊥A. Since μ is a strictly continous linear functional on C(X, E) there can be found a ϕ in C_{0}(X) and a p ∈ N such that:

| ∫ f dμ| < 1 if ∥ϕf∥_{p} < 1 . (3.6)

The set V^{0} of all measures μ for which (3.6) holds and μ⊥A is a convex and weak^{⋆} compact in M(X; E^{⋆}), so by the Krein-Milman Theorem it has an extreme point μ. According to Lemma 3.9. there exists a ϑ in M(X; E^{⋆}) with μ = ϕϑ, and since μ is extreme ∥ϑ∥_{p} = 1. Now choose an h in C(X) with 0 < h < 1 and observe that hμ ; ∥hϑ∥_{p} and

(1−h)μ ; ∥(1−h)μ∥∥(1 − h)ϑ∥_{p} are in V^{0} and

μ = ∥hϑ∥_{p} hμ ; ∥hϑ∥_{p} + ∥(1 − h)ϑ∥_{p} (1 − h)μ ; ∥(1 − h)ϑ∥_{p} ,

from which the following equality is obvious

∥hϑ∥_{p} + ∥(1 − h)ϑ∥_{p} = ∫_{X}(1 − h) d|ϑ|_{p} + ∫_{X} h d|ϑ|_{p} = ∫_{X} d|ϑ|_{p} = 1 .

Since μ is extreme, μ = hμ ; ∥hϑ∥_{p}. Set c = ∥hϑ∥_{p}, then

∫_{X} g dμs = c ∫_{X} g dμs

for each s ∈ E and g ∈ C(X). Thus h is constant on suppμs wthat implies that suppμs is a single point x_{0}, because h can be choosen so that it separates any two points. The identity μ(s + t) = μ(s) + μ(t), which holds for each pair s, t ∈ E, shows that each of the measures μs (s ∈ E) is a point mass at x_{0}. Since μ⊥A, we can conclude that,

0 = ∫_{X} f dμ = μ({x_{0}})f(x_{0}) (f ∈ A) . (3.7)

Since μ is nonzero, μ({x_{0}}) is nonzero either whose kernel, according to (3.7), contains A(x_{0}), contrary to (i).

In this part we always assumed that X is a locally compact, Hausdorff space, although R. Giles and J. Hoffmann had success to generalize both the Stone-Weierstrass theorem and the description of the dual space of (C_{b}(X), β) for completely regular k −spaces. (See [9] and [8].)

28 BALÁZS SZENTES

4. The Korovkin closure

4.1. Motivations of the Korovkin-type theorems. Although the Stone-Weierstrass Theorem guarantees for every f ∈ C[a, b] a sequence of polynomials, which converges to f, it is natural to ask how to construct this sequence, and how can one be sure that this sequence converges to f indeed. One of these sequence was defined by S.Bernstein on the unit interval [0, 1] in ℝ. The nth Bernstein polynomial is defined as

p_{n}^{f}(x) = ∑_{k=0}^{n} \binom{n}{k}f(k ; n)x^{k}(1 − x)^{n−k} .

We can define linear operators B_{n} (n ∈ ℕ) such that B_{n}(f) = p_{n}^{f}, so B_{n} is an operator of C[a, b] into itself, which is in addition positive (i.e. if f ≥ 0 then p_{n}^{f} ≥ 0 ). Our aim is to prove that p_{n}^{f} converges to f uniformly on [0, 1]. It is easy to check that

p_{n}^{1} = 1
p_{n}^{id} = id

p_{n}^{id^{2}} = 1 ; nid + n − 1 ; nid^{2} ,

that proves that if f = 1, id, or id^{2} then (p_{n}^{f})_{n∈ℕ} converges to f uniformly on [0, 1]. We try to show that it is sufficient, that is, the convergence of these three test function guarantees the convergence on every f ∈ C[a, b].

Theorem 4.1 (Korovkin). Let [a, b] be a compact set of ℝ and let (M_{n})_{n∈ℕ} be a sequence of positive linear operators, mapping from C[a, b] into itself. Suppose that (M_{n}f)_{n∈ℕ} converges uniformly to f for the three test functions f = 1, id, id^{2}. Then (M_{n}f)_{n∈ℕ} converges to f on [a, b] for all functions in C[a, b].

We provide a short proof due to H.Bauer (see [1]).

Proof. Let f ∈ C[a, b] arbitrary function with |f| < K ∈ ℝ. Using Heine’s theorem for any ϵ > 0 there can be found a number δ > 0 such that, for all x, y ∈ [a, b], |x − y| ≤ √δ implies that |f(x) − f(y)| ≤ ϵ. If

(x − y)^{2} > δ then (x−y)^{2} ; δ > 1, hence |f(x) − f(y)| ≤ 2K ; δ (x − y)^{2}. Let α

denote 2K ; δ. Therefore

|f(x) − f(y)| ≤ ϵ + α(x − y)^{2}

holds for all x, y ∈ [a, b]. If we fix y we obtain

|f − f(y)| ≤ ϵ + α(id − y)^{2} .

The linearity and positivity of M_{n} imply that

|M_{n}f − f(y)M_{n}1| ≤ ϵM_{n}1 + α(M_{n}id^{2} − 2yM_{n}id + y^{2}M_{n}1) .

Let us evaluate this expression at x = y :

|M_{n}f − fM_{n}1| ≤ ϵM_{n}1 + α(M_{n}id^{2} − 2idM_{n}id + id^{2}M_{n}1) .

¿From the assumption on (M_{n}f)_{n∈ℕ} for the three function f = 1, id, id^{2} we obtain that the expression in the bracket converges to zero, therefore the right-side of the inequality converges to ϵ uniformly on [a, b]. Using the triangle inequality in the form:

|M_{n}f − f| − |f − fM_{n}1| ≤ |M_{n}f − fM_{n}1|

it follows that M_{n}f converges to f on [a, b] uniformly.

This theorem obviously can be applied to the Bernstein polynomials, hence we obtain a proof, that B_{n}f converges to f on any f ∈ C[0, 1].

4.2. The Korovkin shadow. Let X be a locally compact Hausdorff space, and denoted by C(X), the all continous real-valued functions defined on X. X is said to be separated by F ⊂ C(X) (or F separates X) if for all x, y ∈ X there exists an f ∈ F such that f(x) ≠ f(y). Given a linear subspace L ⊂ C(X), we denote by L_{b} the following subset of C(X):

L_{b} = {f ∈ C(X) : ∃h_{1}, h_{2} ∈ L such that h_{1} ≤ f ≤ h_{2}} .

Now we can introduce the following definition

Definition 4.1. The Korovkin closure (or shadow) of L is the set of all continous function f ∈ L_{b} having the following property: for every net (M_{i})_{i∈I} of positive linear maps, where M_{i} : L_{b} → L_{b} such that lim_{i∈I} M_{i}h = h pointwise on X for all h ∈ L, then lim_{i∈I} M_{i}f = f pointwise on X. We denote this closure by Kor(L). If L is not linear then Kor(L) is defined as Kor(Lin{L}).

Remark 4.1. The following inclusions are obvious:

L ⊂ Kor(L) ⊂ L_{b} ⊂ C(X) .

Our results are based on the following well-known theorem which was proved by H. Bauer (see [1] and compare with [4]):

Theorem 4.2.

Kor(L) = {f ∈ L_{b} : sup{h ∈ L : h ≤ f} = inf{h ∈ L : h ≥ f}} .

Let the right-side of the equality above be denoted by L̂. Before proving this theorem we shall prove a lemma which we will use in the theorem later.

30 BALÁZS SZENTES

Lemma 4.3. Let g in L_{b}, x in X and c ∈ ℝ such that:

sup{f(x) : g ≥ f ∈ L} ≤ c ≤ inf{f(x) : g ≤ f ∈ L}

Then there exists a μ : L_{b} → ℝ positive linear functional with:

(1) μ(g) = c and

(2) μ(f) = f(x) for all f in L.

Proof. On L_{b} the map p : h → inf{f(x) : h ≤ f ∈ L} is a positive sublinear functional, which obviously majorizes the linear form Λ : λg → cg defined on the linear space generated by g. Therefore the Hahn-Banach Theorem provides a linear form, namely μ, on L_{b} with μ ≤ p. Hence (1) is satisfied by μ and we claim that (2) holds as well. Indeed,

μ(x) ≤ p(h) = h(x) and on the other hand

μ(h) = μ(−(−h)) = −μ(−h) ≥ −p(−h) = h(x)

which shows (2). If f ≤ 0 then p(f) ≤ 0 because 0 ∈ L, considering that μ ≤ p again, so we arrived at the positivity property.

In the proof of the theorem we use a property of the locally compact Hausdorff spaces what is very similar to the complete regularity property. Therefore we prove the following

Lemma 4.4. Let X be a locally compact topological space, and F ⊂ X closed set such that F ⊂ intK for a given compact set K.

Then F and X \ K can be separated by function.

Proof. Assign to each k ∈ K an F_{k} compact set what is a neigbourhood of k. Let U_{k} ⊂ F_{k} be an open neighbourhood of k, arbitrary. Obviously ⋃_{k∈K} U_{k} ⊃ K. Since K is compact we can find finitely many k ∈ K, namely k_{1}, ..., k_{n}, such that

K ⊂ ⋃_{i=1}^{n} U_{k_{i}} .

Moreover

K ⊂ ⋃_{i=1}^{n} intF_{k_{i}}

and the right-side of the inclusion is clearly relatively compact. Let us denote ⋃_{i=1}^{n} F_{k_{i}} by Y. Y is a compact Hausdorff space, therefore it is normal. In Y, F and Y \K can be separated by function, provided by Urüsszon’s Lemma, which might be extended to X continously as constant on X \ Y.

Using the lemmata above the theorem can be proved:

Proof of Theorem 4.2. First we show that Kor(L) ⊃ L̂. Let (M_{i})_{i∈I} an arbitrary net of positive linear maps such that (M_{i}l)_{i∈I} converges to l on X pointwise. We have to show that for any f ∈ L̂ the (M_{i}f)_{i∈I} also converge to f pointwise on X. Let x ∈ X and l_{1}, l_{2} ∈ L such that l_{1} ≤ f but l_{1}(x) > f(x) − ϵ and l_{2} ≥ f but l_{2}(x) < f(x) + ϵ. There exists an i_{0} ∈ I with |M_{i}l_{j}(x) − l_{j}(x)| < ϵ if i ≥ i_{0} and j = 1, 2. By the positivity of M_{i} we know that M_{i}l_{1} ≤ M_{i}f ≤ M_{i}l_{2}, which implies that l_{1} − ϵ ≤ M_{i}f ≤ l_{2} + ϵ. Which means that |M_{i}f(x) − f(x)| < 4ϵ. If ϵ converges to 0 we obtain the convergence of (M_{i}f) in x.

We shall prove that Kor(L) ⊂ L̂. Obviously enough to show that any g ∈ L_{b}\L̂ doesn’t belong to Kor(L). Using the fact that g ∉ L̂ there exist c, x, such as in Lemma 4.3. and μ satisfying (1) and (2). One can find h_{1} and h_{2} in L with h_{1} ≤ g ≤ h_{2} by the definition of L_{b}. h_{2} − h_{1} ≥ 0 and in x srtictly greater than 0 . Multiplying h_{2} − h_{1} by a proper constant we find a h_{0} in L such that h_{0} ≥ 0 and h_{0}(x) > 1. Fix a neighbourhood base B of x in the weak topology induced by C(X) such that the following two property hold:

(i) h_{0}(t) > 1 for any t ∈ U ∈ B and (ii) L bounded on U. To each U ∈ B we assign a q_{U} ∈ C(X) such that (iii) 0 ≤ q_{U} ≤ 1, q_{U}(x) = 1 but q_{U}(t) = 0 for every t ∉ U. Now let us define M_{U} on C(X) as follows:

M_{U}f ≡ μ(f)q_{U} + f − fq_{U} .

M_{U} is clearly positive and we claim that the range of M_{U} lies in L_{b}. Indeed, q_{U} is in L_{b}, h_{0} dominates it, and fq_{U} is in L_{b} too, because of (iii). Finally take the net (M_{U})_{U∈B} positive linear maps. An easy computation shows that (M_{U}h)_{U∈B} converges to h poinwise for all h ∈ L. (At x : M_{U}h(x) = h(x) because μ(h) = h(x) and if x ≠ y there exists an U ∈ B with y ∉ U, so M_{U}h(y) = h(y).) The net (M_{U}(g))_{U∈B} does not converge to g poinwise, because M_{U}g(x) = c and c ≠ g(x).

Remark 4.2. If X is compact and L is a lattice, then for all f ∈ Kor(L) and ϵ > 0 one can find an l_{ϵ}^{f} in L so that:

sup_{x∈X} |f(x) − l_{ϵ}^{f}(x)| ≤ ϵ .

Proof. By the previous theorem, it is easy to verify that

∀x ∈ X, ∀ϵ > 0, ∃l_{ϵ,x}^{f} such that l_{ϵ,x}^{f}(x) − f(x) < ϵ ; 2 and

f(y) ≤ l_{ϵ,x}^{f}(y) for all y ∈ X .

32 BALÁZS SZENTES

Let U_{x} = {y ∈ X : l_{ϵ,x}^{f}(y)−f(y) < ϵ}. It is obvious that X = ⋃_{x∈X} U_{x} because x ∈ U_{x}. Using the compactness argument one can find finitely many x ∈ X denoted by x_{1}, ..., x_{n}, such that

X = ⋃_{i=i}^{n} U_{x_{i}} .

Let us define l_{ϵ}^{f} as follows:

l_{ϵ}^{f}(y) = ⋀_{i=1}^{n} l_{ϵ,x_{i}}^{f}(y) .

This function clearly satisfies the property above.

4.3. Separating a compact set from a single point.

Lemma 4.5. Let X be a topological space and K ⊂ X be a compact set. Let F ⊂ C(X) with the following property:

(⋆) ∀x, y ∈ X∃f ∈ F such that : f(x) = 0 but f(y) ≠ 0 .

Then ∀z ∉ K ∃f ∈ Lin{f^{2} : f ∈ F } such that:

(1) f(z) = 0

(2) f(x) > 1 for all x ∈ K.

Proof. We know from (⋆) that for all x ∈ X there exists an f ∈ F such that, f(z) = 0 but f(x) ≠ 0. So one can find an f_{x}^{2} ∈ Lin{F^{2}} such that f_{x}^{2}(z) = 0 but f_{x}^{2}(x) = 1.5. Let us define U_{x} for all x ∈ K as the following set:

U_{x} = {y ∈ K : |f_{x}^{2}(y) − 1.5| < 1 2} .

U_{x} is obviously an open set in K and

K = ⋃_{x∈K} U_{x} .

Using the compactness argument, there can be found finitely many x ∈ X namely: x_{1}, ..., x_{n} with:

K = ⋃_{i=1}^{n} U_{x_{i}} .

Now we are ready to define f̂, which separates K from z:

f̂(y) = Σ_{i=1}^{n}f_{x_{i}}^{2}(x) .

f̂(z) = 0 because f_{x_{i}}^{2}(z) = 0 for all i = 1, ..., n, and for any x ∈ K there exists an i ∈ {1, ..., n} such that x ∈ U_{x_{i}}, which means:

f̂(x) ≥ f_{x_{i}}^{2}(x) ≥ 1 .

So f̂ satisfies (1) and (2) properties.

Sometimes it can be difficult to check wether property (⋆) holds, therefore the following lemma might be useful.

Lemma 4.6. Let X, K, z as in the previous lemma and let G ⊂ C(X) be a linear space with the following properties:

(i) ∀x ∈ X ∃f ∈ G such that f(x) ≠ 0 (ii) G separates X.

Then there exists an f ∈ Lin{G^{2}, G^{3}, G^{4}} positive function such that (1) f(z) = 0

(2) f(x) ≥ 1 for all x ∈ K.

Proof. We have to proof, that there exists an f_{x,y} ∈ Lin{F ∪F^{2}} which satisfies property (⋆). By (1) and (2) we know that there exist f, f_{y} ∈ F with f(x) ≠ f((y) and f_{y}(y) ≠ 0. We can find c_{1} and c_{2} in ℝ such that:

c_{1}f(x) + c_{2}f_{y}(x) ≠ c_{1}f(y) + c_{2}f_{y}(y) and c_{1}f(y) + c_{2}f_{y}(y) ≠ 0 .

If c_{1}f(x) + c_{2}f_{y}(x) = 0 let us define f_{x,y} as c_{1}f + c_{2}f_{y}, otherwise let us

define f_{x,y} as c_{1}f + c_{2}f_{y} − c_{1}f+c_{2}f_{y} ; c_{1}f(x)+c_{2}f_{y}(x).

Corollary 4.7. If 1_{x} ∈ F and F linear space which separates X then (⋆) holds. In this case Lin{L} can be replaced by Lin{|f| : f ∈ F }, and f_{x_{i}}^{2} by |f_{x_{i}}|.

4.4. Approximation at one point. In this chapter we try to approximate functions at one point so that the approximating function everywhere greater (or smaller) than the function. It can be convenient to introduce the following notation:

(4.1) L_{0} = {g ∈ C(x) : ∃f ∈ F^{+} such that ∀ϵ > 0

{x ∈ X : |g(x)| > ϵf(x)} is relatively compact}.

Here F^{+} denotes the all positive functions in F.

Lemma 4.8. Let X be as in the previous lemmata, and F ⊂ C(X) linear space with the following properties:

(i) ∀x ∈ X ∃f ∈ L_{0} ∩ F^{+} s.t. f(x) ≠ 0 . (ii) ∀x ∈ X \ K, where K is a compact set, ∃f_{x,K} ∈ F^{+} function s.t. f(x) = 0 but f > 0 on K.

Then ∀ϵ > 0 ∀z ∈ X ∀g ∈ L_{0} ∃l ∈ Lin{F } such that:

∀x ∈ X l(x) > g(x) and

l(z) < g(z) + ϵ .

Proof. Let f_{z} be in L_{0} ∩ F^{+} such that f_{z}(z) ≠ 0 by (i). We can assume that g(z) + f_{z}(z) > 0. Now we can define l_{1}(x) as follows:

l_{1}(x) = δ_{1}f_{g}(x) + δ_{2}f_{f_{z}}(x) + δ_{3}f_{z}(x) ,

where f_{g}, f_{f_{z}} correspond to g and f_{z} by the definition of L_{0}, and

δ_{1}f_{g}(z) + δ_{2}f_{f_{z}}(z) < ϵ ; 2 but δ_{3}f_{z}(z) = g(z) + f_{z}(z) + ϵ ; 4. We try to

show that the set of that points where l_{1} < g +f_{z} is relatively compact. Indeed,

K_{1} = {x ∈ X : l_{1}(x) < g(x)+f_{z}(x)} ⊂ {x ∈ X : δ_{1}f_{g}(x) < |g(x)|}∪

{x ∈ X : δ_{2}f_{f_{z}}(x) < (1 + |δ_{3}|)f_{z}(z)} .

The last two sets are relatively compact which means that K_{1} is also relatively compact. Denote the closure of K_{1} by K. Obviously K is compact set and doesn’t contain z. We are ready to define l_{2}:

l_{2}(x) = l_{1}(x) + max_{y∈K}(|g(y)| + f_{z}(y)) · f_{z,K} .

Where f_{z,K} separates K and z, by (ii) and can be found in F^{+}.

(a) In z: l_{2}(z) = l_{1}(z) > g(z) + f_{z}(z), but l_{1}(z) < g(z) + f_{z}(z) + ϵ.

(b) If y ∉ K: l_{2}(y) ≥ l_{1}(y) > g(y) + f_{z}(y).

(c) If y ∈ K: l_{2}(y) > max_{t∈K}(|g(t)| + f(t)) ≥ g(y) + f_{z}(y). Now let us define l as l_{2} − f_{z}. This function obviously satisfies the conditions of theorem.

Remark 4.3. If 1_{X} ∈ L_{0} ∩ F^{+} then (i) is automatically satisfied.

Remark 4.4. According to the previous chapter, if G separates X then the following sets satisfy (ii):

(a) the linear lattice generated by 1_{X} ∪ G

(b) if 1_{X} ∈ G and G is linear then Lin{G^{2}}.

(c) the vector space generated by G^{2} ∪ G^{3} ∪ G^{4} if for all x ∈ X there exists g ∈ G with g(x) ≠ 0.

4.5. The Korovkin closure in some special case. Using the Theorem 4.2. and Lemma 4.8. we always state that L_{0} ⊂ Kor(F) if F satisfies (i) and (ii) in Lemma 4.8.. If X is compact then L_{0} = C(X). Using the notations of Corollary 4.4. we arrive at the following results:

(1) If 0 ∉ X ⊂ ℝ, G = {id_{X}} and F = Lin{id_{X}^{2}, id_{X}^{3}, id_{X}^{4}} then

Kor({id_{X}^{2}, id_{X}^{3}, id_{X}^{4}}) ⊃ {g(x) ∈ C(X) : g(x) = O(x^{4})} .

In the special case, where X is compact : Kor({id_{X}^{2}, id_{X}^{3}, id_{X}^{4}}) = C(X). (2) If X ⊂ ℝ, G = {id_{X}} and F = Lin{1_{X}, id_{X}, id_{X}^{2}} then

Kor({1_{X}, id_{X}, id_{X}^{2}}) ⊃ {g(x) ∈ C(X) : g(x) = O(x^{2})} .

Moreover if X is compact Kor({1_{X}, id_{X}, id_{X}^{2}}) = C(X).

We remark that with only simple modifications, one can replace the pointwise convergence by uniformly convergence on every compact set, and we can obtain the trigonometric version of the Korovkin Theorem as well. Finally by Remark 4.2. we can easily prove the classical Stone-Weierstrass Theorem.

Theorem 4.9. Let X be a locally compact Hausdorff space, and A ⊂ C_{0}(X) a closed algebra with the following properties:

(i) ∀x ∈ X∃f ∈ A s.t. f(x) ≠ 0 and (ii) X is separated by A.

Then A = C(X).

Proof. Let X_{1} = X ∪ ω the one-point compactification of X. Let us extend all f ∈ A into ω with f(ω) = 0. Let F be the algebra generated by A∪1_{X}. In fact on compact spaces the algebras containing a neutral elements are lattices. It means that we can apply Remark 4.2. considering Remark 4.3. and Remark 4.4.. Hence for every h ∈ C(X_{1}) (where h(ω) = 0) and ϵ > 0, one can find f_{1} ∈ F such that

sup_{x∈X_{1}} |f_{1}(x) − h(x)| < ϵ ; 2 .

Especially in ω:|f_{1}(ω)| < ϵ ; 2. By definition of F: f_{1}(x) = f(x) + c where f ∈ A and c is a constant which smaller than ϵ ; 2. Therefore |f_{1}(x) − f(x)| < ϵ ; 2 for all x ∈ X_{1}. Using the triangle inequality sup_{x∈X} |f(x) − h(x)| < ϵ which implies that h ∈ A = A.

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Department of Applied Analysis, Loránd Eötvös University, Budapest, 1088, Múzeum krt. 6-8, Hungary

  E-mail address: szentes@cs.elte.hu