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Relatively Uniform Convergence and Stone-Weierstrass Approximation

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Relatively Uniform Convergence and Stone-Weierstrass Approximation RELATIVELY UNIFORM CONVERGENCE AND STONE-WEIERSTRASS APPROXIMATION M. Schroder (received 21 February 1985) Introduction Each form of convergence on CY, the set of all continuous real­ valued functions on the space Y, creates its own approximation problem: namely, of describing geometrically the closure of 'nice' subsets of CY such as lattices, algebras or ideals. This problem is well understood in the cases of uniform, compact and continuous convergence. For uniform convergence, it is solved by the basic Stone-Weierstrass theorem, at least for compact spaces. Nanzetta and Plank (1972) took the next step and characterized the uniform closure of ideals in CX, for any real-compact space X. (Since uniform convergence is determined by CY, their result remains completely general and algebraic. In fact, by Gillman and Jerison (1960), each C Y can be realized as C X, where X is the v u u real-compact space HomJCY.) But further extension to vector-lattices or algebras in CX involves their order properties explicitly, a seemingly inevitable drawback. These difficulties simply do not arise for compact convergence, because the Stone-Weierstrass theorem looks after all cases of interest. But against this, its algebraic nature is lost, as compact convergence can seldom be constructed from CY alone. Still less can continuous convergence, a sibling of compact conver­ gence, be created solely from CY. Indeed, through its universal and categorial properties, it sheds even more light on the duality between Y and CY than compact convergence does. But many details of this duality would have remained obscure without an approximation theory. This emerged from E. Binz's work on ideals in 1969 and on certain closed algebras in Math. Chronicle 14(1985), 55-73. 55 1970, to be completed by H.-P. Butzmann in 1974, at least for locally- bounded algebras. See Binz (1975) and Butzmann (1979). However, very little seems to have been written about relatively uniform convergence, an entirely algebraic construct akin to uniform convergence. For instance, Peressini (1967) defined it, and Feldman and Porter (1980) and (1981) used it. Because it seemed to offer a way of lifting the limitation on Butzmann's result, Schroder (1979) began to study its general topological properties (the mistakes noted in Schroder (1982) do not matter here) . These properties encouraged me to attack its approximation problem, and this paper resulted. It should place few demands on the reader beyond some general know­ ledge of Cech-Stone compactifications (see Gillman and Jerison (1960), for instance), as the convergence discussed here is mostly defined by pseudo- metrics. In any case, Binz (1975) gives an adequate summary of convergence in general. Before sketching the paper's layout, I give some abbreviations and make an obvious remark. As uniform, relatively uniform, continuous and compact convergence stand in that order, the coarsest last (so that u > ru > a > k, for short), their approximation theorems ought to agree for compact spaces, when u - k. They do agree. Besides introducing ru, §1 describes a device of Butzmann's which clips onto the Stone-Weierstrass theorem to give a quick proof of the main result in §2. (The reader will be as grateful as I am to Butzmann for his neat idea, which supplants pages of unintelligible calculation needed in early drafts.) For algebras, this result involves their order properties explicitly: an example in §3 shows the depth of this 'defect'. Mainly though, §3 links up with the work of others. For instance, the same example also shows that Butzmann's continuous approximation theorem cannot be extended to all algebras without change. Further, some slightly loose ends in Nanzetta and Plank (1972) are tidied up. The special case of polynomials on Euclidean spaces occupies §4. To get the picture, take a sub-space V of if1 and note that (i) u and ru coincide on CV iff V is compact, by Schroder (1979), theorem 4.3, 56 (ii) the polynomials are u-closed in ClF1, but (iii) they are dense in CV under both c and k. Here too, ru lies between uniform and con­ tinuous convergence, as the polynomials turn out to be rw-dense in CV iff V is locally compact. Besides my mathematical debt, I owe many thanks to H.-P. Butzmann and E. Binz for making my 1981 stay in Mannheim so enjoyable, and to Waikato University for the leave which made it possible. 1. Relatively uniform convergence For typographical convenience and to avoid ambiguity, a clear but non­ standard notation is used for adherence and closure. Namely, if q is a form of convergence on a set Q and if B c Q, then q{B} stands for its q-adherence, the set of all x in Q such that 3 belongs to some filter converging to x under q. The term 'closure' is reserved for the adherence obtained as above from a topology: it then coincides with the usual idea of closure. The easiest way to display the algebraic nature of relatively uniform convergence, ru, is to define it in an algebraic context. So let V be a vector-lattice over the real line R. Given g in V, let B{g) be the absolutely- and order-convex set of all e in V with \e\ 5 \g\ , and let E(g) be the linear space spanned by B(g). Then the Minkowski pseudo-metric obtained from B(g) generates a group topology t{g) on V whose restriction to E{g) converts it into a semi-normed vector lattice. According to Peressini (1967), a sequence (/ ) in V converges relatively uniformly to fQ, if for some real sequence (r ) decreasing to 0 and for some g in V, |/n -/Q | - for all n. In other words, a sequence or more generally, a filter, converges to some limit under ru . iff it converges to the same limit under t(g), for some g in V. These definitions lead directly to the simple description of ru-adherence below. 57 1.1 The ru-adherence of any subset of V is the union of its t(g)~ closures, as g runs through V. Unaware of ru, Schroder (1979) defined mod-fine convergence as the convergence inductive limit of the topologies t(g) restricted to E(g). More concretely, a filter ^ on 7 converges to f iff f € E{g) € <t> and <j> -*• / under t(_g), for some g in V. In particular, <j> f under ru. Conversely, each rw-convergent filter also mod-fine converges to the same limit, largely because E{g) is always t(^)-open in V. Hence, 1.2 mod-fine and relatively uniform convergence coincide. All this remains true for the mod-spaces of Schroder (1979), which are simply linear spaces over the real or complex fields, together with a modulus. Now that the god of generality has been appeased in the cause of algebra, fix a space Y and consider the lattice-algebra CY. For each B c CY, let ah(B) be its algebraic hull, the smallest overset of B closed under addition, multiplication and scalar multiplication (thus ah{B) contains all the constant functions iff B contains the constant function r for some non-zero r). Just as Butzmann's continuous approximation theorem calls for local boundedness, ru-approximation demands something similar too: a subset of CY is called singly bounded if it lies inside E{h) for some h in CY. 1.3 The algebraic hull of any singly bounded set remains singly bounded. Proof. Suppose B c E(h), and let A = ah(B). Since E(h) = ffd^l), suppose /i i 0 as well. Then A c ff(e^), essentially because the exponential swamps all polynomials. To be precise, a typical member g of A is obtained from some real polynomial c(u) = £c^u1 in n varia­ bles say (the sum ranging over a finite collection of multi-indices ct) by substituting the members /n of B for the variables u^, ...,u^. 58 By assumption, y ^ ..., y^ exist so that |/\| 5 v Ji for all i. Now suppose that |u.| < v ,r for all i, and note that \o{u)\ 5 ^ | | y ar^a ^, a polynomial function pO), say. By elementary calculus, 0 < p(r) 5 me on R*, for some m > 0. Consequently, Ig\ = k(£) I S p(h) < meh , h so that g (• E(e ) as desired. □ Next comes Butzmann's device, clearly tailor-made to fit round the Stone-Weierstrass theorem. 1.4 Let 1_ < g € CY, and put = {x € Y :g(x) 2 n} for all n > 0. Suppose also that (i) fn ^ for all n > 0, and (ii) 1/ -f„| 2 /n1 on K for all n > 0. ** n * o 1 n Then (/ ) -*■ fa under ru Proof. For each n > 0, choose a number b with |/ | 5 b^a. Construct (by linear interpolation, for instance) a continuous increasing function b : i?+ R+ such that b(0) = 1 and (Hi) b{n) > for all n. Now let g be the product of g and b o g in CY. By construction, g* > g > 1_. Thus if x € K , then l/n («) - !/« -9*(x)/n, while otherwise, x f. K and * n l/n (*) -/„(*) I S 2(>n+ bQ)g(x) I— g{x)b{n)/n, by (iii) < ^(x)Z?(^(x))/n, because fe increases and <?(x) > n. In short,- |/ -/0 | i j /n, showing 59 that (/ ) converges to fQ under both t{g ) and ru. □ 2. Geometrical hulls The approximation theorems here, like those mentioned in the intro­ duction, use the equivalence defined by B on (an extension of) Y to create a geometrical hull consisting of functions more or less constant on equivalence classes.
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