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) €
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 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. Details follow.
From now on, let X be a real-compact space with Cech-Stone compact- ification X*. As noted before, this assumption causes no loss of generality in the present algebraic context. Until further notice, identify each f in CX with its Stone extension from X* to 5, the one point compactification of R, and let its z-set Z(f) and u-set U{f) be the sets of all x in X* with f{x) = 0 and f(x) = °° respectively.
Each subset B of CX partitions X* into closed sets called 3-classes by means of the equivalence under which y ~3 (B) iff f(y) = /(s) for all f in B. One class, the null-set tf(B) of B, is easily recognised: it is simply the intersection of all Z(/) for f in B. Clearly if B 1 c B^, every S^-class is a union of S2 -classes. Moreover, B and its algebraic hull both generate the same partition.
For W c X* and f € CX, let f respect B-classes on W if it vanishes on N(B) fl W and is constant on MOW for all B-classes M. The two best-known hulls both figure in the Stone-Weierstrass theorem: the tight hull th(B) and the free hull fh(B) of B consist of all functions respecting S-classes on X* and on X respectively. The latter is the closure of ah{B) under compact convergence (for all spaces, not just real-compact ones), while if X is compact then
u{ah(B)} = th{B) = fh(B).
Similarly, the J-classes of an ideal I in CX are singletons except perhaps for N(I). Thus
60 u{I) = {/ € CX : Z(/) -3 = th(I), by Nanzetta and Plank (1972), theorem 2.3.
An analogous hull for ru must lie between these two and, very roughly, it should 'ignore u-sets'. Accordingly> consider the simple hull s/z(S) and the limited hull Vh(B') of B: by definition, f € sh(B) if it respects S-classes on X*\U(h) for some h in CX, while f € lh(B) if it respects B fl E{h) -classes on X*\U(h) for some h in CX. Clearly
2.1 fh{B) o sh(B) o lh{B) U th{B), but nevertheless simple and limited hulls often coincide. Some easy cases appear below: most of them involve cut-offs, that is, the functions
(-r v f) A r j where r > 0 and / € CX.
2.2 The simple and limited hulls coincide for any (i) set of bounded functions, (ii) set closed under cut-offs, ( iii) vector-lattice containing the constant functions. ( iv) ideal,
(V) finitely generated algebra, or (vi) singly bounded set.
Proof. For (vi), let B c E[g) and let f respect S-classes on X*\U(h) . Put k = Ig’l v |fi|. As f clearly respects B f! E(k) -classes on X*\U(k), it belongs to lh(B). Thus sh(B) c lh{B) as desired. Sets of types (i) and (v) are both singly bounded. The former all lie in ff(l), while if b i, ..., b^ generate the algebra A then by 1.3, A c E{g) , where g - e 3 and b = IhJ v ... v \b^\.
Next, for (ii), suppose B is closed under cut-offs. Then B generates the same partition on X* as SQ, if BQ = B fl E(l). Thus any member of sh(B) belongs to sfc(Bo), to lh(BQ) by (i) above, and hence to lh(B). Finally, sets of types (iii) and (iv) fall under those
61 of type (ii). a
These hulls 'do the trick' as the next three results show.
2.3 The ru-adherence of any algebra in CX is its limited hull.
Proof. Take an algebra A in CX, and suppose f € ru{A}. Then for some h, some sequence (/ ) in A converges to f under t(h). Put
£7 - I/I v N . so that (/ ) -*■ / under t(g~) . For each r > 0 , the inequality \f -/[ - r9 holds on X for n > n(r), and extends to X*\U(g) by continuity and density. In particular, |/ | < 2g for n > n(l), and (/ ) -*■ / pointwise on X*\U{g). Hence f respects A 0 E{g)-classes on X*\U{g), since / respects them on X* for all n > n(l). Thus f € lh(A).
Conversely, suppose f Q € lh{A) . Then fQ respects A fl E[h)- classes on X*\U{K), for some h. Assume that h > \fQ\: no problem arises, since use of |ft| v |/ | instead just refines the equivalence ~ and increases the u-set. Let W = X*\U[h), BQ = A fl E{h), = ahCB^ , and g = e 1. Then / , h, g and all members of SQ are real-valued on W, and hence lies inside E{g) fl CW f\ A, by 1.3. Further, for each n > 0, the set - {x i W :g(x) 5 n] is closed in X* and hence com pact. Recall too that f respects B a d a s s e s on W. So by the Stone- Weierstrass theorem on CW, some f € A lies within 1/rz of f n on ’ J n o o Kn. Consequently, (/^) f under ru, by 1.4. That is, fQ € ru{A). o
2.4 The ru-adherenae of any vector-lattice in CX containing the constant functions is its simple hull, or equally, its limited hull.
Proof. The argument proving half of 2.3 actually goes through for any subset B of CX at all: thus ru{B} c lh{B~) . Conversely, let V be a vector-lattice containing 1^ and suppose / respects 7-classes on X*\U{K). Put g = |/ | v |7i| v 1_ and W = X*\U(g), and mimic the other line of argument in 2.3, using the lattice version of the Stone-Weierstrass theorem. □
62 2.5 The ru-adherence, the limited hull, and the simple hull of any ideal I in CX all coincide with
{f € CX :N(I)\Z(f) is far from X).
Proof. Use 2.2 and 2.3. Or see Schroder (1979), theorem 3.3, for a different proof using Nanzetta and Plank (1972), theorem 2.3. □
Clearly for any subset of CX, its limited and simple hulls are both lattice-algebras closed under cut-products and cut-offs. (Schroder (1979) defined the cut-product r % s of two complex numbers r and s to be sgn(rs).(|r|a|s)). Like any continuous binary operation, it carries over pointwise to any Cl. In particular, if f is real-valued and r > 0, then f % r_ is the r-cut-off defined earlier.) Thus the ru-adherence of any algebra or vector lattice containing the constant functions has these properties too. Further, the ru-adherence of any ideal is still an ideal. This all happens mainly because ru makes the algebra operations, % and the lattice operations in CY all continuous.
These results suggest several basic questions. (A) Are all four hulls really needed? (B) Are the inclusions in 2.1 ever proper? (C) If so, how big are the differences? (D) Are simple or limited hulls closed under u or even under ru?
Examples settling (A), (B) and (D) are given in §3, and a partial answer to (C) follows. First, the excess of simple over limited hulls is 'small', meaning that for all B c CX,
2.6 lh[B) c sh{.B) c lh(lh(B)).
Proof. Because both lh{B) and th(B) are lattice-algebras closed under cut-offs, so is their intersection V. Now sh{B) c sh{V) ... as V o B - lh{V) ... by 2.2 (iii) c Ihith^B)) ... as tfc(B) o V - sh{th(B)) ... by 2.2 (iii) = 8h(B),
63 as B and th(B) generate the same equivalence ~. (For the same reason, th(B) and th{th{B)) always coincide.) Thus the sets displayed above all coincide, and hence,
lh(B) c sh(B) = lh(V) c lh{lh[B)). a
By the same yardstick, the excess of free over simple hulls, and of simple over tight hulls, can be 'very large'. Because u{B} c th(B) and th{B) = th(th(B)) for all B, tight hulls are always w-closed. So to justify
2.7 th{B) c sh (B) 2.8 sh{B) c fh(B) t sh(8h(B)). Clearly 2.7 and 2.8 answer the easy part of question (A), by showing that tight, simple and free hulls differ significantly. It is harder to separate limited hulls from these three, but not impossible. See 2.1 and 3.7. 3. Ideals and examples To show where relatively uniform convergence fits into the work of Nanzetta and Plank (1972) on ideals, ideal sets and uniform convergence, I sketch some of their ideas below. Consider a real-compact space X and a compact subset K of X*. Also let zs(K) and zn(K) be the sets of all functions vanishing on K, and on a neighbourhood of K (that is, / € zn{K) iff Z(/) is a neighbourhood of K in X*) . Clearly zn(K) is an ideal with null-set K, while Z8(KJ is a u-closed lattice-algebra, as it coincides with its tight hull. Nanzetta and Plank showed in their (2.1) that X is compact iff 64 u{I} is an ideal, for each ideal I in CX, and (2.3) that if K is the null-set of an ideal I then u{I} = zs{K). In particular, their proof of (2 .1) incorporated the lemma below. 3.1 For any unbounded function g, let K = U(g) and I = zn[K). Then u{I] is not an ideal. They called K an ideal set if zs(K) is an ideal and deduced (2.4) that an ideal I is u-closed iff its null-set K is an ideal set and I = zs(K). Further, they introduced the un-named set n {Z{h) fl X : Z{h) d K}, the closures taken in X*. They stated (3.2) that it is a non-empty ideal set, if K is not far from X, but actually proved more. See 3.2 below. In view of this, it deserves a name, the ideal centre ic(K) of K, say. 3.2 The ideal centre of K is the largest ideal set lying in K. In particular, K is an ideal 3et iff ic(K) = K. Because ic{K) = 0 if K is far from X, it makes sense to regard 0 as the improper ideal set, as 3s(0 ) is the improper ideal CX. The link with relatively uniform convergence can now be made: let I be an ideal, J be its ru-adherence, K = W(I) and L - ic(K) . 3.3 In this notation, (i) N(J) is the ideal set L} (ii) the u-closure and the ru-adherence of J both coincide with zs(L), and (H i ) the ideal zs(L) is both u- and ru-closed. Proof. To show N(J) z> L, take f £ J. By 2.5, Z{f) U U{.g) |Af| 3 K for some g in CX. Put h = //(l_+ z(/) ^ z(/) n x u u{g) n x = z{h) n x 3 L, 65 by definition of ideal centre. Thus N(_J) L. Conversely, suppose that i / Z(«) fl X for some h in CX with Z{h) o K. Then (gh)(x) i 0 for some g in CX, by Gillman and Jerison (1960), exercise 7D. Again by 2.5, h € J. Consequently, gh Z J as well, again because J is an ideal, showing that x fL . In short, each point outside L also lies outside N(J), as desired. For (ii) and (iii), two cases arise. First, if K is far from X then L = 0 and J = CX, which leaves nothing to prove. Otherwise, the ideal set L is non-void, and u{J} is the u-closed ideal zs{L), by NP(2.3). Thus by Schroder (1979), theorem 3.4, u{J} is an z>u-closed ideal which clearly contains J. In particular, u{J} = ru{J) = 2s(I) . □ This raises the natural question: is J itself always u-closed? Example 3.8 shows not. Analogous questions for algebras remain open: take a set B in CX, define S° - B and Bn+1 = for all n, and call B an n-set if / b” = Bn+1 or an °°-set if Bn ^ Bn+1 for all n. I guess that n-algebras exist for all n > 2. (By 3.3, no such algebra can be an ideal.) Finally, an example can be given justifying 2.8. NP pointed out that the closure in X* of any subset of X is an ideal set, and showed further (4.2) that X is LindelOf iff every ideal set in X* can be obtained in this way. Accordingly, let X be real-compact but not LindelOf, take an ideal set K in X* such that K t K fl X - L, and put J = 3s(X) . 3.4 For this ideal I, the relation 2.8 holds. Proof. Being u-closed, I coincides with its simple hull, again by Schroder (1979), theorem 3.4. However, its pointwise closure I coincides with its free hull: thus J = {/ € CX : Z(/) fl X ^ K fl X} = zs(L) , whereas I = zs(K). a The rest of this section describes a LindelOf space X, an algebra 66 A in CX whose simple and limited hulls differ, and an ideal I in CX whose ru-adherence is not u-closed. In addition, A dashes my hope of extending Butzmann's continuous approximation theorem unchanged to all algebras. Let R" be the two point compactification of R and let X be the sub-space of R" obtained by omitting all irrationals. Since the function u -*■ arataniu) embeds both R n and X in the real line, X is Lindelof and hence, real-compact. Consider the Stone projections Q* -*■ X* R" induced by the dense inclusions Q -*■ X -*•R", where Q is the rational line. In going from Q* to X*, all unbounded s-ultrafilters are sent as appropriate to ±«. Similarly, let (p) stand for the pre-image of each irrational p in R under the other projection, and note that CX can be identified with the set of all members of CQ having limits as x -*■ +=° and x -°°. Take A to be the algebraic hull of the branches of the tangent function. To be precise, for each integer I, let be the set of all rational numbers u with Jir 5 u 5 Zir + ir, and put Clearly each belongs to CX, and +hn) • Then let A be the algebraic hull of all these f y Butzmann (1979), Korollar 1.1, in his approximation theorem for con tinuous convergence a, implicitly defined a hull operator bh and proved that o{L} = bh(L) for every locally bounded algebra L in CX. (Of course, Butzmann's result holds for all spaces, not just real-compact ones.) By its geometrical definition, bh is coarser than th. This leads to the diagram below: 3.5 bh(A) ^ a{A) 3 ru{A} U ii th[_A) c sh(A) d lh(A). 67 3.6 Butzmann's hull operator bh does not yield the c-adherence of every algebra. 3.7 .Simple and limited hulls do not always coincide. Proof. Both 3.6 and 3.7 are proved by finding a function belonging to th(A)\c{A} and then using 3.5. The null-set K of A is clearly {0, +<*>,-<»} together with the classes (Zit) for all non-zero integers I. The other A-classes in X* are the singletons {<7} for all non-zero rational q, and (p) for irrational p except multiples of it. Let H(u) - sin(u)/(1+u2) for all u in R ". Since H € CR", its restriction f to X belongs not just to CX, but even to th(A). By definition (see Binz (1975), say), a filter $ on CX converges continuously to f iff ^(vu) f W in R for all u in X, where is the neighbourhood filter of u in X. Suppose / € c{A), so that A € & for some Because fmfn - £. if m t n > each g in A can be written as (/j), where the g^ are all polynomials in one variable, with no constant term. Naturally, almost all of them are zero: then f^ is called an entry in g if the corresponding g^ is non-zero. For each integer I, there is some g in F in which f^ enters. To see this, consider the support I^ of f^ in X, and take a non-zero q in 1^. As On the other hand, because all non-constant polynomials in f^ are unbounded on and because U 1 1^ for all sufficiently large I, the set F{U) is unbounded. This contradiction shows that / cannot belong to c{A}. □ 68 The ideal I is just as easily described. Take the same space X as above, let K = {0,+”,-=} together with the classes (Zir) for all non zero integers I, and put I = zn{K). 3.8 The ru-adherence J of I is not u-closed. Proof. Clearly N(J) o (0,+<*,-<*>} ~ L, say. For each non-zero integer I and each x in X, let g Then g^ € CX, and Z(g^) = K\(lt\). Thus by 2.5, each g^ belongs to J, showing that N(J) c L. All in all, L is the null-set of J (and the ideal centre of K too). Now put g(x) = x/(l +x2) for all x in X. Clearly Z{_g) = L, and so g € u{J}, by NP (2.3). But g f. J , since Z (g) U U(h) contains K iff the compact u-set U{h) includes each (Zi:), and hence meets X at ±°°. □ 4. Polynomial approximation in Euclidean space Consider a sub-space V of fT, and let P be the algebra generated in CV by ^ and the projections v^, __• vm ^rom ^ t0 Which members of CV can be rw-approximated by polynomials? This question is fully answered in §2, by using P-classes in V* to describe ru{P}. But clearly the answer should be translated into more concrete Euclidean terms, if possible. It is. For clarity, I must now distinguish notationally between a function and its extensions: in particular, I stop identifying f in CV with its Stone extension f* : V* -*■ 5. Now put h = 1_ + v \ * + um2’ *et V be the closure of V in 5^, and let s : V* -*• V be the usual quotient map, namely, the Stone extension of the inclusion of V into V. The equivalences on V* defined by s and P are closely related: in fact, they agree outside U(h*). 69 4.1 In the notation of %2, (i) y ~z(P) implies y~2 (3), (ii) s (£/(&*)) = VMp, (iii) U(h*) is s-saturated, that is, it is a union of s-olasses, (iv) if y ~ s(s) and h*{y) i <*> then y ~3 (P), and fy) a P-olass or s-class meets V, it is a singleton. Proof. To start with, let w , ..., w^ be the projections from V to 5. Clearly h*(y) = 00 iff u*(y) = 00 for some j, and further, 0 v*. = w.o s : V* ->■ S for all j. The proof of these, and of their corona te 3 ries (i), ..., (v) above, is left to the reader. a For any f in CV, let Py be the union of all P-classes on which f* is not constant, and define £y to be the set of x in F (1 i/” such that /(u) does not converge to a limit in R as u x with u € V. 4.2 The following are equivalent: (i) f t ru{P}, (ii) Pj. is relatively aompaot in V*\V, and (iii) Ej. is relatively aompaot in 7\7. Proof. "(i) = (ii)" Let (p^) be a sequence in P converging to f under some t(,g) . As £/(/*) U U(g*) U U{h*) is compact in 7*\7, so is its P-saturated hull G, by 4.1. Let X = V*\G. Then each v*. and J consequently, each p* too, is real-valued on X. Further, if a ~ &(P) and a € X then (i) b € X, because X is P-saturated, and (ii) /*(a) = f*[b), because p* -*■ f* pointwise on X and P*(a) = P*(£) for all n. Thus every P-class on which f* is not constant lies outside X (and hence inside G). In short, P^, is relatively compact in V*\V. "(ii) =» (iii)" Suppose Pj. is relatively compact in 7*\7. Much as above, let G be the s-saturated hull of the closure of Pj, U U(f*) U Uih?). Then G is compact in 7*\7. Put X - V*\G and W = s(X). By its definition and by 4.1, W c v fl if1. Also, because X is open and s-saturated in V*, its image W is open in V, and the topology that W inherits as a sub-space of V, coincides with the quotient topology induced by the map s\X : X -+ W. 70 By (ii) and 4.1, /*|X is constant on s-classes, and so it defines a map : W -*■ R which extends f. Moreover, is continuous, from the quotient property of W mentioned above. Thus W misses Ej., or in other words, lies inside the compact set V\W. "(iii) =» (i)" Assume (iii) and choose a set W, open in V fl l/n, containing V, and missing E ~ By definition of E„, for all x in W, J I v the limit of f[u) as u ■* x with u € V, is a real number, j (x) say. The resulting function : W -*■ R extends / and is continuous on W, since V is dense in W and R is regular. Let X be the inverse image of W under s, so that X is an open s-saturated over-set of V in V*, and let / = (/) o (s|l). Since f* :X * R is a continuous extension of /, it coincides with the restriction of f* to X. As a result, f* respects s-classes on X. But because V is LindelOf, the compact set V*\X is far from V, lying in some u-set U{g), by Binz (1975), proposition 83. Thus / € s/j(p) = lh(P) = ru{P}, by 4.1 and definition, by 2.2 (v) and by 2.3 respectively. Note: X* = V*, because V c X c V*. Note: The union Sj. of all s-classes on which f* is not constant can be used equivalently instead of Pj. in 4.2 (ii). So can the analogous set obtained from the quotient map of V* onto the closure of V in the one point compactification of if1. 4.3 The polynomials are 2*u-dense in CV iff V is locally compact. Proof. If V is locally compact then V*\V itself is compact, and so 4.2 (ii) holds for all f in CV. Conversely, suppose V is not locally compact, and let L be its closure in if1. Take a point x of V with no compact neighbourhood in V, and let : n € N} be a base of compact neighbourhoods of x in the locally compact space L. Now choose -*-et 3n as defined in 4.4 below, using v = and put f - \gn/2n . Then (i) f € CV, (ii) each un 6 E but (iii) E^ is not relatively compact in V\V, since x belongs to its closure and to V as well. In all, f £ ru{P}. □ 71 4.4 Let V c FT, and let L be its closure in as above. Suppose v € L\V. Then there is g € CV such that (i) \g\ 5 1, and (ii) g(u) has no limit, as u -*■ v with u € V. Proof. Take a sequence of points in V, such that (2? ) decreases to 0 in R, where rn = \v^~v\. Now define h by linear interpolation on (0 ,2^] so that ^(rn) = n > an(* otherwise, let h(r) = r^/r, for r > 2^. So defined, h is continuous and decreasing on (0,“). Now put g[u) = sin(h(\u -u|)) for all u € V, and verify that g has the proper ties claimed above. □ Comparing 4.3 with Feldman and Porter (1981), example 3, one might see a contradiction: in 4.3, the polynomials are dense in CR while in example 3, they lie in an z>u-closed proper subset V of CR The explanation lies in the different forms of ru used: I used all members of CR to define ru, while Feldman and Porter used only the members of V. REFERENCES 1. E. Binz (1975), Continuous convergence on C(X), Lecture Notes in Mathematics 469, Springer, Berlin. 2. H.-P. Butzmann (1974), Habilitationsschrift, Universitat Mannheim. 3. H.-P. Butzmann (1979), 'Der Satz von Stone-Weierstrass in und seine Anwendung auf der Darstellungstheorie von Limesalgebren', Math. Nachr. 93, 75-102. 4. W.A. Feldman and J.F. Porter (1980), 'The relatively uniform conver gence structure', 46-55 in Proceedings of the conference on convergence structures, Cameron University, Lawton, Oklahoma. 5. W.A. Feldman and J.F. 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