Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds

arXiv:math/0203237v1 [math.DG] 22 Mar 2002 UNIFORM APPROXIMATION OF CONTINUOUS MAPPINGS BY SMOOTH MAPPINGS WITH NO CRITICAL POINTS ON HILBERT MANIFOLDS DANIEL AZAGRA AND MANUEL CEPEDELLO BOISO Abstract. We prove that every continuous mapping from a separable infinitedimensional Hilbert space Xinto Rmcan be uniformly approximated by C∞ smooth mappings with no critical points. This kind of result can be regarded as a sort of very strong approximate version of the Morse-Sard theorem. Some consequences of the main theorem are as follows. Every two disjoint closed subsets of Xcan be separated by a one-codimensional smooth manifold which is a level set of a smooth function with no critical points; this fact may be viewed as a nonlinear analogue of the geometrical version of the Hahn-Banach theorem. In particular, every closed set in Xcan be uniformly approximated by open sets whose boundaries are C∞smooth one-codimensional submanifolds of X. Finally, since every Hilbert manifold is diffeomorphic to an open subset of the Hilbert space, all of these results still hold if one replaces the Hilbert space Xwith any smooth manifold Mmodelled on X. 1. Introduction and main results A fundamental result in differential topology and analysis is the Morse-Sard theorem [19, 20], which states that if f:Rn−→ Rmis a Crsmooth function, with r > max{n−m, 0}, and Cfstands for the set of critical points of f(that is, the points xat which the differential df(x) is not surjective), then the set of critical values, f(Cf), is of (Lebesgue) measure zero in Rm. This result also holds true for smooth functions f:X−→ Ybetween two smooth manifolds of dimensions nand mrespectively. Several authors have dealt with the question as to what extent one can obtain a similar result for infinite-dimensional spaces or manifolds modelled on such spaces. Let us recall some of their results. Smale [22] proved that if Xand Yare separable connected smooth manifolds modelled on Banach spaces and f:X−→ Yis a CrFredholm map (that is, every differential df(x) is a Fredholm operator between the corresponding tangent spaces) then f(Cf) is meager, and in particular f(Cf) has no interior points, provided that r > max{index(df(x)),0}for all x∈X; here index(df(x)) stands for the index of the Fredholm operator df(x), that is, the difference between the dimension of the kernel Date: March 22, 2002. The first-named author was supported by a Marie Curie Fellowship of the European Community Training and Mobility of Researchers Programme under contract number HPMF-CT-2001-01175. 1 2 DANIEL AZAGRA AND MANUEL CEPEDELLO BOISO of df(x) and the codimension of the image of df(x), which are both finite. However, these assumptions are quite restrictive: for instance, if Xis infinite-dimensional then there is no Fredholm map f:X−→ R. In general, the existence of a Fredholm map ffrom a manifold Xinto another manifold Yimplies that Yis infinite-dimensional whenever Xis. On the other hand, one cannot dream of extending the Morse-Sard theorem to infinite dimensions without imposing strong restrictions. Indeed, as shown by Kupka’s counterexample [16], there are C∞smooth functions f:X−→ R, where Xis a Hilbert space, so that their sets of critical values f(Cf) contain intervals and in particular have non-empty interior. More recently, S. M. Bates has carried out a deep study concerning the sharpness of the hypothesis of the Morse-Sard theorem and the geometry of the sets of critical values of smooth functions. In particular he has shown that the above Crsmoothness hypothesis in the statement of the Morse-Sard theorem can be weakened to Cr−1,1; see [3, 4, 5, 6, 7]. C. G. Moreira and Bates have studied some generalizations of the Morse-Sard theorem related to Hausdorff measures and Hausdorff dimensions. They have also shown that the function fas in Kukpa’s counterexample can even be assumed to be a polymonial of degree three; see [8, 17]. Nevertheless, for many applications of the Morse-Sard theorem, it is often enough to know that any given continuous function can be uniformly approximated by a map whose set of critical values has empty interior. In this direction, Eells and McAlpin established the following theorem [14]: if Xis a separable Hilbert space, then every continuous function from Xinto Rcan be uniformly approximated by a smooth function fwhose set of critical values f(Cf) is of measure zero. This allowed them to deduce a version of this theorem for mappings between smooth manifolds Mand Nmodelled on Xand a Banach space Frespectively, which they called an approximate Morse-Sard theorem: every continuous mapping from Minto Ncan be uniformly approximated by a smooth function f:M−→ Nso that f(Cf) has empty interior. However, this seemingly much more general version of the result is a bit tricky: indeed, as they already observed ([14], Remark 3A), when Fis infinitedimensional, the function fthey obtain satisfies that Cf=M, although f(M) has empty interior in N. Unfortunately, even though all the results of that paper seem to be true, some of the proofs are not correct. In this paper we will prove a much stronger result: if Mis a C∞smooth manifold modelled on a separable infinite-dimensional Hilbert space X(in the sequel such a manifold will be called a Hilbert manifold), then every continuous mapping from Minto Rmcan be uniformly approximated by C∞smooth mappings with no critical points. This kind of result might be regarded as the strongest possible one of any class of approximate Morse-Sard theorems, when the target space is finite-dimensional. As a by-product we also obtain the following: for every open set Uin a separable Hilbert manifold Mthere is a C∞smooth function fwhose support is the closure of Uand so that df(x)6= 0 for every x∈U. This result could be summed up by saying that for every open subset Uof Mthere is a function fwhose open support SMOOTH APPROXIMATION BY MAPPINGS WITH NO CRITICAL POINTS 3 is Uand which does not satisfy Rolle’s theorem; one should compare this result with the main theorem from [2] (see also the references therein). Either of these results has in turn interesting consequences related to smooth approximation and separation of closed sets. For instance, every two disjoint closed subsets in Mcan be separated by a smooth one-codimensional submanifold of M which is a level set of a smooth function with no critical points. This may be regarded as a nonlinear analogue of the geometrical version of the Hahn-Banach theorem. In particular, every closed subset of Mcan be uniformly approximated by open sets whose boundaries are smooth one-codimensional submanifolds of M. So far these are some good consequences of our main result, all of them somehow related to Morse-Sard type theorems. But there are some bad consequences as well, perhaps the most noticeable one being that, since the set of smooth functions with no critical points is dense in the set of continuous functions defined on a Hilbert manifold, there are quite large sets of smooth functions for which no conceivable Morse theory could be valid. Let us now formally state our main results. For the sake of a convenient notation in our proofs, when ϕtakes real values we indistinctly use the symbols dϕ(x) = ϕ′(x) to denote the derivative of ϕat a point x, and we reserve dϕ(x) for the derivative of a vector-valued function ϕ:M−→ Rmat a point x∈M. Theorem 1.1. Let Ube an open subset of a separable infinite-dimensional Hilbert space X. Then, for every continuous mapping f:U−→ Rmand for every continuous positive function ε:U−→ (0,+∞), there exists a C∞smooth mapping ψ:U−→ Rmsuch that kf(x)−ψ(x)k ≤ ε(x)and dψ(x)is surjective for all x∈X (that is, ψhas no critical points). We will prove this result in the following section. Let us now establish the announced consequences of Theorem 1.1. One could adapt the ideas in our proof to extend Theorem 1.1 to the setting of Hilbert manifolds but, for simplicity, we will instead use another approach. Indeed, bearing in mind a fundamental result on Hilbert manifolds due to Eells and Elworthy [13] that every separable Hilbert manifold can be C∞embedded as an open subset of the Hilbert space, it is a triviality to observe that Theorem 1.1 still holds if we replace Uwith a a separable Hilbert manifold. Theorem 1.2. Let Mbe a separable Hilbert manifold. Then, for every continuous mapping f:M−→ Rmand every continuous positive function ε:M−→ (0,+∞), there exists a C∞smooth mapping ψ:M−→ Rmsuch that ψhas no critical points and kf(x)−ψ(x)k ≤ ε(x)for all x∈X. Proof. According to the main theorem of [13], there is a C∞embedding of Monto an open subset of the Hilbert space X. Therefore Mis C∞diffeomorphic to an open subset Uof X; let h:U−→ Mbe such a C∞diffeomorphism. Consider the continuous mappings g=f◦h:U−→ Rmand δ=ε◦h:U−→ (0,+∞). By Theorem 1.1 there is a C∞smooth function ϕ:U−→ Rmso that ϕhas no critical 4 DANIEL AZAGRA AND MANUEL CEPEDELLO BOISO points, and kg(y)−ϕ(y)k ≤ δ(y) for all y∈U. Now define ψ=ϕ◦h−1:M−→ Rm. Since his a diffeomorphism it is clear that htakes the critical set of ψonto the critical set of ϕ=ψ◦h. But, as the latter is empty, so is the former; that is, ψhas no critical points either. On the other hand, we have that kf(x)−ψ(x)k=kg(h−1(x)) −ϕ(h−1(x))k ≤ δ(h−1(x)) = ε(x) for all x∈M. As an easy corollary we can deduce our promised nonlinear version of the geometrical Hahn-Banach theorem. We will say that an open subset Uof a Hilbert manifold Mis smooth provided that its boundary ∂U is a smooth one-codimensional submanifold of M. Corollary 1.3. Let Mbe a separable Hilbert manifold. Then, for every two disjoint closed subsets C1,C2of M, there exists a C∞smooth function ϕ:M−→ Rwith no critical points, such that the level set N=ϕ−1(0) is a 1-codimensional C∞ smooth submanifold of Mthat separates C1and C2, in the following sense. Define U1={x∈M:ϕ(x)<0}and U2={x∈M:ϕ(x)>0}; then U1and U2are disjoint C∞smooth open sets of Mwith common boundary ∂U1=∂U2=N, so that Ci⊂Uifor i= 1,2. Proof. By Urysohn’s lemma there exists a continuous function f:M−→ [0,1] so that C1⊂f−1(0) and C2⊂f−1(1). Taking ε= 1/3 and applying Theorem 1.2 we get a C∞smooth function ψ:M−→ Rwhich has no critical points and is so that |f(x)−ψ(x)| ≤ 1/3 for all x∈M; in particular C1⊆f−1(0) ⊆ψ−1(−∞,1/2) := U1, and C2⊆f−1(1) ⊆ψ−1(1/2,+∞) := U2. The open sets U1and U2are smooth because their common boundary N=ψ−1(1/2) is a smooth one-codimensional submanifold of M(thanks to the implicit function theorem and the fact that dψ(x)6= 0 for all x∈N). In order to obtain the result in the above form it is enough to set ϕ=ψ−1/2. A trivial consequence of this result is that every closed subset of Mcan be uniformly approximated by smooth open subsets of M. In fact, Corollary 1.4. Every closed subset of a separable Hilbert manifold Mcan be approximated by smooth open subsets of M, in the following sense: for every closed set C⊂Mand every open set Wcontaining Cthere is a C∞smooth open set Uso that C⊂U⊆W. SMOOTH APPROXIMATION BY MAPPINGS WITH NO CRITICAL POINTS 5 Finally, the following result, which also implies the above corollary, tells us that for every open set Uin Mthere always exists a function whose open support is U and which does not satisfy Rolle’s theorem. Theorem 1.5. For every open subset Uof a Hilbert manifold Mthere is a continuous function fon Mwhose support is the closure of U, so that fis C∞smooth on Uand yet fhas no critical point in U. Proof. For the same reasons as in the proof of Theorem 1.2 we may assume that Uis an open subset of the Hilbert space X=ℓ2. Let ε:X−→ [0,+∞) be the distance function to X\U, that is, ε(x) = dist(x, X \U) = inf{kx−yk:y∈X\U}. The function εis continuous on Xand satisfies that ε(x)>0 if and only if x∈U. According to Theorem 1.1, and setting f(x) = 2ε(x), there exists a C∞smooth function ψ:U−→ Rwhich has no critical points on U, and such that ε-approximates fon U, that is, |2ε(x)−ψ(x)| ≤ ε(x) for all x∈U. This inequality implies that lim x→zψ(x) = 0 for every z∈∂U. Therefore, if we set ψ= 0 on X\U, the extended function ψ:X−→ [0,+∞) is continuous on the whole of X, is C∞smooth on Uand has no critical points on U. On the other hand, ψ(x)≥ε(x)>0 for all x∈U, hence the support of ψis U. 2. Proof of the main result The main idea behind the proof of Theorem 1.1 is as follows. First we use a perturbed smooth partition of unity to approximate the given continuous mapping f. The summands of this perturbed partition of unity are functions supported on scalloped balls and carefully constructed in such a way that the critical set Cϕof the approximating sum ϕis locally compact. Then we have to eliminate all the critical points without losing much of the approximation. To this end we compose the approximating mapping ϕwith a deleting diffeomorphism h:X−→ X\Cϕwhich extracts the critical points Cϕand is as close to the identity as we want. The existence of such a diffeomorphism is guaranteed by a quite elaborated result of West’s [23]. In this way we obtain a smooth mapping ψwhich has no critical points, and which happens to approximate the function ϕ (which in turn approximates the original f) because the perturbation brought on ϕ by the composition with his not very important (recall that his arbitrarily closed to the identity). The following proposition shows the existence of a function ϕwith the above properties. Recall that Cϕstands for the set of critical points of ϕ. 6 DANIEL AZAGRA AND MANUEL CEPEDELLO BOISO Proposition 2.1. Let Ube an open subset of the separable Hilbert space X. Let f:U−→ Rmbe a continuous mapping, and ε:U−→ (0,∞)a continuous positive function. Then there exist a C∞smooth mapping ϕ:U−→ Rmso that (a) Cϕis locally compact and closed (relatively to U); (b) kϕ(x)−f(x)k ≤ ε(x)/2for all x∈U. In fact, when the function ftakes values in the real line, we can obtain a much stronger result which is interesting in itself and might have some applications beyond the problem we are dealing with, as it provides much more accurate information about the structure and location of the critical points of the approximation ϕ. The following theorem shows that any continuous function can be uniformly approximated by C∞smooth functions whose sets of critical points consist of countable union of compact sets which are separated by pairwise disjoint arbitrarily small open sets. Theorem 2.2. Let Ube an open subset of the separable Hilbert space X. Let f: U−→ Rbe a continuous function on X, and ε:U−→ (0,∞)a continuous positive function. Then there exist a C∞smooth function ϕ:U−→ R, sequences (Kn)and (Un)of compact sets and open sets respectively, and a sequence (B(yn, rn)) of open balls which are contained in Uand whose union covers U, such that: (a) Cϕ⊆S∞ n=1 Kn; (b) Kn⊂Un⊆B(yn, rn)for all n, and Un∩Um=∅whenever n6=m; (c) |ϕ(x)−f(x)| ≤ ε(x)for all x, and |ϕ(y)−f(x)| ≤ ε(yn)for every x, y ∈ B(yn, rn)and every n; (d) for every x∈Uthere exists an open neighborhood Vxof xsuch that, either Vx=Unfor a unique n=nx, or else Vx∩Un=∅for all n. Moreover, for any given r > 0, the radii of the balls can be chosen so that rn≤r for all n. Finally, the following restatement of a striking result of West’s [23] ensures the existence of the diffeomorphism h. We say that a mapping gfrom a subset Aof M is limited by an open cover Gof Mif the collection {{x, g(x)}:x∈A}refines G. Theorem 2.3 (West).Let Cbe a closed, locally compact subset of a Hilbert manifold M,Uan open subset of Mwith C⊂U, and Gan open cover of M. Then there is a C∞diffeomorphism hof Monto M\Cwhich is the identity outside U and is limited by G. Assume for a while that Proposition 2.1 is already established, and let us see how we can deduce Theorem 1.1. Proof of Theorem 1.1 For the given continuous mappings fand ε, take a mapping ϕwith the properties of Proposition 2.1. Since ϕand εare continuous, for every z∈Uthere exists δz>0 so that if x, y ∈B(z, δz) then kϕ(y)−ϕ(x)k ≤ ε(z)/4≤ε(x)/2. SMOOTH APPROXIMATION BY MAPPINGS WITH NO CRITICAL POINTS 7 Let G={B(x, δx) : x∈U},M=U, and for the critical set C=Cϕ, use Theorem 2.3 to find a C∞diffeomorphism h:U−→ U\Cso that his limited by G. Define ψ=ϕ◦h. Since his limited by Gwe have that, for any given x∈U, there exists z∈U such that x, h(x)∈B(z, δz), and therefore kϕ(h(x)) −ϕ(x)k ≤ ε(z)/4,that is, we have that kψ(x)−ϕ(x)k ≤ ε(z)/4≤ε(x)/2. Hence, by combining this inequality with (b) of Proposition 2.1, we obtain that kψ(x)−f(x)k ≤ ε(x) (1) for all x∈U. Let us see that ψdoes not have any critical point. The derivative of ψis given by dψ(x) = dϕ(h(x)) ◦dh(x).(2) Since h(x)/∈C=Cϕ, we have that the linear map dϕ(h(x)) is surjective. On the other hand dh(x) is a linear isomorphism (because his a diffeomorphism). Then it is clear that the composition dψ(x) = dϕ(h(x)) ◦dh(x) is a linear surjection from X onto Rm, for every x∈U. Remark 2.4. In the case when f:U−→ Rwe do not need to use the full power of West’s result. Thanks to the more accurate statement provided by Theorem 2.2 we can instead use a much more elementary result that tells us that for every compact subset Kand every open subset Uof Xwith K⊂U, there exists a C∞diffeomorphism h:X−→ X\Ksuch that hrestricts to the identity outside U. In our case, to eliminate the critical points of the approximating function ϕ of Theorem 2.2, we may compose ϕwith a sequence of deleting diffeomorphisms hn:X−→ X\Knwhich extract each of the compact sets of critical points Knand restrict to the identity outside each of the open sets Un. The infinite composition of deleting diffeomorphisms with our function, ψ=ϕ◦∞ n=1hn, is locally finite, in the sense that only a finite number (in fact at most one) of the diffeomorphisms are acting on some neighborhood of each point, while all the rest restrict to the identity on that neighborhood. As in the proof above, it follows that ψhas no critical points (we can use exactly the same argument locally), and still approximates f(recall that each hnrestricts to the identity outside the set Un, on which ϕhas a very small oscillation, and the Unare pairwise disjoint). Proof of Proposition 2.1 We will assume that U=X, since the proof is completely analogous in the case of a general open set. One only has to take some (easy but rather rambling) technical precautions in order to make sure that the different balls considered in the argument are in U. In order to avoid bearing an unnecessary burden of notation, we will make the proof of this proposition for the case of a constant ε > 0. Later on we will briefly 8 DANIEL AZAGRA AND MANUEL CEPEDELLO BOISO explain what additional technical precautions must be taken in order to deduce the general form of this result (see Remark 2.11 below). Let B(x, r) and B(x, r) stand for the open ball and closed ball, respectively, of center xand radius r, with respect to the usual hilbertian norm k·k of X. Case I. We will first consider the case of a real valued function f:U−→ R. Fix ε > 0. By continuity, for every x∈Xthere exists δx>0 so that |f(y)−f(x)| ≤ ε/8 whenever y∈B(x, 2δx). Since X=Sx∈XB(x, δx/2) is separable, there exists a countable subcovering, X= ∞ [ n=1 B(xn, rn/2), where rn=δxn, for some sequence of centers (xn). By induction (and using the fact that every finite-dimensional subspace of Xhas empty interior in X), we can choose a sequence of linearly independent vectors (yn), with yn∈B(xn, rn/2), so that X= ∞ [ n=1 B(yn, rn).(3) Moreover, we have that |f(y)−f(yn)| ≤ ε/4 whenever ky−ynk ≤ rn. (4) Now we define the scalloped balls Bnthat are the basis for our perturbed partition of unity: set B1=B(y1, r1), and for n≥2 define Bn=B(yn, rn)\n−1 [ j=1 B(yj, λnrj); where 1/2< λ2< λ3< ... < λn< λn+1 < ... < 1, with limn→∞ λn= 1. Taking into account that limn→∞ λn= 1, it is easily checked that the Bnform a locally finite open covering of X, with the nice property that |f(y)−f(yn)| ≤ ε/4 whenever y∈Bn. Next, pick a C∞smooth function g1:R−→ [0,1] so that: (i) g1(t) = 1 for t≤0, (ii) g1(t) = 0 for t≥r12, (iii) g′ 1(t)<0 if 0 < t < r12; and define then ϕ1:X−→ Rby ϕ1(x) = g1(kx−y1k2) for all x∈X. Note that ϕ1is a C∞smooth function whose open support is B1, and B1∩Cϕ1={y1}, that is, y1is the only critical point of ϕ1that lies inside B1. Now, for n≥2, pick C∞smooth functions θ(n,j):R−→ [0,1], j= 1, ..., n, with the following properties. For j= 1, ..., n −1, θ(n,j)satisfies that (i) θ(n,j)(t) = 0 for t≤(λnrj)2, (ii) θ(n,j)(t) = 1 for t≥rj2, SMOOTH APPROXIMATION BY MAPPINGS WITH NO CRITICAL POINTS 9 (iii) θ′ (n,j)(t)>0 if (λnrj)2< t < rj2; while for j=nthe function θ(n,n)is such that (i) θ(n,n)(t) = 1 for t≤0, (ii) θ(n,n)(t) = 0 for t≥rn2, (iii) θ′ (n,n)(t)<0 if 0 < t < rn2. Then define the function gn:Rn−→ [0,1] as gn(t1, ..., tn) = n Y i=1 θ(n,i)(ti) for all t= (t1, ..., tn)∈Rn. This function is clearly C∞smooth on Rnand satisfies the following properties: (i) gn(t1, ..., tn)>0 if and only if tj>(λnrj)2for all j= 1, ..., n −1, and tn< rn2; and gnvanishes elsewhere; (ii) gn(t1, ..., tn) = θ(n,n)(tn) whenever tj≥rj2for all j= 1, ..., n −1; (iii) ∇gn(t1, ..., tn)6= 0 provided (λnrj)2< tjfor all j= 1, ..., n −1, and 0 < tn< rn2. Moreover, under the same conditions as in (iii) just above we have that ∂gn ∂tn (t1, ..., tn) = ∂θ(n,n) ∂tn (tn) n−1 Y i=1 θ(n,i)(ti)<0,(5) since no function in this product vanishes on the specified set, while for j < n, according to the corresponding properties of the functions θ(n,j)we have that ∂gn ∂tj (t1, ..., tn) = ∂θ(n,j) ∂tj (tj) n Y i=1,i6=j θ(n,i)(ti)>0.(6) If we are not in the conditions of (iii) then the corresponding inequalities do still hold but are not strict. Let us now define ϕn:X−→ [0,1] by ϕn(x) = gn(kx−y1k2, ..., kx−ynk2). It is clear that ϕnis a C∞smooth function whose open support is precisely the scalloped ball Bn. As above, let us denote by Cϕnthe critical set of ϕn, that is, Cϕn={x∈X:ϕ′ n(x) = 0}. Since our norm k·k is hilbertian we have that, if x∈Cϕn∩Bn, then xbelongs to the affine span of y1, ..., yn. Indeed, if x∈Bn, ϕ′ n(x) = n X j=1 ∂gn ∂tj (kx−y1k2, ..., kx−ynk2) 2(x−yj) = 0,(7) 16 DANIEL AZAGRA AND MANUEL CEPEDELLO BOISO This function ϕjis of the form ϕj(x) = P∞ n=1 αj n(x)ϕj n(x) P∞ n=1 ϕj n(x)= lim n→∞ fj n(x), where fj n(x) = Pn k=1 αj k(x)ϕj k(x) Pn k=1 ϕj k(x),for all x∈ n [ i=1 Bj i, the domains of the fj nform increasing towers of open sets whose union is X, and, for each x∈Xthere is some open neighborhood Vj xof xand some nj x∈Nso that ϕj(y) = fj n(y) for all y∈Vj xand all n≥nj x(see Fact 2.6). Now define the mappings ϕ:X−→ Rmand fn:Tm j=1 Sn i=1 Bj i−→ Rmby ϕ(x) = (ϕ1(x), ..., ϕm(x)),and fn(x) = (f1 n(x), ..., fm n(x)). By the choice of the εjand the construction of the functions ϕj, it is clear that kϕ(x)−f(x)k ≤ ε/2,for all x∈X, that is, ϕapproximates fas is required. Fact 2.10. If x∈Cfn∩Tm j=1 Sn i=1 Bj ithen x∈ A[yj i: 1 ≤i≤n, 1≤j≤m]. Proof. This is a consequence of Fact 2.8. Indeed, according to Remark 2.9, each dfj n(x) is a nontrivial linear combination of the vectors (x−yj k) (with k= 1, ..., n). So, for each jand each x∈Sn i=1 Bj iwe can assign numbers βj 1(x), ..., βj n(x) such that at least one of them does not vanish, and dfj n(x) = n X k=1 βj k(x)(x−yj k).(12) Suppose now that x∈Tm j=1 Sn i=1 Bj iand that the linear map dfn(x) : X−→ Rmis not surjective (that is, xis a critical point of fn); this means that there are numbers γ1(x), ..., γm(x), not all of them zero, such that m X j=1 γj(x)dfj n(x) = 0.(13) Then, by combining (12) and (13) we get that m X j=1 n X k=1 γj(x)βj k(x)(x−yj k) = 0,(14) where not all of the numbers γj(x)βj k(x) vanish. Since the vectors yj kare all linearly independent, it follows from (14) that xis in the affine span of the vectors yj kwith j= 1, ..., m;k= 1, ..., n. SMOOTH APPROXIMATION BY MAPPINGS WITH NO CRITICAL POINTS 17 As dϕ is continuous, it is obvious that the set of critical points Cϕis closed in U. Now we can easily show that Cϕis locally compact as well. Indeed, take x∈X. For every j= 1, ..., m we know that there exists a neighborhood Vj xof x and some nj x∈Nso that ϕj(y) = fj n(y) for all y∈Vj xand every n≥nj x. Fix n=nx:= max{n1 x, ..., nm x}, and take Wxan open bounded neighborhood of xso that Wx⊂Vx:= Tm j=1 Vj x. Then we have that ϕ(y) = (ϕ1(y), ..., ϕm(y)) = (f1 n(y), ..., fm n(y)) = fn(y) for all y∈Vx, and in particular Vx⊂Tm j=1 Sn i=1 Bj i.Now, according to Fact 2.10, it follows that Cϕ∩Vx=Cfn∩Vxis contained in an affine subspace of dimension nm. In particular Cϕ∩Wxis compact, because it is closed, bounded, and is contained in a finite-dimensional subspace. Remark 2.11. Let us say a few words as to the way one has to modify the above proofs in order to establish Proposition 2.1 when εis a continuous positive function. At the beginning of the proof of Case I of Proposition 2.1, before choosing the δx, we have to take some number αx>0 so that |ε(y)−ε(x)| ≤ ε(x)/4 whenever ky−xk ≤ 2αxand then we can find some δx≤αxso that |f(y)−f(x)| ≤ ε(x)/8 whenever y∈B(x, 2δx). In particular, after choosing the rn=δxnas in the proof of Case I above, we have that |f(y)−f(yn)| ≤ ε(yn)/8,and ε(yn)≤4 3ε(y) (15) for all y∈B(yn, rn). Then we can go on with the proof, with appropriate modifications, to construct the functions ϕand fn. Some obvious changes must be made in the definition of the functions anand αn. Fact 2.5 now tells us that |ϕ(y)−f(yn)| ≤ ε(yn)/4 (16) for all y∈B(yn, rn). Then, by combining (15) and (16) we get that |ϕ(y)−f(y)| ≤ |ϕ(y)−f(yn)|+|f(yn)−f(y)| ≤ ε(yn) 4+ε(yn) 8=3 8ε(yn)≤ε(y) 2 for all y∈B(yn, rn) and, since these balls cover X, this proves that |ϕ(y)−f(y)| ≤ ε(y)/2 for all y∈X. In Case II it is enough to define the functions εj(x) = ε(x)/√4m, for j= 1, ..., m. The rest of the proof applies just replacing εjand εwith εj(x) and ε(x), and making some obvious minor modifications as in Case I. Proof of Theorem 2.2 The proof of this result is based on that of Case I of Proposition 2.1. We will have to select the numbers λnwith more care, and make sure that the boundaries of the balls considered have a nice transversality property. An argument similar to that of Remark 2.11 shows that there is no loss of generality in assuming that εis constant. 18 DANIEL AZAGRA AND MANUEL CEPEDELLO BOISO Suppose that we are at the beginning of the proof of Proposition 2.1 and we only know that X=S∞ n=1 B(yn, sn/2), where sn=δxn, for some sequence of linearly independent vectors (yn), and |f(y)−f(yn)| ≤ ε/4 provided ky−ynk ≤ 3 2sn. The following lemma shows that we can slightly move the radii snso that, for any finite selection of centers yn, the spheres that are the boundaries of the balls B(yn, sn) have empty intersection with the affine subspace spanned by those centers. Lemma 2.12. We can find a sequence of positive numbers (rn)with sn≤rn≤3 2sn so that, if we denote Sn=∂B(yn, rn)then, (i) for each finite sequence of positive integers k1< k2< ... < km, A[yk1, ..., ykm]∩Sk1∩... ∩Skm=∅. (ii) for any n, k ∈N,yn/∈Sk. Proof. We will define the rninductively. For n= 1 we may take r1∈[s1,3 2s1] so that r1does not belong to the countable set {ky1−ykk:k∈N}; this means that yk/∈S1for any k∈N. On the other hand, it is obvious that {y1}∩S1=∅. Assume now that r1, ..., rnhave already been chosen in such a way that the spheres S1, ..., Snsatisfy (i) and (ii), and let us see how we can find rn+1. For any finite sequence of integers 0 < k1< ... < kj≤n+ 1, let us denote Ak1,...,kj=A[yk1, ..., ykj]. For simplicity, and up to a suitable translation (which obviously does not affect our problem), we may assume that yn+1 = 0, so that Ak1,...,km,n+1 is the m-dimensional vector subspace of Xspanned by yk1, ..., ykm. Now, for each finite sequence of integers 0 < k1< ... < km≤n, consider the map Fk1,...,km:Ak1,...,km,n+1 −→ Rm defined by Fk1,...,km(x) = kx−yk1k2−rk1 2, ..., kx−ykmk2−rkm 2. Note that DFk1,...,km(x) = 2(x−yk1), ..., 2(x−ykm) and therefore rankDFk1,...,km(x)< m if and only if x∈ Ak1,...,km. By the induction assumption we know that Sk1∩... ∩Skm∩Ak1,...,km=∅, hence it is clear that rankDFk1,...,km(x)=mfor all x∈Sk1∩...∩Skm∩Ak1,...,km,n+1. This implies that Mk1,...,km:= Sk1∩... ∩Skm∩Ak1,...,km,n+1 SMOOTH APPROXIMATION BY MAPPINGS WITH NO CRITICAL POINTS 19 is a compact m−m= 0-dimensional submanifold of Ak1,...,km,n+1, and in particular Mk1,...,kmconsists of a finite number of points (in fact two points, but we do not need to know this). Therefore M=[Mk1,...,km (where the union is taken over all the finite sequences of integers 0 < k1< ... < kn≤n) is a finite set as well. Now we have that I:= sn+1,3 2sn+1\{kzk:z∈M}∪{kyjk:j∈N} is an uncountable subset of the real line, so we can find a number rn+1 ∈I. With this choice it is clear that Sk1∩... ∩Skm∩Sn+1 ∩Ak1,...,km,n+1 =Mk1,...,km∩Sn+1 =∅ for all finite sequences of integers 0 < k1< ... < km< n + 1, and also yj/∈Sn+1 =∂B(0, rn+1) for all j∈N. Therefore the spheres S1, ..., Sn,Sn+1 satisfy (i) and (ii) as well. By induction the sequence (rn) is thus well defined. Now define Bn,ϕn,ϕ,fn, as in Case I of the proof of Proposition 2.1. All the properties shown in the proof of 2.1 about the functions fnand ϕ(in particular Facts 2.5 and 2.8) are independent of the way we may choose the numbers λjin the definitions of Bjand ϕj. Now we only have to see how we can select those numbers λjso as to have more control over the set Cϕof critical points of ϕand thus prove the statement of Theorem 2.2. We will define the numbers λnand the open sets Un inductively. First step. Define ϕ1as above and put f1(x) = α1(x) for all x∈B1=B(y1, r1). Set µ2= 1/2, K1=Cf1∩B1={y1}, and U1=B(y1, µ2r1). Second step. Fix λ2∈(µ2,1), and define B2,ϕ2, and f2as above. According to Fact 2.8, we have that Cf2∩B2⊂ A[y1, y2], and Cf2∩(B2\B1)⊆ A[y2]. We claim that there must exist some µ3∈(λ2,1) so that Cf2∩B2∩B1⊂B(y1, µ3r1). Otherwise there would exist a sequence (xj) in Cf2∩B2∩B1so that kxj−y1kgoes to r1as jgoes to ∞. Since Cf2∩B2⊂ A[y1, y2], we may assume, by compactness, that xjconverges to some point x0∈∂B(y1, r1) = S1. If x0∈B(y2, r2) then f′ 2(x0) = 0 (by continuity of f′ 2), and x06=y2(because y2/∈S1by ii) of Lemma 2.12), so f′ 2(x0) = α′ 2(x0)6= 0, a contradiction. Therefore it must be the case that x0∈∂B(y2, r2) = S2. But then x0∈S1∩S2∩A[y1, y2], and this contradicts Lemma 2.12. 20 DANIEL AZAGRA AND MANUEL CEPEDELLO BOISO So let us take µ3∈(λ2,1) such that Cf2∩B2∩B1⊂B(y1, µ3r1). Choose also some ν2∈(µ2, λ2). In the case that y2∈B1, let us simply set U2=B(y2, r2)∩B(y1, µ3r1)\B(y1, ν2r1), and K2=Cf2∩B2∩B1⊂U2. In the case that y2/∈B1, find δ2∈(0, µ3r2) so that B(y2, δ2)⊂B2\B1, and set U2=B(y2, r2)∩B(y1, µ3r1)\B(y1, ν2r1)∪B(y2, δ2), and K2=Cf2∩B2∩B1∪{y2} ⊂ U2. Clearly, we have that Cf2⊆K1∪K2, and U1∩U2=∅. Third step. Now choose λ3∈(µ3,1) with λ3>1−1/3, and define B3,ϕ3, and f3 as above. We have that f3and f2coincide on (B1∪B2)\B3. On B3, according to Fact 2.8, we know that Cf3∩B3∩B2∩B1⊆ A[y1, y2, y3]; Cf3∩(B3∩B2\B1)⊆ A[y2, y3], and Cf3∩(B3∩B1\B2)⊆ A[y1, y3] ; (17) Cf3∩(B3\(B1∪B2)) ⊆ A[y3]. Again, there must be some µ4∈(λ3,1) so that Cf3∩B3∩(B1∪B2)⊂B(y1, µ4r1)∪B(y2, µ4r2). Otherwise (bearing in mind the local compactness of A[y1, y2, y3]), there would exist a sequence (xj) in Cf3∩B3∩(B1∪B2) so that (xj) converges to some point x0and (xj) is not contained in B(y1, µ4r1)∪B(y2, µ4r2) for any µ4<1. Since a subsequence of (xj) must be contained in one of the sets listed in (17), we deduce that the limit point x0must belong to one of the following sets: S2∩S1∩A[y1, y2, y3]; S2∩A[y2, y3]\B1; S1∩A[y1, y3]\B2, Now we have two cases: either x0∈B3, or x∈∂B3. If x0∈B3then f′ 3(x0) = 0 (by continuity of f′ 3), and x06=y3(because y3/∈S1∪S2by (ii) of Lemma 2.12), so it follows that f′ 3(x0) = α′ 3(x0)6= 0, a contradiction. On the other hand, if x0∈∂B3then x0∈S3as well, and now one of the following must hold: x0∈S3∩S2∩S1∩A[y1, y2, y3]; x0∈S3∩S2∩A[y2, y3]; x0∈S3∩S1∩A[y1, y3], but in any case this contradicts Lemma 2.12. Hence we can take µ4∈(λ3,1) so that Cf3∩B3∩(B1∪B2)⊂B(y1, µ4r1)∪B(y2, µ4r2). SMOOTH APPROXIMATION BY MAPPINGS WITH NO CRITICAL POINTS 21 Take ν3∈(µ3, λ3). Now two possibilities arise. If y3∈B1∪B2, let us define U3="B(y3, r3)\ 2 [ j=1 B(yj, ν3rj)#\"2 [ j=1 B(yj, µ4rj)#, and K3=Cf3∩B3∩(B1∪B2)⊂U3. If y3/∈B1∪B2, since y3/∈S1∪S2we can find δ3∈(0, µ4r3) so that B(y3, δ3)⊆ B3\(B1∪B2), and then we can set U3="B(y3, r3)\ 2 [ j=1 B(yj, ν3rj)\2 [ j=1 B(yj, µ4rj)#[B(y3, δ3), and K3= [Cf3∩B3∩(B1∪B2)] ∪{y3} ⊂ U3. Notice that U3does not meet U1or U2, and Cf3⊆K1∪K2∪K3. N-th step. Suppose now that µj,λj,νj,ϕj,Bj,fj,Kj,Ujhave already been fixed for j= 1, ..., n (and also µn+1 has been chosen) in such a manner that fjagrees with fj−1on (B1∪... ∪Bj−1)\Bj, and Kjand Ujare of the form Kj=Cfj∩Bj∩(B1∪... ∪Bj−1) (18) and Uj=B(yj, rj)\j−1 [ i=1 B(yi, νjri)\hj−1 [ i=1 B(yi, µj+1ri)i(19) in the case that yj∈B1∪... ∪Bj−1, and are of this form plus {yj}and B(yj, δj) respectively when yj/∈B1∪...∪Bj−1; assume additionally that Uj∩Uk=∅whenever j6=k, that Cfj⊆Sj i=1 Ki, and that λj>1−1/j. Let us see how we can choose λn+1,µn+2,νn+1,Kn+1 and Un+1 so that the extended bunch keeps the required properties. Pick any λn+1 ∈(µn+1,1) so that λn+1 >1−1/(n+ 1), and define ϕn+1,Bn+1 and fn+1 as above. We know that fn+1 agrees with fnon the set (B1∪...∪Bn)\Bn+1. On Bn+1, according to Fact 2.8, we have that Cfn+1 ∩Bn+1 \ m [ j=1 Bkj⊆ A{y1, ..., yn+1}\{yk1, ..., ykm} for every finite sequence of integers 0 < k1< k2< ... < km< n + 1. We claim that there exists some µn+2 ∈(λn+1,1) so that Cfn+1 ∩Bn+1 ∩(B1∪... ∪Bn)⊆ n [ i=1 B(yi, µn+2ri). 22 DANIEL AZAGRA AND MANUEL CEPEDELLO BOISO Otherwise there would exist a finite (possibly empty!) sequence of integers 0 < k1< k2< ... < km< n + 1, and a sequence (xj)∞ j=1 contained in "Cfn+1 ∩Bn+1 ∩ℓ \ j=1 Bij#\m [ j=1 Bkj⊆ A[yi1, ..., yiℓ, yn+1] (where i1, ..., iℓare the positive integers less than or equal to nthat are left when we remove k1, ..., km), such that (xj) converges to some point x0∈Si1∩... ∩Siℓwith x0/∈Sm j=1 Bkj. If x0∈Bn+1 then f′ n+1(x0) = 0 (by continuity of f′ n+1), and x06=yn+1, so we easily see that f′ n+1(x0) = α′ n+1(x0)6= 0, a contradiction. If x0∈∂Bn+1 then x0∈Sn+1 as well, and in this case we have x0∈Si1∩... ∩Siℓ∩Sn+1 ∩A[yi1, ..., yiℓ, yn+1], but this contradicts Lemma 2.12. Therefore we may take µn+2 ∈(λn+1,1) so that Cfn+1 ∩Bn+1 ∩(B1∪... ∪Bn)⊆ n [ i=1 B(yi, µn+2ri). Choose any νn+1 ∈(µn+1, λn+1). As before, now we face two possibilities. If yn+1 ∈ Sn i=1 Bi, let us define Un+1 ="B(yn+1, rn+1)\ n [ i=1 B(yi, νn+1ri)#\"n [ i=1 B(yi, µn+2ri)#,and Kn+1 =Cfn+1 ∩Bn+1 ∩(B1∪... ∪Bn). If yn+1 /∈Sn i=1 Bi, since yn+1 /∈Siwe may find δn+1 ∈(0, µn+2rn+1) so that B(yn+1, δn+1)⊆Bn+1 \Sn i=1 Bi, and then we can add this ball to the above Un+1, and the point {yn+1}to that Kn+1, in order to obtain sets Un+1,Kn+1 with the required properties. By induction, the sequences (ϕn), (fn), (Un), (Kn), (λn), (µn), (νn) are well defined and satisfy the above properties. From the construction it is clear that Un∩Um=∅whenever n6=m, and Cfn⊆ n [ j=1 Kj for all n. As observed before (see Fact 2.6), for each x∈Xthere exists an open neighborhood Vxof xand some nx∈Nso that ϕ(y) = fnx(y) for all y∈Vx. Bearing these facts in mind, it is immediately checked that Cϕ⊆S∞ n=1 Kn.The other properties in the statement of Theorem 2.2 are immediately deduced from the above construction. 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Universidad de Sevilla. Sevilla, SPAIN. E-mail addresses: daniel aza[email protected], [email protected]