Smooth Banach Spaces and Approximations Citation Wells, John Campbell (1969) Smooth Banach Spaces and Approximations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/85QX-YM62. https://resolver.caltech.edu/CaltechTHESIS:01252016-133502160 Abstract If E and F are real Banach spaces let C p,q (E, F) O ≤ q ≤ p ≤ ∞, denote those maps from E to F which have p continuous Frechet derivatives of which the first q derivatives are bounded. A Banach space E is defined to be C p,q smooth if C p,q (E,R) contains a nonzero function with bounded support. This generalizes the standard C p smoothness classification. If an L p space, p ≥ 1, is C q smooth then it is also C q,q smooth so that in particular L p for p an even integer is C ∞,∞ smooth and L p for p an odd integer is C p-1,p-1 smooth. In general, however, a C p smooth B-space need not be C p,p smooth. C o is shown to be a non-C 2,2 smooth B-space although it is known to be C ∞ smooth. It is proved that if E is C p,1 smooth then C o (E) is C p,1 smooth and if E has an equivalent C p norm then c o (E) has an equivalent C p norm. Various consequences of C p,q smoothness are studied. If f ϵ C p,q (E,F), if F is C p,q smooth and if E is non-C p,q smooth, then the image under f of the boundary of any bounded open subset U of E is dense in the image of U. If E is separable then E is C p,q smooth if and only if E admits C p,q partitions of unity; E is C p,p smooth, p ˂∞, if and only if every closed subset of E is the zero set of some C P function. f ϵ C q (E,F), 0 ≤ q ≤ p ≤ ∞, is said to be C p,q approximable on a subset U of E if for any ϵ ˃ 0 there exists a g ϵ C p (E,F) satisfying sup/xϵU, O≤k≤q ‖ D k f(x) - D k g(x) ‖ ≤ ϵ. It is shown that if E is separable and C p,q smooth and if f ϵ C q (E,F) is C p,q approximable on some neighborhood of every point of E, then F is C p,q approximable on all of E. In general it is unknown whether an arbitrary function in C 1 ( l 2 , R) is C 2,1 approximable and an example of a function in C 1 ( l 2 , R) which may not be C 2,1 approximable is given. A weak form of C ∞,q , q≥1, to functions in C q ( l 2 , R) is proved: Let {U α } be a locally finite cover of l 2 and let {T α } be a corresponding collection of Hilbert-Schmidt operators on l 2 . Then for any f ϵ C q ( l 2 ,F) such that for all α sup ‖ D k (f(x)-g(x))[T α h]‖ ≤ 1. xϵU α ,‖h‖≤1, 0≤k≤q Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics and Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): De Prima, Charles R. Thesis Committee: Unknown, Unknown Defense Date: 18 November 1968 Funders: Funding Agency Grant Number NSF UNSPECIFIED Record Number: CaltechTHESIS:01252016-133502160 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:01252016-133502160 DOI: 10.7907/85QX-YM62 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 9545 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 25 Jan 2016 23:49 Last Modified: 06 May 2024 22:49 Thesis Files Preview PDF - Final Version See Usage Policy. 16MB Repository Staff Only: item control page

SMOOTH BANACH SPACES AND APPROXIMATIONS Thesis by John Campbell Wells In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1969 (Submitted November 18, 1968) ii ACKNOWLEDGEMENTS I wish to thank my advisor, Professor Charles DePrima, for many rewarding discussions. His help in writing this dissertation was invaluable. Much gratitude is also owed to Professor Felix Browder who stimul a ted my interest in the topic of this paper, to Professor Robert Bonic who provided much encouragement and to Professor Ted Petrie who sug g ested several interesting problems. I also wish to thank the National Science Foundation for various fellowship grants and the California Institute of Technology for financial assistance as a teaching assistant. iii ABSTRACT If E and F are real Banach spaces let cP' q(E, F) Os q s p s a:1, denote those maps from E to F which have p continuous Frechet derivatives of which the first q derivatives are bounded. A Banach space E is defined to be cP,q smooth if Cp'q(E,R) contains a nonzero function with bounded support. This generalizes the standard cP smoothness classification. If an LP space, p ::<: 1, is Cq smooth then it is also Cq,q smooth so that in particular LP for p an even integer is Ca:1,a:1 smooth and LP for p an odd integer is cP-l,p-l smooth. In general, however, a cP smooth B-space need not be cP,P smooth. c 0 is shown to be a non-C 2 ' 2 smooth B-space although it is known to be Ca:1 smooth. It is proved that if E is Cp,l smooth then c (E) is 0 cP,l smooth and if E has an equivalent cP norm then c (E) 0 has an equivalent cP norm. Various consequBnces of Cp,q smoothness are studied. If f E Cp'q(E,F), if F is Cp,q smooth and if E is non-Cp,q smooth, then the image under f of the boundary of any bounded open subset U of E is dense in the image of U. If E is separable then E is Cp,q smooth if and only if E admits Cp,q partitions of unity; Eis cP,Psmooth,p< ,if 00 iv and only if every closed subset of E is the zero set of some cP function. Osq<psco, is said to be C p,q approximable on a subset U of E if for any e: > 0 there exists a g E. cP(E,F) satisfying k sup xEU ,Osksq k ID I f(x) - D g(x)jj It is shown that if E is separable and Cp,q smooth and if f E Cq(E,F) is C p,q approximable on some n eighborhood of every point of E, then F is Cp,q approximable on all . of E. In general it is unknown whether an arbitrary function in c 1 (P 2 ,R) is c 2 1 app.roximable and a n example ' l · h may no t b e c , · of a function l·n c1 cx02 , ·R ) w11c 2 1 approximable is given. A weak form of C , q~l, to functions in co' q 2 Cq(.R ,R) is proved: Let [U } be a locally finite cover of a 2 p and let [T } be a corresponding collection of Hilberta. 2 Schmidt ope rat ors on P • Then for any f E Cq( j! 2 ,F) with Dqf locally uni form ly continuou s , there ex i sts a 2 Cco(P. ,F) such that for all a sup llDk(f(x) - g(x))['ra.hJll xEU a , II hi\ s;l , Osksq s 1. g E v TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ii ABSTRACT iii INTRODUCTION 1 Chapter I DIFFERENTIAL CALCULUS 5 II SMOOTHNESS CLASSES 11 III DIFFERENTIABLE FUNCTIONS ON c (E) 19 IV ZERO SETS OF cP FUNCTIONS 26 V Cp,q PARTITIONS OF UNITY 33 VI SMOOTH APPROXIMATION 41 VII WEAK Cp,q APPROXIMATION ON P. 2 46 0 BIBLIOGRAPHY 58 1 INTRODUCTION The central theme of this dissertation is the study of Frechet differentiable functions on real Banach spaces with bounded derivatives. If E and Fare real Banach spaces and 0 ~ q ~ p :=;;cc, a map f from E to F will be said to be of class Cp,q(E,F) if f has p continuous Frechet d e rivatives, the first q of which are bounded. Contrary to the finite dimensional c a se, an infinite dimensional Ba nach space need not have any nontrivial Cp,q functions. We define a B-space to be Cp,q smooth if there exists a nonzero function in cP,q(E,R) with bounded support. This generalizes the standard concept of cP smoothness and Cp,O smoothness is equivalent to cP smoothness. In the first chapter we provide the necessary preliminary material on differential calculus in Banach spaces. Several of the results of Bonic and Frampton [l'] concerning cP smoothness are valid for cP,q smoothness. One property in particular is that a map in Cp,q(E,F) has an "analytic" property if F is cP,q smooth and E is non Cp,q smooth: the values of Fon a bounded open subset U of E are uniquely determined by its values on the boundary of U. This, and other results, including a summary of the 2 Cp,q smoothness of various B-spaces, is contained in Chapter II. An LP space is shown to be Cq,q smooth if it 00 00 is Cq smooth so that LP for p an even integer is C ' smooth. In Chapter III we show that a cP smooth B-space need not be cP,P smooth by proving that c 00 0 , the C smuoth B-space of sequences of real numbers converging to zero, 2 2 p,l is not C ' smooth. In addition we show that if E is C smo o th, then c (E),(the B-space of sequences in E converg0 ing to zero), is Cp,l smooth. N.H.Kuiper constructed an 00 equivalent C norm for c 0 and we generalize this by proving that c (E) has an equivalent cP norm if E ha s an equiva0 lent cP norm. In Chapter IV the existence of a cP function with a prescribed zero set is studied. The main . result is that a separable B-space ~ is cP'Psmooth,p<oo,if and only if every closed subset of E is the zero set of some function. Secondly, if E is only cP smooth and A is a closed subset of E, we give a sufficient condition on A to insure that it is the zero set of some cP function. If a B-space E admits cP partitions of unity, then cP(E,F) is dense in CO(E,F) for any F, but in g eneral the existence of cP partitions of unity on E is unknown. Bonic and Frampton in [l] proved that if E i s separable then E admits cP partitions of unity if and only if E is 3 cP smooth. We generalize this by proving in Chapter V that a separable Cp,q smooth B-space admits Cp,q partitions on unity. Along with this we study the Cp,q_ness of sums of the form E ai~i(x),where [~i} is a Cp,q i partition of unity. In Chapter VI we examine the problem of smooth approximation. We say that a map f E Cq(E,F) is C , p,q 0 ~ q < p :!>: approximable on a subset U of E if for any 00 , e: > 0 there exists a gECP(U,F) such that !IDkf (x) - Dkg(x) II < sup e: xEU,O:!>:k~q We say that f is strongly C p,q approximable on U if it satisfies the above condition with e: replaced by an ar- bitrary positive continuous function e(x). In general the C approximability of an arbitrary Cq function on an p,q infinite dimensional Banach space is unsolved. We prove, however, that if a B-space E is Cp,q smooth and separable, and if f E Cq(E ,F) is Cp, q approximable in some neighborhood of every point of E, then f is strongly C approxip,q mable on all of E. In the last part of the chapter we prove a theorem that suggests that the Ex. \x.l \ . l from P. 2 to R might not be on any open subset of p2 • l Coo,q c 1 function cr(x) c 2 1 app r oximable ' The last chapter is devoted to a weak form of approximation to functions defjned on R2 • Let [U"} u. 4 be a l ocall y finite cover of f 2 and let (Ta} be a collec- tion of Hi lbert-Schmidt operators on i 2 • Then we show that for any f E Cq( i 2 ,F), with Dqf (x) locally uniformly continco 2 uous, there exists a gEC (f ,F) such that for all a. sup \\Dk(f(x)-g(x))[Ta.hJI\ xEUa., \I h\\ ~l, Os;ks;q s; 1. 5 CHAPTER I DIFFERENTIAL CALCULUS We will define the two most important types of derivatives on Banach spaces. For the proofs of the theorems of this section, refer to [5],[8],[10], and [17]. From here on all Banach spaces will be assumed to be real. Definition. If E and F are Banach spaces, a continuous k-multilinear map T from E into F is a continuous map from E x ••• x E into F satisfying T[h 1 , •• ,ahi+bhi, •• hk] = aT[h 1 , •• hi, •• hk] + b'I1[h 1 , •• hi, •• hk] for all real a, b and l::::isk. Definition. If E and F are Banach spaces, Lk(E,F),k~l, will denote the set of continuous k-multilinear maps from E into F. We will write L(E,F) for L 1 (E,F). If T E Lk(E,F) then the norm,\\T\\ ,is defined as sup .. \\T[h 1 , • • h 1J\\. l\hi !lsl,isk Lk(E,F) with the above norm is a Banach space and from here on Lk(E, F) will be as~co umed to have the topology induced by the above norm. There is a canonical isomorphism,*, between Lk(E,Lp(E,F)) and Lk+p(E,F) 6 given by (*T)[h 1 , •. hk+p] = (T[h 1 , .. hk])[hk+l'"'hk+p] and with this isomorphism we will regard Lk(E,LP(E,F)) and Lk+p(E,F) as identical. R em~rk~ If TELk(E,F) then T[h] will be the shorthand not a tion for T[h, •• h]. Definition. L~(E,F) will denote the set of all continuous symmetric k -multilinear maps from Ex ••• xE into F. Definition. If f is a map from a Bana ch space E into a Banach space F, f is said to have a Gateaux derivative at x in direction h if GDf(x)[h] = lim f(x+th) -f( x) t -0 t exists. It is immediate from the definition that GDf(x)[h] is homog e neous in h(i.e. GDf(x)[ah] aGDf(x)[h]) a lthoug h GDf(x)[h] may be nonline ar in h or may be linear but u nbounded. P r op osit io n 1.1 If f:E-F and f h as a Gateaux derivative a t all points on the s egment [x,x+h] in direction h, then for any wEF*(the dual space of F),(f(x+h)-f(x),w) (GDf(x+ Th)[h] ,w) where O< T < l Proposition 1.2 and T depends on w. If f:E -F and f bas a Gateaux derivative at a ll points on the segment [x,x+h] in direction h, then \\ f(x+h)-f(x) \is sup \iGDf(x+Th)[hJ\\ O<T<l I 7 Proof. Pick w in Prop. 1.1 such that Ii w\\ = 1 and l\f (x+h)-f(x)\\ = (f(x+h)-f(x),w). Under certain conditions GDf(x)[h] is a bounded linear function in h: Proposition 1.3 Suppose that f:E~F and that GDf(x)[h] exists for all h and for all x in a neighborhood of x • 0 Suppose that for all fixed h, GDf(x)[h] is continuous at x o as a function of x and that GDf(x )[h] is contin0 uous in h. Then h-GDf(x )[h] is a bounded linear map 0 from E into F. Definition. If E and F are B-spaces and f:E-F, then f is said to be Frechet differentiable at xEE if there exists a Df(x)E L(E,F) such that lim su t-0 11hn:s;t lif(x+h)-f~x~-Df(x)[hJU. = O. h Df(x) is then said to be the Frechet derivative of f at x. If Df(x) exists at x, then clearly Df(x)[h] = GDf(x)[h] for all h. Df(x) is invariant within the set of equivalent norms. Proposition 1.4 If f is Frechet differentiable at x then f is continuous at x . Proposition l._2 If f:E-F has a Frechet derivative at all points on the segment [x,x+h], then for any wEF* 8 ((f(x+h)-f(x),w) = (Df(x+Th)[h],w), where O<T<l and T depends on w. Proposition l.6(Mean Value Theorem) If f:E-F has a Frechet derivative at all x in the segment [x,x+h], then l\f(x+h)-f(x)l\ s: sup llDf(X+Th)[hJll. s: sup llDf(X+Th)U ·llhll O<T<l O<T<L Proposition 1.7 • If f is a map from an open subset U of a B-space E into F , if f has a derivative at x and if llf(x)-f(y)\I s: M(llx-yl\), then Proposition 1.8 Suppose that ~\Df(x)I\ s: M. f:E~F and that GDf(x)[h] exists and is bounded and linear in h for all x in a nei g hborhood of some x 0 • If GDf(x)[h], considered as a map from E into L(E,F), is continuous at x 0 , then f has a Frechet derivative at x 0 and Df(x 0 )[h] Definition. of x 0 = GDf(x 0 )[h] • If f :E ~F and Df(x) exists in a neighborhood and if the map Df(x) from E into L(E,F) is differen- tiable at x 0 , then we say that f is twice differentiable at x 0 and we write D2 f(x) 0 = D(Df(x 0 )). Note that D2 f(x) 0 2 E L(E,L(E,F)) '.'?' L (E,F). Inductively we say that DP exists at x 0 if nP- 1 f(x) exists in a neighborhood o f x 0 and if Dp-lf(x) is dif f erentiable at x • We write nPr(x ) = 0 0 1 D(DP- r(x )) and again note that nPf(x ) E L(E,Lp-l(E,F)) 0 - 0 LP(E,F). The following proposition is a generalization of the formula a2 f axay a3 f ay ax 9 If DPf(x 0 ) , p~2, exists and nP- 1 f(x) Proposition l~ is continuous in a neighborhood of x 0 , then DPf(x 0 ) E LP(E F) s ' • Definition. If O~p<m and if U is an open subset of a Banach space E, we will say that f E cP(u,F) if f:U---F and DPf(x) exists and is continuous in U. We say that f E 00 C (U,F) if f E cP(U,F) for all p. Note. From here on the words derivative, differentiation etc, will refer to the Frechet derivative unless otherwise stated. DPf will always denote an el ement of LP(E,F~ Proposition 1.10 Let E,F,G be B-spaces and let Ube an open subset of E, V an open subset of F. If fECP(U,V) and gECP(V,G) then fog E cP(U,G). Suppose that fn E c 1 (U,F) where U is an Proposition 1.11 open subset of a B-space E and that g:U~F, G:U--7-L(E,F). Suppose that for every point x 0 E U there is a neighbo r hood Nx 0 of x 0 contained in U such that fn(x) and Dfn(x) converge uniformly to g(x) and G(x) in NXo. Then gEC 1 (U,F) a nd Dg(x) = G(x) for all x EU. Let ucE be a convex Proposition l.12(Taylor's Formula) neighborhood of x 0 and suppose that f E cP(U,F). Then p-1 f(x +h) o = I; k=O + R (x p 0 ,h) 10 where R (x ,h) p 0 = l H=Bp-1 J0 p 1 . Dpf(x +th)[h]dt • 0 For every e>O there is a 6 >O such that if \lhll < 6 \IRP(x 0 ,h)\I s: then ellh\IP • Proposition l.13(Inverse Function Theorem) Let E a nd F be Banach spaces and suppose that f E cP(U,F) where U is E E. Suppose that Df(x 0 ) is an isoo morphism from E into F. 1rhen there exists a nei g hborhood a neighborhood of x V of x 0 a nd a cP map g from f(V) onto V such that gof and fog are the identity maps on V and f(V). 11 CHAPTER II SMOOTHNESS CLASSES In this chapter we introduce a new smoothness classification for Banach spaces. Thi s generalizes the usual cP smoothness classes. Definition. If E and F are Banach spaces, and U is an open subset of E and Osqspsro, then Cp,q(U,F) will denote those functions fin cP(U,F) for which sup llDkf(x)li< xEU,Osksq 00 • Definition. A Banach space E will be said to be Cp,q smooth if Cp,q(E,R) contains a non-trivial function with bounded support (sometimes called a Cp,q bump function). The standqrd concept of cP smoothness places no boundedness restrictions on f or its deriv3.tives: Definition. A Banach space is said to be cP smooth if cP(E,R) contains a non-trivial function with bounded support. It is easy to check that any fECP(E,R) can 00 be composed with a suitable function in C (R,R) to yield a function in cP,O(E,R) which has th e same supp ort as f. Hence a Banach space is Cp,O smooth. cP smooth if and only if it is 12 We will prove in the next chapter that there exi s ts a c 2 smooth B-space which is not c 2 ' 2 smooth so that Cp,q smoothness is more r e trictive th an cP smoothness. If E is Cp,q smooth then any B-space equi val e nt to E and any closed subspace of E is again Cp,q smooth. Also if p;:>:p' a nd q;:>:q', then Eis cP',q'smooth. Several basic theorems proved by Bonic and Frampton in [l] concerning cP smoot hness are ge ner a lized below for Cp,q smoothness. The following basic p ro posi tion will be essential in manipulating Cp,q functions: Proposition 2.1 If fECp,q(E,F) and g E Cp,q(F,G) , then gof E cP,q(E,G). Proo~ The proof can be obtained by inducti o n from the following formula, known as Fas di Bruno 's formula: i=l a1+ •• +ak= j k! (Df) a1. ··(Dkf)ak a 1 !· ·a1c ! a 1 +2a 2 + • • kak = k Proposition 2.2 A Banach space E is Cp,q smooth if and only if the norm topology on E is equivalent to the topolo gy induc ed on Eby the functions cP,q( E,R ). Proof. The proof i s identic a l to the proof of Prop. 2 of [l]. Proposition 2.3 Suppose that E is a Banach space with equiva lent norm a such that a E Cp,q(U-[O},R), where U is an open nei g hborhood of O. Then E is Cp,q smooth. Proof. For some r, U contains the ball 00 00 [xi a.(x)<r}. Construct a g E C t ~ r/2 and g(t) = 0 if r E Cp'q(E,R). Also, g(a.(O)) ~ ' (R,R) with g(t) = 1 if 1. Then by Prop. 2.1,g(a.(x)) = 1 and g(o.(x)) has bounded support. Remarks. Q.E.D. It is not known whether the converse to Prop. 2.3 is true, even for q = 0 (i.e. does a cP smooth space have an equivalent cP norm?). If o.(x) is an equivalent norm for E, then l:x.Cx+h) - a.(x)\ ~ a.(h) ~ Kl\hll forsomeK, so by Prop. 1.7 if Da.(x) exists then \IDa.(x)ll ~ K. Also Dk(a.(x)) = Dk(a.(~) )r 1 -k and hence if Dka. is bounded on bounded sets it is bounded everywhere. Definition. If f E cP(E,F) and q ~ p,then by llfllq ,we will denote sup llDkf(x) 11 xEE,O::s;k::s;q • Proposition 2.4. Suppose that F is Cp,q smooth but that E is not Cp,q smooth. Suppose that U is a bounded open subset of E and that au and U are the boundary and closure of u. Then any f E cP,q(U ,F) and fEc 0 cu ,F) has the property that f(aU) is dense in f(U). Proo~ The proof uses the same argument as the argument in the proof of Prop.4 of [l]. Suppose that f(x) is not contained in the closure of f(BU) for some x in u. Then by the hypothesis we can find a ~ E Cp,q(F,R) with ~(f(x)) = 1 and ~(x) = 0 in some neighborhood of f(aU). 14 Let g(y) = ~(f(y)) if y E U and g(y) = 0 otherwise. Then g is nonzero, has bounded support and by Prop.2.1 g E Cp,q(E,R). This contradicts the non-Cp,q smoothness of Q.B.D. ~. It follows that if f 1 , r 2 are two functions in Cp,q(E,F) which agree on the boundary of U, then they Remarks. agree on all of U. Thus Cp,q functions on a non-Cp,q smooth B-space have a type of"analytic" property: the values of the function on a bounded open set are uniquely determined by the values on the boundary. rrhe fol1owing two problems were posed by Bonic and Frampton for non- cP smooth B-spaces and they can also be acked for nonCp,q smooth B-spaces: suppose that E is non-Cp,q smooth, that F is Cp,q smooth and that U is a bounded open subset of E, then what continuous functions on au are boundary values of functions in Cp,q(E,F)? Also, given f E Cp,q(U,F) and f E c0 (U,F) how can f be determined from fl~U? The following is a summary of the Cp,q smoothness and related properties of various Banach spaces: 00 a) All finite dimensional B-spaces are C 00 ' smooth. b) Restrepo([l5]) has proved that a separable B-space, E, has an equivalent c 1 norm on E-[O} if and only if E* is separable. By the remarks following Prop. 2.3, all Banach spaces with an equivalent 1 norm in c 1 (E-[O},R) are c • 1 smooth. Hence a 15 separable B-space with a separable dual is cl,l smooth. £l Bonic and Reis in [3] have shown that if E has a c 2 norm away from zero and the dual norm in E* is also of class c 2 away from zero, then E is a Hilbert space. Ql L 2 (S,L:,µ) , where(S,L:,u) is a positive measure • o:> o:> space, is C ' smooth. It is easy to check that D(\lx\1 2 )[h] = 2(x,h), D2 (\\x\\ 2 )[h 1 ,h 2 ] = 2(h 1 ,h2) 2 and Dk( \\xi\ ) = 0 for k > 2. Hence \\xii E Cro(L 2 (S,L:,µ)-(O} ,R) and all the derivat ives of \\xi\ 2.3, are bounded on bounded sets. Hence by Prop. 'J LL(S,L:,µ) is Cro,ro smooth. tl Bonic and Frampton in [l] have completely classified the cP smoothness of LP(s,L:,µ) for p ~ 1. Their results are as follows. LP is c 00 smooth if p is an even inte ge r. LP i s nr-l smooth if p is an odd integer. This means tha t there e x ists a CP -1 bump function satisfying II Dp-l f (x+h)- nP- 1 f(x)\\ s O(\\x \\) for all x. If p i s not an integer let [p] be the greatest int ege r less than p. Then LP is D[p] p-[p] smooth. This means that there exists a C[p] bump function sat isfying \\D[P]f(x+h) - D[P]f(x) II s O( l\hp-[pl If). If pis an od d integer, pP and hence any infinit e dimensional LP space is not nP smooth. ThJH mcu n s there does 16 not exist a cP-l bump function f such that nPr(x) exists for all x. Final ly if p is not an integer then f P and any infinite dimensional LP space is not C~~EpJ smooth. This means there does not exist a c[p] bump function f satisfying UD[p]f(x+h)-D[p]f(x) D[p]f(x)\I s; oCilhllp-[p]). By Prop.2.5 below, LP is C00 ' 00 , cp-l,p-l or C[p],[p] smooth if pis an even integer, odd integer or a non-integer respectively. f2. We show in the next section that c 0 (E)(i.e. the B-space of sequences in E converging to 0 is Cp,l smooth if E is cP, 1 smooth, that c (E) has a 0 cP norm if E has a cP norm and that c (i.e c (R) ) 0 0 2 2 is not c ' smooth. This example shows that a cP smooth B-space need not be cP,P smooth. Proposition 2.5 00 LP(s,L,µ) is C 00 ' smooth, cP-l,p-l smooth or C[p],[p]smooth if pis an even integer, odd integer or non-integer respectively. Proof. Let a.(f) = (llfll 'f =JI f(x) I Pdµ(x). Then it can be shown(refer to [l]) that . 1 k k J~rlf(x) l p- (sgn f(x)) ·h 1 (x)··hk(x)dµ(x) for k < [p] and = By H8lder's inequality, pt for p an even integer. IDk(a(f(x))[h , •• ,hkJI is 1 ~ ~i \lf\lp-kllh1il • ·llhkll • 'r herefore Dka. and hence Dkl\x\\ is bounded on bounded sets for k < [p] and if p is an even integer, for all k. Hence by Prop. 2.3 the result is proved. Definition. A family of functions (cpn.} E cP(E ,R+) will be called a cP partition of unity if every point of E has a neighborhood on which all but a finite number of cp 's vanish a. and ;; co = 1 • tr ' a. Definition. A Banach space E will be said to "admit cP partitions of unity" if for every open covering [U13} of E there is a cP partition of unity (cpa.} such that the support of each cp a. is contained in some U13 • Eroposition 2.6~ If E is a separable space then E admits cP smooth B- cP partitions of unity. Remark. Every metric space is paracompact and hence all 0 B-spaces admit c partitions of unity. It is not known if separability can be dropped from Prop.2.6, in particular, it is not known whether any non-separable Hilbert space admits c 1 partitions of unity . l) Refer to Bonic and Frampton [l]. A stronger version of this theorem is contained in Chapter V. 18 Proposition 2.7 Suppose that E and F are B-spaces and tha t E a dm i ts cP partitions of unity. Then g iven fECO(E,F) and e(x) E c 0 (E,R) with e(x) > O, there exi s ts a gECP(E,F) such that II f(x)-g(x) I\ < e(x) for all x in E. Proof . For every x in E find nei ghborhoods 2 Nx1 and Nx2 of x such that inf e(y) ~ e(x)/2 and if yENx y EN~ then \lf(y)-f(x)I\ < e(x)/4~ Now (N~n N~} covers E and by the hypothesis we can find a cP partition of unity (cpa.} supp orted by (N~n N~}. Pick, for each a.,an xa. in the support of cpa. and define g(x) = ~ f(x 0 )cpa. (x). 'rhen g(x) is a n element of cP(E,F) and \\f(x)-g(x)\I \I lJ (f(x)-f(x ) ) cp ( x ) (a\xEsupp ~a.} a. a. I\ E l\f(x)-f(x )\j~ (x) (a.\xEsupp ~a.J a. a. ::;; ~ (iif(x)-f(x~)I\ (a.lxEsupp cp a. J + \lf(x~) - f(xa. )\\) cpa.(x) where x a.' is a p oint in E such that supp cp a.c (Nx1 ,n Nx2 ,} • a. a. The last summa tion is ~ e(x). E e(x ' )/2 •cpa.(x) (a.\xEsup p cpa.} a. Q.E.D. 19 CHAPTER III DIFFERENTIABLE FUNCTIONS ON c (E) 0 Definition. If E is a Banach space, then c (E) denotes 0 the Banach space of all sequences X = (x.} with x. in l l E and \\xi\\ --- O. The norm on c 0 (E) is defined as llX\\ sup i \\x.l \I Bonic and Frampton in [l] and [2] proved that if E is cP smooth then c (E) is also cP smooth. In this 0 chapter we prove several strong er results. We show in Theorem 3.2 that if E is cP,l smooth then c (E) is also 0 cP,l smooth and in Theorem 3.1 that if E has a cP norm then c (E) also has a cP norm. In Theorem 3.3 we show that 0 c is not c 2 ' 2 smooth. This is the first example of a 0 cP smooth Ba nach space which is not also cP,P smooth. There is an important class of spaces equivalent to c (E). Sup pose that K is a compa ct subset of Rn 0 and that O < o. < 1. If f E c 0 (K,E) define llflla. Let Ca.(K,E) Aa.(K,E) 6 > = sup xly II f(x)-f(y) II /( llx-yll )a. [f E CO(K,E)\ l\f\la< co} and let [f E Ca(K,E) \ for any e: > 0 there is a 0 such t hat IIf(x) - f(y) II ~ e: \I x-yll a.whenever l/x-y//~<5} • 20 Then Bonic, Frampton and Tromba in [4] h ave p roved that Aa(K,E) i s equivalent to c (E). 0 N.H.Kuiper has shown that c 0 has a n equiva lent c 00 norm(refer to fl]). We give th e foll owing genera li- zation of th a t Th e orem~ result: Suppose that E has a cP norm, \\ x\\, away fr om zero. Then c (E) also has a cP norm away from zero. 0 Proof. t hat h(t) h is decreasing, h(t) = 1 for t 0 for t s 1, h(3/2) = 1/2, >- 2 and h(t) is concave downwa rd for 3/2. Now if X s t 00 First construct an h in C (R,R) such 00 = TT h(\lx 1 \I) !f;(X) i=l !/J locally depe nds only on a finite number of x. 'sand hence l !f;E cP(c (E),R). NO}r 0 let G = (X\<J;(X)~Y,, }. We sh ow that G i s convex. To do this suppose that !/J(X) and !/J(Y) are >- Y,, and s up p ose that \\xi\\s 1 and l\y 1 J\s 1 for i and t hat O < t < 1. 'rhen > N 00 TT h( II txi + ( 1-t )y ill) !/;( tX+( 1-t )Y) i=l 00 >- rr h(t\lx 1 \I + c1-t)\\:yi\\) i=l N IT h(t\lx. \I + c1-t)\ly. \I) i=l l l Now \\x \\ and \\y \i are s 3 /2 for a ll i,henc e 1 1 by th e concavity of h the last product is 21 N N i=l k=O ~ TT (th ( \\x.1 \I ) + ( 1-t ) h ( 11 y 1. 11 ) ) = ~. t N- k ( 1-t ) k ak where a = k ~ Tfh(\lx.11)· 1 ( FE~ iEF '-k ~k and where Tr h(llY-11)) 1 i~F i~N is the set of all subsets of 1,2, •• N having k members. m Now if b. > 0 then l a. l m l/m ~ b. ~ ( 1T b .\ i=l i=l ~ l • Hence ~ (Fl[" ( il};,h( llxJ) · ~h( llY11\))) l{) 1Ih( 11 I\) ) Nk (N) l/(~) -: N-k (N) N ( (1Ih( I\ xi \I) ) l < k · ( (N-k)/N ~ Then ~(tX (1/2) yi k/N .(1/2) ~ 1/2 N E tN-k(l-t)k•l/2 1/2. Hence k=O G is convex. Let a.(X) be the Minkowski functional of G. Then + (1-t)Y) k ) a.(X) is implicitly determined by the equation ~~x)) = 1/2. Since a.(X) locally depends on only a finite number of variables, we can apply the finite dimensional implicit function theorem to conclude that a.(X) E cP(c (E),R). Then a.(X) is an equivalent norm because G 0 is bounded and contains an open neig hborhood of O. Corollary 3.1 by Prop. 2.3, c Remark. c 00 0 has an equivalent C norm and therefore 00 0 is C ' l smooth. Although a.(x) has a boun::1ed derivative, rjJ itself 22 does not. In fact any function, F(X), of the form co F(X) = TT ch( 11 x. II ) ) 1 l where h(t) = 1 for \ti s 1, h(t) = 0 for ltl ~ 2 is not of class cP, 1 (c (E),R). To show this it suffices 0 to consider E R. Let a be the largest number between 1 and 2 = 1. For any M choose such that h(a) ~ (h(a+l/2M) )n < and let x0 n such that (n a's, O's } and x1 = [n (a+l/2M) s, O's }. Then we have F(X 0 ) = 1, 1 F(X ) <~and 1 ~ s l/2M. By Prop. 1.7 l\X -X \\ 0 1 IFCX 1 ) -F(X 0 )1 ~ \\X 1 -X 0 \I· s~p l\DF(X)\I l/2M· sup i\DF(X)\\ x Hence sup \\DF(X)\\ :;::: M , and since M is arbitrary, x sup\\DF(X)\\ x = co • It is possible to construct a nontrivial c"" ' 1 function on c 0 without evaluating a Minkowski functional as the following ex a mple shows. Example. Let 00 00 h E C ' (R,R), h(t);:::O, h(t)=O if \t\;:::}1. and .}1. J-%h(t)dt= 1. Define % % l}i. •. • l}i.h(yl). •h(yn)F( (xl +yl' • .xn+yn,xn+l' •• })dy •.. dy 0 1 23 where F(X) = inf \IX-YI\. l\Y\\:s:l F is continuous because II X-Yll • Suppose that I :xm I s; J4 if m > n(X) • \I F(X )-F(Y) ll s: \IX' -X\\ :s: )4, llY\\ s: 14 and x'm = x'm form :s: n(X), then Now if F(X' +Y) = F(X+Y) • Hence when II Z-X\\ s; }4, Cfln(X) ( Z) depends 00 only on the first n(X) coordinates and therefore is C • Also llX'-X\\ :s: J4, \\Yll s: J4 and y 1 = •• = .yn = 0 imply that :B'(X'+Y) = F(X'). Hence cpm(X') and \IX' -XI\ = cpn(X)(X') when m;?: n(X) J4 • The above implies th a t s; = lim c+>n(X) cp(X) n~ CX> exists and is C~ s; .}4 J-1'- •••. r}'{i. -14 h(y 1 ) • ·h(yn)llX-Z\I dy 1 •. dyn = \IX-Zll • Hence lc.p(X) - cp(Z) I :s: l\X-Zll which gives \\Dcp(X)I\ s; 1 for ali X. Finally let r(t) r E CCX>(R,R), Os;r(t):s:l, r(t)=l if ts; 0 and = 0 if }4s:t. Then r(cp(X) E CCX>' 1 (c ,R), r(cp(O)) 0 = 1 and the support of r(cp(X) is contained in the unit ball. Theo e rm 3.2 If Eis Cp,l smooth then so is c (E). 0 1 Proof. First find an f in cP' (E,R) such that f(x) = 1 if \lxll :s: 1, f(x) = 0 if 2 :s: \\xi\ and 0 s; f(x) s; 1. Define a map T from c 0 (E) into c 0 as follows: If X = (x ,x , ••• } E c 0 (E) let T(X) = (l-f(x 1 ),l-f(x ), •• ,J. 1 2 2 Then since T locally depend s on a finite number of coordinates, T E cP(c 0 (E) ,c 0 ) . Also \IT(X) - T(Y)\\ 24 = s~p\f(xi) - f(yi)I l $ sup\\Df(x)j\ x • suplx.-y. \ = l\fi\ 1 ·l\X-Y!i . l l • l Hence T E cP,l(co(E),co). By Cor. 3 . 1 we can find g in Cco,l(co,R) with g(O) = 1 and g(X) = 0 i f 'l'hen g(T(X)) E cP' 1 (c (E),R) and g(T(O)) = 1 \\ X\\ ".?: 1 0 . and g(T(X)) = 0 if i\ X\\ >- 2. Q.E.D. The following theorem will imply t ha t a C2 ' 2 bump function cannot be found for c 0 • We a ctually prove a slightly stronger result. Le t f E c 1 (c ,R) with Df(X) uniformly Th eo r e m 3.3 0 continuous . Then th e support of Proof. in f i s unbounded. If not, then there would exist an f c 1 (c 0 ,R) such that f(O) = 1, f(X) o if \\x\\ ".?: i and Df is uniformly contj_nuous. Pick N such that \\ HH sl/N implies llDf(X+H) - DF'(X) \\ by Prop.1.5 there is a T $ 1/2. Now if \\H\\ s l/N then with O<T<l such tha t f(X+H) - f(X) = Df(X+TH) [HJ so that \f(X+H) - f(X) - Df(X)[H] I= \Df(X+TH)[H] - Df(X)[H] I s Y,,\\H\I • Let A be the set of all X in c 0 such that 2N-l of the first 2N components of X have abso lute value l/N, the r e maining 25 component of the first 2N components has absolute value less than or equal to l/N and all the other components are zero. Since A is connected and X in c 0 even, for all there exists a Y in A such that Df(X)[Y] = O. IJ.1 herefore we can pick inductivel;y H , •• HN E A such that 1 Df(H 1 +···+Hk-l)[Hk] = 0 and such that H1 +•••+ Hk has at le a st 2 N-k components equal to k I N. But then \\H 1 + ••• + HNI\= 1 and \f(H 1 +••+HN) - f(O)\ s N 6 lf(H 1 +••+Hk) - f(H 1 +··+Hk_ 1 ) - Df(H 1 +··+Hk-l)[Hk]\ k=l N s 6 72\IHkll = Y,, which is a contradiction. Q.E.D. k=l Corollary 3.2 2 2 smooth. c 0 and c 0 (E) are not C ' Proof. Any function in C2 • 2 (c ,R) has a uniform0 ly continuous first deriv3tive. Corollary 3.3 of c 0 Suppose that U is a bounded open subset and that f E c 0 ,R), f E c 1 (u ,R) and Df(X) is uniform- cu ly continuous on u. Then f(aU) is dense in f(D). 26 CHAPTER IV ZERO SETS OF cP FUNCTIONS In this chapter we consider the problem of finding a cP function with a prescribed zero set. It will be shown that for separable Banach spaces, cP,P smoothness is a necessary and sufficient condition that every closed set be the locus of zeros of a cP function. If a B-space is cP smooth but not Cp,p smooth it will be shown that the probl e m can still be solved for a special class of closed sets. Theorem 4.1 Let E be a separable cP,P smooth Banach space. Then every closed subset ,A, of Eis the zero set of some cP,P function. Proof. 1, h(x) = 1 if \lxll s that 0 s h(x)s 2 s llxll First construct an h E cP,P(E,R) such 1 and h(x) = 0 if • Let x. be a dense countable subset of the coml plement of A and let d(x. ,A) denote the distance from x. l to A. Define f.(x) l l h( x-xi ) 2d{x. ,A) and let l = sup\IDkf.1 (x) \I xEE rhen define gn(x) if n > m > k then 1 and N p max l. , k Sp M.k l 27 s 6n ( sup p=m+l xEE Hence the Dkg (x) 's converfi.; e uniformly to conti nuous n functions g(k)(x). Repeated appl ic atio n of Prop. 1.11 gi ve s Dg(k)(x) = g(k+l)(x). He nce g(O)(x) E cP(E,R). Th a t g(O) i s a lso in cP,P(E,R) foll ows easily. If xEA then f.(x) = 0 for all i and h e nce g (O)(x) =O . If x¢A l then find a n xi such that d(x,xi) < ~d(x 1 ,A). Then f. (x) > 0 which implies g(O)(x) > O. Q.E.D. l Corollary 4.1 Let A and B be disjoint closed sub sets of a cP,P smooth sep arable Ba n a ch space. Th en there exists a cP function F such th a t 0 s F(x) s 1 and F(x) = 0 or 1 if and only if x E A or B. Proof. By Urysohn' s Lemma there i n CO ( E , R ) sa t i s f y i ng i ~ an f 0 s f ( x ) s 1 and f ( A) = 0 , f ( B ) = 1 • Apply Prop. 2 .7 to obtain an fl E cP(E,R) wi th \f(x)-f (x)\ 1 < 1/3. By t he t heore m ther e exist s g i(x) E cP(E,R),i=l,2, ~ with gi(x) 0 and gi(x) 0 if and only if x E A or B for i = 1 or 2. Now find a ~ E C 00 00 ~((x\xsO}) ~(3( f = 0 and ~((x\x~l}) ' = (R,R) with Os~(t)sl, 1. Then r 2 (x) 1 (x)-l/ 3 )) h as v alues betw ee n 0 a nd 1 and has v a lue 0 on A and 1 on B. We can th e n take F(x) to equa l 28 Q.E.D. Theo re m 4.2 Let Ebe a B~n a ch space which is not cP,P,p<oo, smooth. Then there exists a closed subset of E which is not the zero set of any Proof. cP function. Le t Bi be a sequence of disjoint op e n b al ls in E of radii l/i converg ing to some point x E such t hat distanc e( B.l ,B.) > 0 if i I ,1 and suppose that f E cP(E,R) with f(x) if x EA. Then letting g.(x) l = j • Let 0 in A =E -U• B.l l 0 if and only f(x) when x E B.l and g(x) = 0 when x ¢ B.l , we have that the g.l (x) 's are of cla s s cP a nd have bounded supports. By the non cP,P smoothness of E, sup llDPgi (x) II xEB. = 00 • It then follows that Dpf(x) is J. not continuous at x 0 which is a contra diction. Corollary 4. 2 There exists a closed subset o f c Q.E.D. 0 which is not the zero set of any C2 function. Proof. c 0 is not c 2 ' 2 smooth by Cor . 3.2 • By Theorem 4.1 and 4.2 a separabl e Banach spa c e , E, is cP,P smooth if a nd only if every closed subset of E is the zero set of a cP function. Theore m 4 .1 may be true for nonseparable B-sp a ce s but t his appears to b e a difficult problem . Indeed, if every clo sed subset of a n onsepa r ab l e Hilbert space H wa s th e zero set of a cP 29 function, then H would admit cP partitions of unity. To see this let [U } be any locally finite cover of H. a. By assumption we can find fa. E cP(H,R) such that fa.(x) ~ O and fa.(x) f a. > 0 if and only if x E Ua.. The n ~a.(x) = (x)/ 6f (x) is a cP partition of unity refining [Ua.} . a. a. Another question t ha t we can pose is: Given disjoint subset s A and B of a B-space E with distance(A,B) > O, does there exist a Cp,q function f such that f(A)=O and f(B)=l ? An equivalent question is: Given a subset A o> O, does there exist a Cp,q function f with of E and a f(A)=l and f(x) = 0 if distance(X,A)~ o? If A is convex and t he space i s uniformly convex the answer is yes for p=q=l as we show in the corollary to the next theorem. 'rheorem 4.3 Suppose that E is a uniformly convex B-space 1 and that \\x\I E c (E-(O} ,R). 'rhen if A is a closed convex 1 subset of E, d(x) = distance(x ,A) E c (E-A,R). Proof. A well known consequence of uniform convexity is th a t there exists a unique closest point p(x) in A to xEE and th at p(x) is continuou s. Now suppose that xf_A. Since the norm is c1 , llx-(p (x )+h)\I= llx-p(x)\\-Dl\x - p(x)\\[h] + o(\ih\\). For all h with p(x)+h EA we have,by the definition of p(x), that II x-( p(x) +h) II ~II x-p(x) JI , hence D\\x-p(x)\\ [h]s;O. The hyperplane L = [y\D\\x-p(x)\I (y-p(x) )=0} is therefore a supporting hyperplane for A at p(x). Hence for all \\h i\ <p(x), d(x+h,L)s;d(x+h)s;d(x+h,p(x)). Thi s gives 30 II x-p(x) II +Dll x-p(x) II [ h] s;d(x+h) s;\I x-p(x) II +Dllx-p(x) II [ h] +o( II hll). Hence ld(x+h) - d(x) - Dllx-p(x)li[h] I s; o(\ihll) so that Dd(x) = Dllx-p(x)ll • Since p(x) is continuous,Dd(x) is continuous. Remark. Uniform convexity implies tha t E is re flexive and hence by statement b) of ChapterII,if E is separable and 1 uniformly convex then llxll E c (E - (O} ,R). Corollary 4.3 If A is a convex subset of a uniformly convex B-space E and if 6> O, there exists an f E C1 ' 1 (E,R) with f(A) = 1 and f(x) 0 if distance(x,A) = :<! 6 . Proof. Find a g E Cm'm(R,R) with g(t) t s; 6 /3 and g(t) = 0 if t = 1 if ~ 2 ~ /3 . Then god(x,A)EC 1 ' 1 (E,R) and satisfies the boundary c onditions. If a separable B-space is cP smooth but not cP,P smooth, the n cert ai n closed subsets may still be t he zero sets of cP functions. Definition. A subset A of a B-space E will be said to have a cylindrical boundary if for all x E aA there exists 0 0 a neighborhood Nx of x such that An Nx (A is the interior 0 of A) is weakly open in Nx (i.e. An Nx Nxn W for some weakly open W). Theorem 4 . 4 Let Ebe a cP smooth s eparable B- s pace. 'rh e n any closed sub s et A whos e compl eme nt has a cylind- rical bou!ldary is the zero set of some cP function. 31 Lemma 4.1 Let A be a weakly open subset of a separable B-space. Then the complement of A is the zero set of a c"" function. Proof. We can write =n i w. A where l n. n {x\ \yi(k)(x) \<l} l w.l E E* • k=l EC (R,R) with a(t) = 0 if \t\ ~ 1 and O<a(t)s:l n. if It I < 1. Define g.l (x) 111 a( Yi(k) (x)) • Then k=l 00 00 g.(x) E C ' (E,R) and g.(x) > 0 if and onl;y if x E W.• Let 00 CJ l l Let MJ.k sup\\Dkg .(x)ll and let N ,J P xEE CD f(x) l 1 = "'u""" p=l 2PN p g (x) p sup M.k • Define j,ks;p J . o d as in • We can app 1 y t 1De same me th · the proof of Theorem 4.1 to show that the derivatives of all the partial sums converge uniformly. Prop. 1.11 gives 00 00 that f E C ' (E,R). Clearly f(x) > 0 if and only if xEA. Q.E .D. Proof of Theorem 4.4 Nx such that Nx n cl there exists a Then For each x in E fjnd an is weakly open in Nx • By Prop. 2.6 cP partition of unity {~i} CA fl supp ~ i refining {Nx}. = supp ~in Wi for some weakly open set 00 Wi. Using Lemma 4.1 we find fi(x) E C (E,R) such that f.(x) l ~ 0 and f.(x) > 0 if and only if x E Wl.• Defining. l F(x) =6f.(x)~.(x), we have th~t F E cP(E,R) and F(x)=O i l l if and only if x E A. Q.E.D. 32 Remarks. a) An example in p 2 of an open set with cylin- drical boundary which is not itself weakly open is C = [x\ \xii < 1, for all i}. For any y E R2 suppose that \y. I <~for i > n(y). Then if Bis the open l ball of radius~· about y, Bn C = Bn £xi \xi\<l,i~n(y)}. Hence C has a cylindrical boundary. Since C cont a ins no linear subspace it is not weakly open. b) Open sets with cylindrical boundaries are closed under finite intersections and finite unions 2 but not under countable unions. Again consider R and let en By a) each Cn has a cylindrical boundary but we show that Ucn does not. Let U be any neighborhood of zero and suppose that U contains the open ball of radius r. Find n such that l/n < r/2 and suppose t hat W is any weak neighborhood of e 1 /n,([ei} is the orthonornal basis). Then W contains a set of the form N [x\(yj,(x-e 1 /n)) < l}, j=l, ••• m • Find k such that \y~\ < 2n/3. Then the point e 1 /n + 3ek/2n lies in both N and U but is not contained in UC n • Hence UC n does not have a cylindrical boundary at 0. c) We pose the following question: Suppose that E is cP smooth but not cP,P smooth. Then if A is the zero set of some cP function, is a(CA) cylindrical? 33 CHAP'l'ER V Cp,q PARTITIONS OF' UNITY Definition. frn l 'f' ex. } will be called a Cp,q partition of unity on a Banach space E if [cpa.} is a partition of unity and cpcx. E cP,q(E,R) for each ex.. Definitiog..!. A B-space E will be said to "admit Cp,q partitions of unity" if for every open cover [U 13 } of E there [UR}• t-' exists a Cp,q partition of unity [cp} supported by a. ' We show in this chapter that a separable cP,q smooth B-space admits Cp,q partitions of unity. We also show that while [cpi} may be a c 2 partition of unity for P. 2 , there exists a bounded sequence of real numbers [a.} l such that ~. a.cp.(x) ¢ C2 ' 2 (P. 2 ,R). We begin with a lemma. l l l Lemma 5.1 Let E be a separable cP,q smooth B-space and suppose that [Ucx.} is an open cover of E. Then there exists four countable locally finite open covers [V~}, [V?} l l , [V{} and [V{l of E and maps gi E Cp,q(E,R) such that: 1) vi vi ' vi 2) [V~} refines l 3) 0::;; gi (x)::;; 1, gi c:: c= v ~ ' v~ c v~ ' i ~ 1 • (U } and is locally finite. a. (Vf) == l and gi (GV~) == O. 1 ) > o for j == 1,2,3. 4) If q ~ 1 then dist(v0l ,cv0+ l 34 Proof. Using the Cp,q smoothness, find for every x E: E a cpx E: cP' q(E ,R) such that 0 s cpx s 1, cpx(x) =l Ua. Let and support cpx is contained in some lylcpx(y) > ~}. Then Ax = (Ax} covers E and since E is Lindelof, we can extract a countable subset of (Ax} which also covers E. Denote the elements of this subset by A. = l 00 fy\cp. (y)>Y,d. Now we can find f. E: C '""(R,R), j ?:2, such that J l 1 1 1 if t . ?: ~ and t . s 2 + -J. , i < j , f .(tl, •.• ,t .) J J ,) l ' 11 12 . . 0 if t j s 2 -J ,and t i>-'2+-j,i<J. Now let for j ?: 2. it (x) 1 = cp (x) and 1 ':{t.(x) J = f.(cp (x), •• ,cp.(x)) 1 J J Define v~l (x\ it. (x) > 14} v?l (x\ ':Iti(x) > ~} v~l (x\ ':It.l (x) > }4} v~l (x\ ':It.l (x) > O} l Property 1) follows from the definition . Since V~c supp cpi , l (V1) refines (Ua}. To show that (Vi} covers E suppose that x EE and that i(x) is the first integer for which cp.(x) >-~.Such Then 'l'i(x) = 1 and hence x E vi(x) so l an integer exists because the A. 's cover E. again suppose that x EE l (Vi} covers E. Now and find an inte g er n(x) such that cpn(x)(x) > ~. Then there exists,by the continuity of cpn(x)' a neighbohood Nx of x and an ax >~ such that 35 inf cp ( )(y) ~a • Pick k large enour-;h so tltat 2/k< a -Yr:. n x x x EN 'Y x l 2 Then cpn(x)(y) > '2 + k for y E Nx and hence 'l'j(y) = 0 for yENx v1:J = 0 for <j ~ k so th;:it [V~} is l and j ?:'. k. 1rherefore N () x 00 locally finite. Finally take some h E C (R,R) with h(t) if ts;}'{., h(t) = l = o if t:21{. and Osh(t)s;l. Defining gi(x) h('lt. (x)) we have that g. E Cp'q(E,R) and that property 3) l l 1.5. holds. Property 4) follows from Prop. Q.E.D. Theorem 5.1 A separable Cp,q smooth Banach space admits cp,q partitions of unity. Proo~ Let [Ua.} be any open cover of E and use Lemma 5.1 to get four locally finite covers [V~},j =1,2,3,4 l refining [Ua.} and maps gi E Cp'q(E,R) satisfying the conditions of the lemma. Let f 1 (x) = g (x) and 1 fi(x) gi(x)(l-g (x))···(l-gi-l(x)) for i>l. Then fi E Cp'q(E,R) 1 V~l and supp f.l (x) c: supp g.l (x) c .. , hence every point of E has a neighborhood on which all but a finite number of f. Is vanish. Now since [xlg. (x) = l}:;) l l v?, and cv?} covers l l E, for every x, n 1T(l-g. (x)) i=l l "" O, for some n. n Hence 6 f.(x) = 1 - lim TT (1 - g.(x)) = 1 and [f.(x)} i=l l n-Ho i=l l l is a Cp,q partition of unity refining [Ua.l Q.E.D. 36 If E is Cp,q smooth and separable then by the above theorem we can find a Cp,q partition of unity [ cµ. } l such that diam.(supp qi.) are uniformly bounded. We ask the l question: Does there exist a Cp,q partition 0£ unity (cpi} such that (5.1) 6 a.cp.(x) E cP,q(E,R) for all bounded real a.? . l l l l If E = Rn the answer is yes: Theorem 5 .2 00 Suppose that d > 0. Then there exists a C 00 ' partition of unity (qi } on Rn such that diam(supp cp.) <d 1 l ER, E a.cp.(x) E and for every bounded sequence a. i l l l Ceo ,oo(Rn ,R) • Proof. Find hE Cco(R,R) with h(t) > 0 if \ti< 1 and h(t) = 0 if It\ :2: 1. Write x = [x 1 , ... ,xn} and let L be the lattice of points [dk1/2 n, ••• ,dkn/2 n} where k 1 , •• kn are integers. Label these points by xi, i = 1,2, ••• and define n TT h(x. j=l J Then the supports of the f. 's cover Rn, f. l E Cco'co(Rn,R) l and diam(supp fi) = d for all i. Finally let CX> - • f.(x)/ 6 f.(x). Then cp.(x) =cp.(x+xJ-x 1 ) l l 1 J 1 exists an Mk such that ~i(x) = • so tL.at there s~p\\Dkcpi (x)\\ <Mk< co for all i. Now any point of Rn is covered by the supports of at most (2n+l) cp'.s. Hence if la. l~M, \\Dk6a.cp.(x)\ls 1 1 l l Q.E.D. 37 When E is infinite dimensional then no such canonical construction is possible. In fact if the supports of the partition functions have uniformly bounded diameters then by Lebesque's Covering Theorem for any N there a re points of E at which more than N of the partition fu nc t i ons are non-zero. This seems to sugg est that the answer to the question is false for q ~ 1. We show below that this is the case for E =i 2 and q = 2. The first theorem i s of interest in itself. We consider the contin2 2 uous map CJ: f -7f defined by CJ(x) = L: Ix . le . ,where e i l l i is the orthonormal basis, and show that CJ(x) i s not unifo rmly ap p roximable by a C2 ' 2 function . We will look at CJ(x) a gain in th e next chapter when we study cP ap pro ximations. Theorem 5.3 Suppose that B i s a ball of radius d and 2 2 2 center z, t hat f E c ( i , i ) and that sup\\f(x)-CJ(x)\\ xEB ~ a < d ~3/6. 'rhen sup\\D 2 f(x)I\ = xEB 2 Proof. Since a c function with sup\\D 2 f(x)\\ < oo x EB has a uniformly continuous deriva tive on B, the theorem Cl). will fol J ow from the following lemma; Let B be the cl osed ball in p 2 of radius d 2 and center z. Suppose that f E c 1 U ,£ 2 ) and that Df is Lemma 5.2 uniformly cont i nuous on B. Then sup\\f(x)-CJ(x)\\ 2! d "1?/6. xEB 38 Let x . and f.(x) denote (x,e.) and . 1 1 Proof. 1 (f(x),e.) where (e.} is an ort h onormal b as is . For an 1 1 ()) 2 a rbitra ry 1 > e: > 0 choose k such that 6 z. < e: and pick k+l l 6 < e: s uch th a t x,y E B and l\x - y\\ < 6 implie s l\Df(x) - Df(y)\I < e:. Let N be the g re a test integer les s 2 2 2 2 2 2 2 th an or e qual to (d - e: )/o so that d - 2e: < o N < d - e: 2 • k 6 Denote by z ' . Then if we let F be the N dimen- ziei i=l sional box (yl IY·l - z!l I s 6 for i = k+l, •• ,k+N and \y. - z! I = 0 otherwise} we will have F c l l B. By the mean value theorem(Prop.1.5) if yEF and Yk+l'"'yi' " 'Yk+N are fixed then f .(y ) =a +by.+ e(y.) l 1 where a = f . (y - y .e . ) l 11 l , b = ( D f ( y - y . e . ) [ e . ] , e . ) and 1 1 l l ((Df(y-y.e. + e.(y.)y.) - Df(y-y.e.) )[y.J,e.) 11 l 11 11 1 1 for some 0< e. (y.) < 1. l Since IY· I< 6 l l ~ \e(yi)I s e:lyij. Now a simple calculation shows that £~a+ byi +e(yi) - !Yi I )2 dyi > o 3 (~ - '>e) , Hence > 2 JFl\f(y) - u(y)\1 dyk+l' .• ,dyk+N • Therefore sup \l:f(x) - u(x) 11 xEF 2 39 (d > 2 - 2€ )( 1~ - 2 2€) • Hence sup[I f (x) - cr (x) II. xEB J 2 - 2€ 2 )( 1~ - 2€) • Since sup\lf(x) - cr(x)\\ > (d xEF trary, supl\f(x) - cr(x)\\ xEB :2: <!: € is arbi- Q.E.D. d/2"13 • c2 partition of unity Now suppose that (c.pi} is a 2 and that diam(supp cp. )< d for all i. If we pick l for p . points xlEsupp c.p. and let b. l s: l i = cr(x) then \ltJi b.cp. (x) - cr(x)\\ = 1 1 II~ (cr(xi )cpi (x) - <J(x)cpi (x)) II ~Uxi - xllc.pi (x) s: Hence by d. l Theor~m 5. 3 if B is a l ball of radius r > 2~ d,then sup l\D 2 ~b.cp. (x)ll xEB i l l (5.2) Let a i bi if ::: supp cpi n B ~ = CD 0 and a.l Then the a. IS are bounded and 1 o ::: 0 o °!J herwise, ~ a.q:i. (x) = i l 1 2J. b.cp. (x) 1 1 1 when xEB and therefore (5.3) sup\\D xEB 2 ~ a.cp. (x)ll 1 1 CD i I • The next theorem will show that (5.3) also holds when the ai's are a suitable bounded real sequence. Theorem 5 .4 Let {cpi) be a cP, p <!: 2, partition of unity on p 2 and suppose th a t diam ( supp cpi) < d for al 1 i. Then there exists a sequence (ai} of bounded real numbers such that sup \\Dk 2J a .cp. (x)\\ x i 1 1 == CD for 2 s: k s: p • 40 Proof. Choose bi BJ.= [x\ = a(xj) and r > 2/3 d as above and let 2 l\x-2reJ.llsr}. Then sup \ID 2Jb.cp.(x)\I 1 1 xEB. i = oo so we J can pick j < 2 hj E P with llhj\I = 1 such that yj E Bj and I ;G. b.1 D2cp.1 (yj )[hj ,hj] I . Let F.J = [i I cp.1 (yj) > O}. 1 Then if j I j I ,F. ,J n F.J a. = 2 . . . sign(D cp. (yJ)[hJ ,hJ]) if i E F. = 0 1 = I 0 • Now define J 1 if i ~ F j for any j • Then 1r 3 ~ llb.D 2cp.(yj)[hj,hjJll 'EF j 1 1 > j/3r. 1 2 is arbitrary, sup \ID 6 a.cp. (x)ll 1 1 llxll s;3r i Frop. 1.6 the Theorem follows. Since j = oo and by Q.E.D. 41 CHAPTER VI SMOOTH APPROXIMATION Definition. Suppose that E and F are Banach spaces, t h at U is an open subset of E and t hat f E Cq(U ,F). 1.I1hen if Osq<ps;oo we will say th a t f is Cp' q . given E: ap~roximable . I' on U if > 0 there exi.sts a g E cP(u ,F) such that sup llDkf(x) - Dkg(x)ll < e: • We will say that f is xEU,Os:ksq strongly C approximable on U if given any e(x) E p,q c 0 (u ,R+) there exists a g E cP(u ,F) such that for x in U, sup llDkf(x) - Dkg(x)ll < e(x). In both cases th e functions Osks:q g will be called C approximations. p,q It is well known that if E is finite dimensional then every f E Cq(E,F) ,( g "2: 1), is strongly C approxip,q mable on E. When E is infinite dimensional but separable, 0 Prop. 2.7 implies that every f E c (E,F) is strongly C p,O approximable if and only if E is cP smooth. However when q > 0 it is not known whether there exist any infinite dimensional Banach spaces such that every Cg function is Cp,q approximable. In particul a r, it is not known whether every c 1 function on ~> c parable fhlbert ~: pace i~3 c2,l approximable. 42 The theorem below will show that if a Cq function on a separable Cp,q smooth B-space is locally cp,q approximable, then it is strongly approximable on the whole space. This would be an essential theorem in constructing Cp,q approximations on manifolds modeled on Cp,q smooth Banach spaces. Theorem 6.1 Let Ebe a separableCp,q smooth B-spac e and let F be another B-sp a ce. Let f E Cq(E,F) and s uppose that for every x in E there is a nei ghborho od Nx of x such that f is C approximable on Nx. Then f is strongly C p,q p,q approximable. Proof. If e(x) > O,let {U } be an open cover of E refining a. [N } and such that inf e ( x) > 0. Apply Lemm a 5 .1 to get four x xEU a. locally finite subcovers [V~} refining [Ua.} and functions g. l E cP,q(E,R) satisfying the conditions of the lemma. Let e:l. inf e(x) and let M. = llC1-g 1 (x))···(l-g. 1 (x))g.(x)I\ • 4 l ll q xEV. l By the hypothesis, th ere e x ist s an h. (x) E cP,q(V~ ,F) with l (6.1) l sup \\Dkf(x) - Dkh. (x)\I < e:. /(2q+iM.) 4 l l l xEV. ,O::;;k::;;q • l Now define r 0 (x) = f(x) and fi(x) = f(x)(l-g 1 (x))··· (1-gi(x)) + h 1 (x)g 1 (x) + h 2 (x)g 2 (x)(l-g 1 (x)) + ••• + hi(x)gi(x)(l-g 1 (x))•••(l-gi_ 1 (x)) 2 2 If x E v 1 U •.• UVi then for i>l. (l-g 1 (x)) · · ·(1-gi(x)) O,hence 43 1 ) fi(x) E c p,q( v 11 u ... uvi,F (6.2) i ~ 1. ' 4 Also if x %. vi then gi(x) = 0 so that 4 fi(x) = fi-l(x) when x~ v.l (6.3) • Now using (6.2) and (6.3) and the fact that [V~} and U tv{l cover E, for every x there is a neighborhood of x and an integer x kx such that f. l+ 1 (y) = f.(y) l for y E ux and i > kx and f i (y) E cP(ux ,F). Hence h(x) = lirn f.(x) exists and h(x) E cP(E,F) • Now . l l~CD fi (x) - fi-l (x) = (hi (x) - f(x)) ·(1- g 1 (x)) • · ·(1-gi-l (x))gi (x) sup \IDk(f. (x) - f. (x) )II 1 1 i4 xEVi and hence f(x~ll· ~ (~) sup llD1\h.1 (x) - s j=O J xEV.4 i suvp 4 11Dk(C1-g (x)) •• (l-gi_ 1 (x))gi(x))ll 1 xE . s l ~ (~)€./2q+iM. • M. i l i s € ./2i for ksq. Using this and 1 j=O J ( 6. 3) we have for Qsksq , \tDkf(x)-Dkh(x) II = l\Dkf(x)-DkfN(x) II for some N, and this is N e(x). < s E 4 [j\xEV.,jsN} \\Dk( f .(x)-f. J J- 1 (x))ll J E l/2j < e(x). Hence f(x) is strongly C j=l p,q approxirnable. Q.E.D. Consider separable Hilbert space, p2 , with orthonormal basis [e.} l = • Write x = ~ x.e. and define cr(x) E \x. \e. as in Chapter V and i l l i ~(x) l = l L). x. \x. j. 'I 'hen i l l 44 II cr(x) - cr(y)IJ s: \Ix - y\\ and 16. (x.l +y.) Ix.l +y.l I- 6. x.l Ix.l I l l - (2cr(x),y>I l I~. y?I = [ly\\ 2 • Hence DL:(x) = 2cr(x) and l s l L(x ) E c 1 (P 2 ,R). We observe that cr(x) is nowhere differentiable • To show this let x E p2 and suppose th a t cr is differentiable at x. Then there exists a 6 such that when l\y\\ < 6 n such tha.t :1!: ,\lcr(x+y) - cr(x) - Dcr(x)[yJ\l < l!Yll /8. Choose I xn I < 6/4 and let y = 6en • Then 36/4 + 36/4 - 26/4 = 6/2. On the other hand II cr(x+y) + cr(x-y) - 2cr(x)\J = II cr(x+y) - o-(x) - Du(x)[y] + cr(x-y) - cr(x) - Dcr(x)[-yJ\J s; l\cr(x+y) - cr(x) - Dcr(x)[yJ\I +II cr(x-y) - cr(x) - Dcr(x)[-yJ\I s: 6/4, contradiction. We pose the question: Is there any b e tter c2 l ' approximation to L(x) on the unit ball than a constant From Theorem 5. 2 it follows that L:(x) is not c 2 l approximable by C2 ' 2 functions on any ball. The function? ' following theorem shows that if\IL:(x)-g(x)\\ a ball of radius R, where g E c 2 < R/2 on ce 2 ,R), then g can not have a decomposition of the form g(x) - Theorem 6.2 1 Suppose that G(x) E c 1 6 g.(x.). i l l ce 2 'P 2 ) and that G(x) = ~ h.(x.)e .• Then if Bis a ball of radius R, i l l l supl\ G(x) - cr(x) \I ~ xEB R/2. 45 Let B h ave center Proof. -- 6 R > e: > 0. P ick n s uch that ~ (6.4) e\lx-bl\ /R. jh.(x.)-h.(b.)i 1 1 a. ,i =n+l a < J and supp ose t hat n e: and l e t b = ~ a.e .• 1 J t! \Ix - b\\ < o implies Now find 6 such th a t -DG(b)[x-bJ\I 2 00 1 \\ G( x ) - G(b) when \x.1 - b.1 \ < o Thus dhi(b . )(x.-b.)j < dx. 1 i i , ~lxi-bil. 1 < 6 Choose N larg e enough so that and let z = n+N R - e e . • The n l\zl\ = R - e: so that (b ± z) EB. :0 j=n+l N J By applying (6.4) with i = n+l, •• ,n+N we obtain l!G(b+z) + G(b-z) - 2G(b)ll ~ (6.5) l\G(b+z) - G(b) - DG(b)[zJ\I. + l\G(b-z) - G(b) -DG(b)[-z) II~ 2el\zl\/R < 2e • Since a(b+z) = a(b-z) we have llG(b+z) - cr(b+z)ll + l\G(b-z) - cr(b - z)ll + 2\!G(b) - cr(b)\I - 2cr(b+z)!l + 21\G(b) - cr(b)ll ?-'. llG(b+z) + G(b-z) , which by (6.5) is ll2G(b) - 2cr(b+z)I\ - 2e + 2llG(b) - cr(b)I\ - 2e = 2llz\\ - 2e: = :2: :2: 21\cr(b+z) - cr(b)\\ 2R ·- 4e. Therefore either llG(b+z) - cr(b+z)\\, \IG(b-z) - cr(b-z)il Hence sup \\G(x) - a(x)\I xEB theorem is proved. :2: or \lG(b) - cr(b)I\ is ~- e. Since :2: ~ - e. e is arbitrary , the 46 CHAPTER VII WEAK C APPROXIMATION ON p,q £2 As stated in Chapter VI, it is unknown whether 2 every c 1 function on £ is c 2 1 approximable. In this ' chapte r we show that C approximation ca n be performed p,q . . on £2 provided we use a weaker approximation condition on the derivatives. The approximation is first done 00> co 2 local ly and then the C ' smo othness of R is used to build up a global approximation. We first point out that the usual finite dimen00 sional technique of convoluting a cP function with a C function having a small bounded support(i.e. letting f(x) = Jf(x+y)cp(y)dµ(y)) to obtain a C approximati on, co' p Rn 2 fails on £ • There is of course no translation invariant 2 borel measure on £ but we might hope that given f E 2 Cq(R ,F) there would exists a probability measureµ on P2 such that f(x) = Jf(x+y)dµ(y) is of class cP, p>q. This , however,is not the case and we sket ch a proof for q = 1. Define co (1-cos ~nx) n ----· n where x ~ n xnen. Then it is not hard to show that 47 F (x) E c 1 (p 2 ,R) and that F 1 (x) is nowhere second differ1 entiable. Suppose now that µ is a probabilty me asure on p 2 i 1 (x) = JF 1 (x+y)dµ(y). with bounded support and define Then J~ ( c = where 1 - co spfi y n) In d µ ( y ) c ( Jc o srn: y n d µ y ) ) = and Now ~ < °" 2 + 2 c ( Is i rum y n d µ y) ) ( fsinmy dµ(y) \ Os:a n 1 - n y /2 n ) ~ ~ Jc1-ny~/2)dµ(y) s:2 where y n = JllYl\ 2 dµ(y) <°"follows lim sup an • \ :r cosfti y:dµ(y) Jcosmyndµ(y) Jy~dµ(y). CX> From/~ 'Yn n lim inf nyn = 0 which gives 1. Since the an's do not approach O, the same method of proving F (x) is nowhere second differentiable 1 can be used to show that F1 (x) is nowhere second differ- entiable. This ca n be generalized. Define FP(x) = iS (1 - cosVri xn)/ n(p+l)/ 2 • n=l 2 Then FP E cP(p ,R) and FP(x) is nowhere p+l differentiable. If µ Fp (x) = is any probability measure on 1 2 and if we define JF (x+y)dµ(y), then F is nowhere p+l differentiable. p p 48 In the constructions to follow we will need two propositions about measures on Banach spaces. The first proposition is well known. We recall that a probabili ty measure u on E measure satisfying Proposition 7.1 is a positive regular Borel µ(E) = 1. Let µ be a probability measure on a complete metric space Q . Then for every e: > 0 there of Q such that µ(K )~l - e. e: E CO(E,F) where E and F are Banach spaces exists a compact subset K Lemma 2·1 Let f e: and let K be a compact subset of E. Then lim t~o Proof. sup hEK, llYll ~t l\ f(y+h) - f(h) II = 0 • Suppose e: > O. For every h EK find Rh such that liy-hl[ <Rh implies l\f(y)-f(h)ll < r./2. Let {B(hi ,Rh.)} be a 1 finite subcover of the cover {B(h,Rh)}, where B(h,Rh) is the ball with center hand radius Rh. Let 6 be the Lebesque number of {B(hi ,Rh.)}. Then for every h E K and y E: E with l \\yl\ ~ 6 we have h,y+h E B(h.l ,Rh . ) for some i. Hence lif(h+y) 1 f(h)\I ~ lif(h+y) - f(hi)il + \\f(h) - f(hi)ll ~ e:/2+e/2 = e:. Q.E.D. Proposition 2 .2 Suppose that µ is a probability measure on a B-space E with compact support K and suppose that f E cP(U,F), p;;? O, where U is an open subset of E. Then if V is an open subset of U such that the algebraic sum V+K is contained in u, g(x) = Jf(x+y)dµ(y) E cP(v,F) and Dkg(x) = JDkf(x+y)dµ(y) for 0 ~ k ~ p • 49 Proof. there is a 6 > 0 Suppose x EV and e: > O. By Lemma 7 .1 such that ll zll < 6 implies supllf(x+y+z) - f(x+y)\I < e:. yEK But then JJzll < 6 implies l~ g (x+z) - g(x)\I Ji\f(x+y+z) - f(x+y)\Jdµ.(y) s s e: • Hence g E c 0 (v ,F). Assume that g(x) E Cq(V ,F) for q < p and Dkg(x) JDkf(x+y)dµ(y) for 0 s ks q. We show that g(x) E Cq+l(V ,F) and Dq+l g (x) = Jnq +lf(x+y)dµ(y). For any x in V, lim t-o sup \i(Dqf(x+ty+z)-Dqf(x+z;) - Dq+lf(x+z) [ty] )/tlJ \ly\l=l,zEK = lim sup ((Dqf(x+ty+z) - Dqf(x+z) - Dq+lf(x+z)[ty])/t ,w). t-'>O lly\\ =l, zEK wEF*, !lw\I sl Now by Prop. 1.5, (Dqf(x+ty+z) - Dqf(x+z) ,w) = (Dq+lf(x+z+Ty)[ty] ,w) for some O < T < t so the la s t limit is :.,;; ((Dq+lf(x+z+Ty) - Dq+lf(x+z) )[y] ,w) lim SUJ? t - o II Yll =1, zEK, wEF*, I\ wll sl O<T<t = \l(Dq +lf ( x+z+ Ty) - Dq+l f (x+z ))Cy] II lim sur t - o iiY ll =l, zEK,O<T<t 0 by Lemma 7.1. Hence lim SUJ? II Dqg(x+tyf - Dqg(x) t - o llYll=l - (Jnq+lf(x+z)dµ(z)) [yJ\I = O , so that Dq+lg(x) exists and equals JDq+lf(x+z)dµ(z) . Q. E.D . 50 Corollary 7 .2 Let µ be any probability measure on a B-space E and suppose that f(x) E cP,P(E,F). Then g(x) = Jf(x+y)dµ(y) E cP,P(E,F) and Dkg(x) JDkf(x+y )dµ(y) for k :s; p. Proof. Ke with = By Prop. 7.1 there exists compact sets µ(Ke) ~ 1- e • Define ge(x) JKf(x+y)dµ(y). 'rhen = € by Prop.7.1 , for ks p, Dkge(x) = JKDkf(x+y)dµ(y) and this € implies that Dkge (x) converges uniformly as e-+ 0 to .Dk f(x+y)dµ(y) J . k forks p. So by Prop. 1.11, D g(x) exists and Dkg(x) = JDkf(x+y)dµ(y). g(x) is in cP,P(E,F) because llDkg(x)!I ~ Jl!Dkf(x+y)lldµ(y). Q.E.D. Consider now separable Hilbert space £ 2 and let (e.} be an orthonormal basis. We will define for each l nonnegative sequence (a . } , with L) a.l 2 < oo , a probability . 1 measure l µA , A= (ai}. Let 7J(t) be a fixed function in 00 00 C (H,R) satisfying 77(t) = 0 if !ti~ 1 and J11(t)dt = 1. -co Define for each positive integer n an integral on c0 , 0 ci 2 ,R) as follows: A!( f ( x) ) = fr' (~.) JH'l=l fr' 7J (~i~) f ( i=l . ~, y i e i) dµ~ ( y) i=l i n where H' is the space spanned by n e. (ill~i~n,a.>0] l , µ." is n the standard Lebesque measure on H~ and Tf' and L)' denote the product and summation over only those i's for which 51 a. > o. (x \ I xi I s; ai}. For any f E c 0 ' 0 ce 2 ,R) find o such that l Let K denote the compact Hilbert cube K = z,y EK and \\z.-y\l<o implies \f(z)-f(y)\ <e. Then if CD we take N such that 2 a . < 6 2 we have for m <!: n <!: N, E . N+ 1 1 l= I I AA( f(x)) - AA( f(x)) I s; rrm'/1 )l rrm ry(~i) f ( E'y. e. )dµ' (y) m n ~a. H' a. i l m i= s; m (l TT' -a. ) . l= 1 t 1f17-(y ") m, , m 1'. 1 a. l 1'. m i m, n, lf(Ey.e.)-f(:Ey.e.)\dµ'(y) . i i i i m = Hence € • A~(f(x)) exists and we define this limit to lim n-+CD be AA(f(x)). The function a l AA is clearly linear, bounded positive a nd satisfies K is c ompact, AA(l) = 1. Since supp AA c AA is an inte g ral. By the Riesz Representa- tion Theorem there is a unique probability measure 1 2 such that measures K and µ A on AA(f(x)) for all f E C0,0(.~ 2 ~H). Jr(x)dµA(x) In the proof of the next theorem we will use the µA to mollify cP functions on 1 2 • We recall that a Hilbert-Schmidt operator T on 1 2 is an element of L(J. 2 ,1 ) 2 CD satisfying E i,j=l (Te. , Te.) < l J CD • 52 Theorem 7.1 Suppose that f E cP'P(U,F),l5:p<oo, where U is an open subset of i 2 , that Dpf(x) is uniformly contin- uous on U, that Vis an open subset of U with dist(V,CU)>O, and that T is a Hilbert-Schmidt operator on 12 • Then 00 there exists a g(x) E C (V,F) satisfying SUJ? xEV, II hi\ 5:1, O~k~p Proof. \IDk(f(x) - g(x))[Th]\I ~ 1. Let T = SW be a polar decomposition for T, where S = VTT• and W is a partial isometry. Then S is positive definite self-adjoint Hilbert-Schmidt and if we denote the unit ball by B, then T(B) c (7.1) SW ( B ) c::: S( B) Assume that the orthonormal basis [e.} is a set l of eigenvectors for Sand that Se = a. .e . • Then a. ~ 0 1 1 i l 2 and ~ a. . < °". Now Dkf(x) is uniformily continuous on U i l for k 5: p so we can find µA o < dist(V,CU) and sup \\Dkf(x) - Dkf(y)I\ ~ 1/2\IT\\. x ,yEU, \lx-yll < o, O~k~p (7 .2) Let o > 0 such that t = min(l, o/22Jo..2), a. =ta.., A = fa.} and define i l l l as above. Letting K be tbe compact set l (x\ \x.l I 5: a.} , l we have diam K < o , supp µ A c K and V + K c U. Now let 1 k M = sup l~kTJ(t) \dt and use Prop. 2.? to obtain g(x) E k~p J -1 dt 2 C (f ,F) such that 00 53 ( 7. 3) supllf(x) - g(x)ll s tq/2Mq x Let Jf(x+y)dµA(y) and g(x) = Jg(x+y)dµA(y), then f (x) 0 f by Frop.7.2, E cP(V,F) and gEC \V,:B'). (7.2) giv es Suppose now that x EV, that i 1 , •• ik, k :s:; p, are integers with > 0 for jsk and N = max(i 1 , • • • a. lj ik).Then II ax. • • ax. (f(x) - g(x)) H 11 lit n (-1 ) J TfYJ n , (Y IT' :..J.). [ f(x + En,y .e . ) ·-1 a. J,1 a.J H' n J J n,y .e.) J dµ' Cy) - g(x + E n J J (7.5) :s:; lim n- a> n k ( n y.) i k J- J II n 0 l :..J.) II f Cx+ "'LJy.e """' J J. ) lT ._ ' (a.)JH,ay . . . ay . .1T' _1 1J (a J- 1 ,J n 11 (which by (7.3) is) s 8k N , ( -1 ) J (Tf N, Tf (~)) • tq/2Mq dµN' (y) 1T . a . H' ay. • • ay. a. J= 1 J n 11 k • M • i" J 54 It follows now from (7.1) and (7.5) that ,._,. !ID k (f(x) - g(x))[Th]\I s: sup llD k (f(x)-g(x))[ShJll N sup xEV, II hll s:l ~ N xEV, 11 h s:l co s: sup II "E/ l\h\\s:l ii-. i1c=l y,,. Combining this with ( 7 .4), we obtain for 0 s: ks: p, sup llDk(f(x) - g(x))[ThJll s: sup II Dk( f (x) - f(x)) [Th] II xEV, II hll s:1 xEV, II hll s:1 + k N ~ sup llD (f(x) - g(x))[ThJ\I xEV, IIhll s:l s:Y,,+Y,, Remark. s: k ~ supllD (f(x)-f(x))\l ·llTll + Y,, xEV =l. Suppose that the f Q.E.D. in Theorem 7.1 has the p roperty that for any e > O there exists a ge: E c°",Pcv ,F) such that \\f(x) - g e: (x)\l 0 s:e: and l\g e: II p s: M, where M is independent of e:. Then the conclusion of the theorem would be true if the operator •r were only assumed to be ~ompact. L(f To show this assume T compact and find Pin 2 ,£ 2 ) with finite dimensional ran~e and such that 55 11 '11 - P\\ < 1/2( \1 f\[ p + M) • Apply the theorem to . get a g E cmce 2 , F) wi th supl\Dk(f(x)-g(x))[2PhJll < 1. Since xEV, \I hll sl, ksp g(x) Jg(x+y)dµA(y) where g is a Cm O approxim a tion , to f and since by assumption we can take l\g\\ follows t hat we can assume l\gll sup\\Dk(f(x) - g(x))[ThJ\1 + s p p ~ M, it s M . Therefore sup\\Dk(f(x) -g(x))[(T-~h) II s up llDk(f(x)-g(x))[PhJ[l s ( lifll p +M)\IT-Pll +~ s 1. xEV, II hll sl , ksp We now give a global formul a tion of Theorem 7.1. The proof is si milar to the pro of of Theorem 6.1 in which Lemma 5.1 played a key role. Theorem 7.2 Let f E cP(£ 2 ,F),l~p<=, and suppose that DPf(x) is uniformly continuous in some neighborhood of every point of £2 • Then for any locally finite cover (Uo.} of £2 and collection (Ta.) of Hilbert-Schmidt opera- tors on £2 there exists a g(x) E Cm(£ 2 ,F) sup a. sup llDk(f(x) - g(x) )[Ta.hJU • xEUa.,llhl\sl,Osksp Proof. = 1 such that VT a. T*a. As in the proof of Theorem 7.1,let so that Sa. is self-ad,joint positive definite Hilbert-Schmidt and Ta. (B) c Sa. (B), wh e re B is the unit ball. 56 For every x in 1 2 find a ball B(x,R ) of radius R a bout x x x such that B(x,R ) intersects only a finite nur~er of U 's a. x and DPf(x) is uniformly continuous on B(x,Rx). Now since 1 2 is Co:>'o:> smooth, we can apply Lemma (Vr}, j=l,2,3,4, and functions (B(x,Rx/2)} to obtain covers o:> g.(x) EC ' o:> l 2 (f ,R) such that 1) dist( v~,cv~+l) > l (V~} l 2) 5.1 to the cover o, l covers p = 1,2,3 j 2 3) (V~} is lo cally finite and refines (B(x,R /2)} . x l o ~ g.l ( x) ~ 1, g.l ( x) ( v?) = 1 and g. ( x) (CV;;) = o. l l l 4) Now define cp (x) if i > 1 and M . = l s..l = g 1 (x), cpi (x) = ( l-g 1 (x)) • • • (1-gi-l (x))g;i (x) I\ cp . \I l • p If we 1 et z; so. = ta. IuJ1 v{1 01 (note that the sum is over a finite number of a.' s) then S. is l positive definite self-adjoint Hilbert-Schmidt and S (B) c a. S.(B). Set l s~l 2P+iM .(max( 1, II s. !I) fs. l and use Theorem 7.1 l l , observing th a t f(x) E cP,P(B(x,Rx),F) and dist(V~ ,B(x,Rx) ) > O, to obtain functions h. E Ca:J(V~ ,F) l l l satisfy ing (7.6) sup l\Dk(f(x) - hi (x))[S~hJll 4 xEV i, II hi\ ~1, k$;p Define r (x) 0 = f(x), •.• , fi(x) ~ 1. f ( x) ( 1-p;l ( x) ) •.. ( 1-rr,. ( x) ) + l 57 (7.7) Also (7.8) f.(x) = l For every x E 1 4 f. 1 -1 (x) when x¢V . 1 2 there is a neighborhood N of x and an x integer n such that N c v 1 u • . . uv 1 and N nV~ = x n 1 x 1 Hence by (7.7) and (7.8) we can define g(x) ::: Now f i (x) - f i-l (x) 0 for i>n. lim i_,..co = (hi (x) - f (x) )cpi (x), hence k sup l\D (fi(x)-fi_ 1 (x))[Sih]\\ x,\lh\lsl ,ksp (7 .9) s ~ (k) supl[Dn(h.(x)-f(x) )[S.hJI\ . supl\Dk-nq:i.(x)[S.h]\I 1 1 1 1 n-O n - $ x ' II hil ~ l ' ks;p x' II hll sl 'ksp ~ (k) l/(2p+iM . llS. \\P) • M. 11s. t1k-n s; l/2p+i s; l/2i n 1 i i i · n= O by (7.6) and (7.8). Therefore if x E Uu supl\Dk(f(x) - g(x) )[TuhJI\ \I h\\ sl, ksp s supllDk(f(x) - g(x))[SuhJll \\h\I s:l ,ksp 58 BIBLIOGRAPHY [ l] [ 2] [ 3] [4J [ 5] [ 6] [ 7] [ 8] [ 9] [ 10] [ 11] [ 12] [ 13] [ 14] [ 15] [ 16] [ 17] R.Bonic and J.Frampton, Smooth Functions on Banach Manifolds, J.Math.Mech. 15(1966),877-8'9"S' , Differentiable Functions on Certain ~~B-an __a_c~h-Spaces, Bull.Amer.Math.Soc.71(1965), 393-395. R.Bonic and F.Reis, A Characterization of Hilbert Space, Acad.Bras.-de Citn. 38(1966),~39-241. R.Bonic, J.Frampton and A.Tromba,~- Manifolds, Notes of the 1968 Conference on Global Analysis, Berkeley. Dieudonn~,J., Foundations of Modern Analysis, Academic Press, New York, 1960. Dunford,N. and Schwartz,T., Linear Operators~ Vol I, General Theory, Interscience, New York, 1~5~ J.Eells, A Setting for Global Analysis, Bull.Amer. Math.Soc., 72(1966), 751-807. Hille,E., and Phillips,R., Functional Analysis and Semi-Groups, Amer. Math. Soc. Colloq. Publ., Vol.31, Amer. Math. 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