Smooth approximations of norms in separable Banach spaces
aa r X i v : . [ m a t h . F A ] A ug SMOOTH APPROXIMATIONS OF NORMS IN SEPARABLEBANACH SPACES
PETR H ´AJEK AND JARNO TALPONEN
Abstract.
Let X be a separable real Banach space having a k -times con-tinuously Fr´echet differentiable (i.e. C k -smooth) norm where k ∈ { , . . . , ∞} .We show that any equivalent norm on X can be approximated uniformly onbounded sets by C k -smooth norms. Introduction
The problem of approximation of continuous mappings by more regular func-tions, such as polynomials or C k -smooth functions, is one of the classical themes inanalysis. If the underlying spaces are infinite-dimensional the additional difficultyis in the lack of compactness and measure as available tools. Yet, many importantand rather general results can be obtained even in this setting, as shown in themonograph by Deville, Godefroy and Zizler [DGZ] and the references therein. Inparticular, if X is a separable real Banach space admitting a C k -smooth bumpfunction, k ∈ IN ∪ {∞} , then every continuous mapping f : X → Y , Y a realBanach space, can be uniformly approximated by means of C k -smooth mappings[BoFr]. The pioneering result in this area is due to Kurzweil [Kurz] who proved asimilar result in the real analytic setting (provided X has a separating polynomial),see also [Fry1], [CH].A natural version of the problem in the setting of norms is a question whetherevery equivalent renorming of a (separable) real Banach space can be uniformlyapproximated on bounded sets by means of C k -smooth norms, provided that X admits an equivalent C k -smooth norm. Of course, from the above mentioned re-sults we know that such approximations are possible by means of C k -smooth func-tions, but the additional requirement of convexity cannot be achieved using thesetechniques. This problem was posed explicitly in [DGZ] (for the case of a separableHilbert space, see pp. 206-207). In [DFH1] and [DFH2] the problem was solved inthe affirmative for separable Banach spaces under some additional assumptions.The result was shown to hold if X is either a polyhedral space (e.g. c ), or if it isa superreflexive space with a Schauder basis and the highest derivative D k k · k isbounded (e.g. ℓ ) . These papers left open not only the case of an abstract separa-ble Banach space, but also many concrete spaces such as c ⊕ ℓ (which admits a C ∞ -smooth renorming, but the C -smooth approximations of norms were open).The main result of the present note is a complete solution of the problem for allseparable Banach spaces given in Theorem 2.8. The proof follows the main ideas in[DFH1] with an essential new ingredient contained in Lemma 2.4. The trick is to Date : October 29, 2018.2000
Mathematics Subject Classification.
Primary 46B03; 46T20; Secondary 47J07; 14P20.
Key words and phrases.
Fr´echet smooth, C k -smooth norm, approximation of norms,Minkowski functional, renorming, Implicit Function Theorem, Nash function.The first author was supported in part by Institutional Research Plan AV0Z10190503 andGAˇCR P201/11/0345. This paper was prepared as the second author enjoyed the warm hospitalityof the Czech Academy of Sciences in Spring 2011. The visit and research was supported in partby the V¨ais¨al¨a foundation. replace a given C k -smooth norm on X by a perturbation using a quadratic polyno-mial in order to get a new C k -smooth renorming which preserves C k -smoothnesswith respect to quotients with a finite-dimensional kernel. For the sake of conve-nience we will give a full proof of our theorem, providing also the arguments forsome known auxiliary results (e.g. Lemmas 2.6 and 2.7). We also correct a minormistake in the proof of the main results in [DFH1], which consisted of overlookingthat the seminorms used in the proof are not smooth at the points of their kernels.The exact place of this omission is indicated in our proof of the main theorem.1.1. Preliminaries.
We denote by X , Y and E real Banach spaces. The closedunit ball and the unit sphere of X are denoted by B X and S X , respectively. We willwrite B ( x, r ) = x + rB X for x ∈ X , r >
0. For suitable background information ongeneral Banach space theory and notations we refer to [FHHMZ]. However, nextwe will recall some basic concepts for the sake of convenience.Recall that a mapping f : X → Y is Fr´echet differentiable at point x if thereexists a bounded linear mapping T x : X → Y such thatlim t → + sup h ∈ tS X k f ( x + h ) − f ( x ) − T x ( h ) k t = 0 . In such a case we denote by Df ( x )[ z ] = T x ( z ) , the Fr´echet derivative of f at x . We denote by Df : X → L ( X ; Y ) the Fr´echet derivative as a function actingfrom the space X with values in the space of bounded linear operators L ( X ; Y ).If x Df ( x ) is defined and continuous then f is termed as being continuouslyFr´echet differentiable , or in the class C . In the case where Df ( x ) is continuouslyFr´echet differentiable with respect to x , we say that f is (Fr´echet) C -smooth, andso on. If f is C k -smooth for all k ∈ IN , then it is C ∞ -smooth. When we say thata norm is smooth we mean that it is that away from the origin. As customary, wedenote the class of real analytic functions by C ω . Here we will study such functionsonly in the finite-dimensional setting. Given a map f : X ⊕ Y → E , we denote by D f ( x, y ) the Fr´echet differential taken with repect to the second coordinate y .Given a subset A ⊂ X , a retraction ρ : X → A is a closest point mapping if k x − ρ ( x ) k = dist( x, A ) holds for all x ∈ X . The closed linear span of A is denotedby [ A ].Recall that a family { ( x i , x ∗ i ) } i ∈ IN of elements of S X × X ∗ is a biorthogonalsystem if x ∗ i ( x j ) equals to 1 if i = j and 0 otherwise. It is called an M -basis if[ x i : i ∈ IN ] = X and [ x ∗ i : i ∈ IN ] ω ∗ = X ∗ . If additionally [ x ∗ i : i ∈ IN ] = X ∗ , then { ( x i , x ∗ i ) } i ∈ IN is a shrinking M -basis.Recall that a closed convex bounded (CCB) subset C of a Banach space X is a body if it has a non-empty interior. The polar C of C is given by C = { f ∈ X ∗ : | f ( x ) | ≤ , x ∈ C } . The following fact is well-known and can be found in [Dieu].
Theorem 1.1 (Implicit Function Theorem) . Let
X, Y, Z be Banach spaces, U ⊂ X ⊕ Y be an open set and F : U → Z be C -smooth. Let x , y ∈ U and D F ( x , y ) be a linear isomorphism of Y onto Z. Then there is a neighbourhood V of x in X ,and a unique mapping u : V → Y , such that u ( x ) = y , and for all x ∈ V itholds that ( x, u ( x )) ∈ U and F ( x, u ( x )) = F ( x , y ) . Moreover, if F is C k -smooth, k ∈ { , , . . . , ∞ , ω } , then so is u. Results
Preparations.
Recall some elementary facts from [HP].
MOOTH APPROXIMATIONS OF NORMS IN SEPARABLE BANACH SPACES 3
Definition 2.1.
We say that a function f : ℓ ∞ (Γ) → IR is strongly lattice if f ( x ) ≤ f ( y ) whenever | x ( γ ) | ≤ | y ( γ ) | for all γ ∈ Γ . Definition 2.2.
Let X be a vector space. A function g : X → ℓ ∞ (Γ) is said tobe coordinate-wise convex if, for each γ ∈ Γ , the function x g γ ( x ) is convex.Similarly g is coordinate-wise non-negative if g γ ( x ) ≥ for all γ ∈ Γ . Lemma 2.3.
Let f : ℓ ∞ (Γ) → IR be convex and strongly lattice. Let g : X → ℓ ∞ (Γ) be coordinate-wise convex and coordinate-wise non-negative. Then f ◦ g : X → IR is convex.Proof. Let a, b ≥ a + b = 1. Since g is coordinate-wise convex and non-negative, we have 0 ≤ g γ ( ax + by ) ≤ ag γ ( x ) + bg γ ( y )for each γ ∈ Γ. The strongly lattice property and the convexity of f yield f ( g ( ax + by )) ≤ f ( ag ( x ) + bg ( y )) ≤ af ( g ( x )) + bf ( g ( y ))so f ◦ g is convex. ⊓⊔ Lemma 2.4.
Suppose that ( X, k · k ) is a separable Banach space with a Fr´echet C k -smooth norm, k ∈ { , , . . . , ∞} , and { ( x i , x ∗ i ) } i ∈ IN is an M-basis of X . Then k · k can be approximated by norms ||| · ||| such that on ( X, ||| · ||| ) the quotient norms ||| · ||| X/ [ x ,...,x n ] , n ∈ IN , are Fr´echet C k -smooth.Proof. The required norms ||| · ||| are given by ||| x ||| = k x k + ε ∞ X i =1 − i x ∗ i ( x ) , ε > . It is clear that these norms approximate the original one uniformly (on k·k -boundedsets) as ε tends to 0. We observe immediately that the functional ||| · ||| is C k -smooth and therefore ||| · ||| is C k -smooth away from the origin. From now on wewill work with the space ( X, ||| · ||| ).Write E = [ x , . . . , x n ]. Our strategy is to apply the Implicit Function Theoremto establish the C k -smoothness of the (single-valued) closest point mapping ρ : X → E . Then the quotient norm ||| · ||| X/E will be C k -smooth away from the origin, sinceit can be written as ||| ˆ x ||| X/E = ||| x − ρ ( x ) ||| , which is necessarily C k -smooth map as a composition of C k -smooth mappings x − ρ ( x ) and ||| · ||| .Fix x ∈ X and let d = dist( x, E ). By compactness the set of closest points T r>d B ( x, r ) ∩ E is non-empty. Suppose that it contains two points, say a, b ∈ E .Then for 1 ≤ i ≤ n we have that x ∗ i ( x − a + b ) < x ∗ i ( x − a ) + x ∗ i ( x − b ) unless x ∗ i ( a − b ) =0. Thus ||| x − a + b ||| < ||| x − a ||| + ||| x − b ||| if a = b . Hence the set of closest points isa singleton. In the sequel we shall denote by ρ the closest point mapping X → E .Note that for x ∈ X and y ∈ E we have ρ ( x + y ) = B ( x + y, dist( x + y, E )) ∩ E = y + ( B ( x, dist( x + y, E )) ∩ E )= y + ( B ( x, dist( x, E )) ∩ E ) = y + ρ ( x ) . Next, we wish to verify that ρ is C k -smooth. Let L : X ⊕ E → X be a boundedlinear mapping L ( x, y ) = x − y .Let G : X ⊕ E → ( X ⊕ E ) ∗ be given by (using the chain rule) G ( x, y ) = D ( ||| · ||| ◦ L )( x, y ) = D ( ||| · ||| ) ◦ L ( x, y )= D ||| x − y ||| ∈ ( X ⊕ E ) ∗ = X ∗ ⊕ E ∗ . MOOTH APPROXIMATIONS OF NORMS IN SEPARABLE BANACH SPACES 4
Let P : X ∗ ⊕ E ∗ → E ∗ be the canonical projection. Note that since dim( E ) = n ,there is a natural identification I : E ∗ → E, I n X i =1 a i x ∗ i ! = n X i =1 a i x i . We let now F : X ⊕ E → E be given by F ( x, y ) = I ◦ P ◦ G ( x, y ) . Note that F ( x, y ) = 0 if and only if y = ρ ( x ). Indeed, F ( x, y ) = 0 if and only if P ◦ G ( x, y ) = 0 which means that y → ||| x − y ||| attains its minimum, due to thestrict convexity of the last function. Using the coordinates, if y = P ni =1 a i x i then F ( x, y ) = ∂∂x ||| x + n X i =1 a i x i ||| , . . . , ∂∂x n ||| x + n X i =1 a i x i ||| ! . Recall that ||| · ||| is at least C -smooth. So, we may put D F = DF | E = ∂ ∂x i ∂x j ||| x + n X i =1 a i x i ||| ! i,j =1 ,...,n For a fixed x the above map D F can be considered as a (symmetric) Hessianmatrix M ∈ IR n × n of the restriction of ||| · ||| to the affine finite-dimensional space( x + E ) in X . (See [BoWa, p.39] for the properties of the Hessian and convexity).We claim that M is invertible. Indeed, invertibility follows easily from the thefact that M is positive-definite. The functional ||| · ||| decomposes in a naturalway into two parts, namely, k x k + ε P ∞ n +1 − m x ∗ m ( x ) and ε P ni =1 − i x ∗ i ( x ) . Thefirst part is a convex C -smooth function so that the corresponding Hessian matrix M is positive-semidefinite. The Hessian matrix M of the latter one is a strictlypositive diagonal matrix, because of the definition of the biorthogonal functionals,thus positive-definite. By linearity, we obtain that M = M + M is positive-definite.We conclude that D F ( x, y ) is an isomorphism E → E for any x ∈ X, y ∈ E .Recall that |||·||| is a convex function so F ( x, y ) = 0 exactly at the pairs of points( x, y ) = ( x, ρ ( x )). Fix x ∈ X . We apply the Implicit Function Theorem to findopen neighborhoods V ⊂ X of x and U ⊂ X ⊕ E of ( x , ρ ( x )), and a C k -smoothmapping u : V → E such that F ( x, u ( x )) = F ( x , ρ ( x )) = 0 for ( x, u ( x )) ∈ U .Necessarily u ( x ) = ρ ( x ), as ρ is single-valued. Since x was arbitrary, we concludethat ρ is C k -smooth on the whole of X . ⊓⊔ Remark 2.5.
The smooth approximation of the quotient
X/E in the previousproof works for E = ℓ as well, essentially by the same argument.The following lemma is similar to Lemma 4.1 [Zp] and to the first part of Theorem1 [Ben]. We give a proof for the reader’s convenience. Recall that a Banach space X with a separable dual admits a shrinking M -basis, see [HMVZ, p.8]. Lemma 2.6.
Let X be a Banach space with separable dual X ∗ and let { ( x i , x ∗ i ) } i ∈ IN be a shrinking M -basis of X . Let W ⊂ X be CCB body such that ~ ∈ int( W ) and < ε < . Then there exists a ω ∗ -compact subset F ⊂ W such that:(1) ε W ⊂ conv ω ∗ ( F ) ⊂ ε W (2) For each integer i the set x i ( F ) is finite.Proof. Let d = inf {k g k : g ∈ ∂W } and T i = { f ( x i ) : f ∈ W } for i ∈ IN.
Eachset T i is bounded and thus there exists a εd · i k x ∗ i k -net C i in T i . Put MOOTH APPROXIMATIONS OF NORMS IN SEPARABLE BANACH SPACES 5 A = ( n X i =1 a i x ∗ i ∈
11 + ε W : n ∈ IN, a i ∈ C i ) , F = A ω ∗ . It is obvious that x i ( F ) = x i ( A ) ⊂ C i where i ∈ IN . Thus condition 2. is satisfied.Clearly conv ω ∗ ( F ) ⊂ ε W . Next, we will show that ε W ⊂ conv ω ∗ ( F ). Let us take f ∈ ε W andrecall that ε < . Since ε W is convex, by using the geometric Hahn-Banachtheorem the task reduces to checking that there exists h ∈ F such that k f − h k < d ε < dε (1 + )(1 + ) < d ε (1 + 2 ε )(1 + 4 ε ) = d (cid:18)
11 + 2 ε −
11 + 4 ε (cid:19) . (1)Since span { x ∗ i } is dense in X ∗ , there exists g = n X i =1 b i x ∗ i ∈
11 + 2 ε W , such that k f − g k < ε d . We have b i ∈ T i and there exists a i ∈ C i such that | b i − a i | < εd · i k x ∗ i k , i ∈ IN.
Hence k g − n X i =1 a i x ∗ i k < d ε , and a straight verification shows that h = n P a i x ∗ i ∈ ε W . Thus, h ∈ F by theconstructions of h and the set A ⊂ F . Now, k f − h k ≤ k f − g k + k g − h k < d ε , so that (1) holds, and the proof is complete. ⊓⊔ The following Lemma is close to some results from [Zp] also. We will use thenotations from Lemma 2.6. In addition put M n = [ x i ] n ⊥ , n ∈ IN.
Lemma 2.7.
For arbitrary ε > there exists a sequence of points { g k } in theset F , a sequence of integers { n k } , n k → ∞ , and a decreasing sequence { F α } of ω ∗ -closed subsets of F such that after reindexing it holds that(1) S k ∈ IN (( g k + M n k ) ∩ F k ) = F .(2) diam(( g k + M n k ) ∩ F k ) < ε .Proof. We will use the following well-known property of ω ∗ -compacts in a separabledual space: for every ε > g ∈ F and ω ∗ -neighborhood G of g such that G ∩ F = ∅ and diam( G ∩ F ) < ε. Indeed, we apply the fact that X isAsplund and X ∗ has the RNP, see [FHHMZ, Ch. 11.2] for discussion.Condition (2) in Lemma 2.6 suggests that in a sense F resembles a Cantor set.Because of the structure of the set F , the sets ( h + M n ) ∩ F, h ∈ F, n ∈ IN form abase of ω ∗ - topology on F and each such a set is both closed and open subset of( F, ω ∗ ). Moreover, the family ℑ = { h + M n : h ∈ F, n ∈ N } contains countably many (different) sets and obviously each non-empty ω ∗ - compactsubset I ∩ F , I ∈ ℑ , has the same structure as F . For each ordinal α we define sets F α and ( h α + M n ( α ) ) by transfinite induction as follows: F = F, F α +1 = F α \ ( h α + M n ( α ) ) , MOOTH APPROXIMATIONS OF NORMS IN SEPARABLE BANACH SPACES 6 where ( h α + M n ( α ) ) is a member of the family ℑ such that ( h α + M n ( α ) ) ∩ F α = ∅ and diam(( h α + M n ( α ) ) ∩ F α ) < ε. If α is a limit ordinal we put F α = ∩ β<α F β . Since F is separable and ω ∗ -compact, there exists a countable ordinal η such that F η = ∅ and F η +1 = ∅ . It is clear that [ α ≤ η (cid:0) ( h α + M n ( α ) (cid:1) ∩ F α ) = F. Let us reindex the countable family { h α + M n ( α ) } α ≤ η into { h k + M n k } ∞ k =1 . Sincefor each integer q there exist only finite many members h + M n of the family ℑ such that n ≤ q , it follows that n k → ∞ for k → ∞ . ⊓⊔ Fr´echet C k -smooth approximation of norms. Next we give our main result.
Theorem 2.8.
Let ( X, k · k ) be a separable Banach space. Let k ∈ IN ∪ { + ∞} and k · k be C k -smooth. Then every equivalent norm on X can be approximateduniformly on bounded sets by C k -smooth equivalent norms.Proof. For k = 1 the above result is known, see [DGZ, p.53]. Since X is Fr´echetsmooth the dual is separable (see e.g. [FHHMZ, Thm. 8.6]) and therefore X admitsa shrinking M-basis { ( x i , x ∗ i ) } i ∈ IN . By Lemma 2.4 we may assume without lossof generality that the quotient norms k · k X/ [ x ,...,x n ] , n ∈ IN , are Fr´echet C k -smooth. Denote by W = B ( X, k·k ) , P n : X → [ x i ] n the projection given by P n ( x ) = P ni =1 x ∗ i ( x ) x i . Put Q n ( x ) = x − P n ( x ). Using previous lemmas we obtain sequences { g j } j ∈ IN ⊂ F and { M n j } j ∈ IN and put C = [ j ∈ IN (cid:0) g j + ε · B X ∗ ∩ M n j (cid:1) ⊃ F. (2)By taking into account that ε is arbitrary, both above and in condition (1) of Lemma2.6, we observe that sets of the form conv ω ∗ ( C ) are sufficient in approximating thepolar W ◦ . In fact, for technical reasons we will approximate C with yet another set D to be defined shortly.Write E j = [ x i : 1 ≤ i ≤ n j ] and Y j = [ x i : n j + 1 ≤ i ] for j ∈ IN andobserve that X = E j ⊕ Y j , as E j is finite-dimensional. We will denote by q n j the corresponding quotient mapping X → X/E j and we note that ( X/E j ) ∗ =( M n j , k · k ∗ ). Observe that k q n j ( x ) k X/E j = k q n j ( Q n j ( x )) k X/E j for x ∈ X by the definition of the quotient norm. Also, putting q n j ◦ Q n j defines an isomor-phism Y j → X/E j . Unfortunately, the map k q n j ( · ) k X/E j is C k -smooth only awayfrom its kernel. This fact has been overlooked in [DFH1] (where our Q n is denotedas P n ), so the proof therein in not entirely correct. Therefore we need to smoothenup this mapping.Next we will give norms that are smooth away from the origin and approxi-mate the seminorms k q n j ( · ) k X/E j . Let N be a norm on IR satisfying the followingconditions:(a) ( IR , N ) is C ∞ -smooth,(b) N ( − a, b ) = N ( a, b ) = N ( b, a ) for a, b ∈ IR ,(c) (1 , − / , (1 , , (1 , / ∈ S ( R ,N ) .Indeed, this can be accomplished by taking a Minkowski functional of a suitablesum of two functions, similarly as in (6) to follow. Define norms N j : X = E j ⊕ Y j → [0 , ∞ ) by N j ( x ) = N j ( P n j ( x ) , Q n j ( x )) = N ( a j k P n j ( x ) k , k q n j ( Q n j ( x )) k X/E j ) MOOTH APPROXIMATIONS OF NORMS IN SEPARABLE BANACH SPACES 7 where a j ց c ) these norms coincide with the mapping x a j k P n j ( x ) k around the kernel of q n j . Also, N j coincide with the mapping x
7→ k q n j ( Q n j ( x )) k around the kernel of P n j . These facts together with the C k -smoothness of all themappings involved in the definition are responsible for N j being C k -smooth awayfrom the origin. Note that N j ( x ) ≥ k q n j ( x ) k X/E nj , so that B X ∗ ∩ M n j ⊂ N ◦ j for j ∈ IN.
Since a j → j → ∞ we getdist H ( N ◦ j , B X ∗ ∩ M n j ) → j → ∞ . (3)The above limit involves the symmetric Hausdorff distance. Fix ε > D = [ j ∈ IN (cid:0) g j + εN ◦ j (cid:1) ⊃ C ⊃ F. (4)We show first that D is a ω ∗ -compact set. Let { h m } m ∈ IN ⊂ D , h m ω ∗ → h ∈ B X ∗ . Ifinfinitely many of h m are in one of the sets g j + εN ◦ j , then of course h ∈ D . So letus assume that h m ∈ g j m + εN ◦ j m where m ∈ IN , j m → ∞ as m → ∞ . Then, by (3)we may write h m = g j m + z m + εu m where u m ∈ B M njm , m ∈ IN , and k z m k → m → ∞ . Since j m → ∞ , we have n j m → ∞ , according to Lemma 2.7, and therefore ω ∗ − lim u m = 0. Thus h = ω ∗ − lim g j m ∈ F ⊂ D , which completes the proof of ω ∗ -closedness of the set D .Write Q ◦ = conv ω ∗ ( D ) and Q = { x ∈ X : f ( x ) ≤ f ∈ Q ◦ } . The latterset is a closed convex set in X with a non-empty interior and it is used here toapproximate W . Let us denote by F Q the Minkowski functional of the set Q .For each j ∈ IN we define F j ( x ) = sup { f ( x ) : f ∈ g j + εN ◦ j } = g j ( x ) + εN j ( x ) . (5)Observe that F j are positively homogeneous, 2-Lipschitz for small enough ε , andby the right hand side they are C k -smooth away from the origin.Pick x ∈ Q such that sup f ∈ Q ◦ f ( x ) = 1. Then necessarily sup f ∈ D f ( x ) = 1. Bythe ω ∗ -compactness of D there is f ∈ D such that f ( x ) = 1 and of course f ( λx ) = λ for λ ∈ IR . This means, by recalling (4) and (5), that F Q = max j F j . We introduce a strongly lattice function F : ℓ ∞ → [0 , ∞ ] as follows F ( x k ) = X φ k ( x k ) , (6)where φ j : IR → [0 , ∞ ) are C ∞ -smooth convex and even functions such that φ k [ − − kk + 2 δ k , kk + 2 δ k ] = 0 , φ k (1 + k + 1 k + 2 δ k ) = 1 , φ k (1 + δ k ) = 3 . (7)Let 0 < λ <
1. Fix a decreasing sequence δ n →
0, such that λ n = 1 + nn +2 δ n δ n = 1 − δ n ( n + 2)(1 + δ n ) (8)is an increasing sequence and equals to λ for n = 1. Let h k = (1 + δ k ) F k . ByLemma 2.3 we obtain that G ( x ) = X φ k ( h k ( x ))is convex wherever it is well-defined. Next, we will check that the formula for G locally contains only finitely many non-zero summands on the set G − [0 , MOOTH APPROXIMATIONS OF NORMS IN SEPARABLE BANACH SPACES 8 let x be such that F Q ( x ) = 1, and suppose F k ( x ) = 1. If G ( λx ) = 1 then φ k ◦ h k ( λx ) ≤
1, thus h k ( λx ) ≤ k +1 k +2 δ k , so F k ( λx ) ≤ k +1 k +2 δ k δ k , and hence λ ≤ k +1 k +2 δ k δ k . (9)Put ρ = δ k / (4( k + 2)(1 + δ k )). Now, if k y − λx k < ρ , then by using the facts that F n is 2-Lipschitz, F Q ( x ) = 1 and (9) we obtain F n ( y ) ≤ λF n ( x ) + 2 ρ ≤ λ + 2 ρ ≤ k +1 k +2 δ k δ k + 2 ρ = 1 − ρ for n. Thus h N ( y ) ≤ (1 + δ N )(1 − ρ ) ≤ NN + 2 δ N , provided that N is large enough. By recalling (7) we observe that G ( y ) is a finitesum for k y − λx k < ρ . Thus G is C k -smooth.Finally, let us check that G − ([0 , Q . Indeed, if | F Q ( x ) | = 1, then G ( x ) ≥ G − ([0 , ⊂ Q . If x ∈ Q is such that | F n ( x ) | ≤ λ n for n , then G ( x ) = 0. Since λ n is an increasing sequence, we get that { x ∈ Q : F Q ( x ) ≤ λ } ⊂ G − (0) . Since λ < ⊓⊔ Similarly to [DFH1] we conclude the following fact.
Corollary 2.9.
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