Polynomial algebras and smooth functions in Banach spaces

Polynomial algebras and smooth functions in Banach spaces

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Available online at www.sciencedirect.com

ScienceDirect Journal of Functional Analysis 266 (2014) 1627–1646 www.elsevier.com/locate/jfa

Polynomial algebras and smooth functions in Banach spaces ✩ Stefania D’Alessandro a,b , Petr Hájek b,c,∗ a Department of Mathematics, Università degli Studi, Milano, Italy b Mathematical Institute, Czech Academy of Sciences, Žitná 25, 115 67, Prague 1, Czech Republic c Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague,

Zikova 4, 160 00, Prague, Czech Republic Received 15 April 2013; accepted 19 November 2013 Available online 5 December 2013 Communicated by G. Schechtman

Abstract Let An (X) be the algebra of polynomials on a real Banach space X, which is generated by all continuous polynomials of degree not exceeding n. Let m be the minimal integer such that there is a non-compact m-homogeneous polynomial P ∈ P(m X; 1 ). Then n  m implies that the uniform closure of An (X) does not contain all polynomials of degree n + 1, and hence the chain of closures An (X), n  m is strictly increasing. In the rest of the note we give solutions to three problems concerning the behaviour of smooth functions on Banach spaces posed in the literature. In particular, we construct an example of a uniformly differentiable real valued function f on the unit ball of a certain Banach space X, such that there exists no uniformly differentiable function g on λBX , for any λ > 1, which coincides with f in some neighbourhood of the origin. © 2013 Elsevier Inc. All rights reserved. Keywords: Polynomials in Banach space

According to the fundamental Stone–Weierstrass theorem, if X is a finite dimensional real Banach space then every continuous function on the unit ball BX can be uniformly approximated by polynomials. For infinite dimensional Banach spaces the statement is false, even if we ✩

ˇ P201/11/0345, Project Barrande 7AMB12FR003, and RVO: 67985840. Supported in part by GACR

* Corresponding author.

E-mail addresses: [email protected] (S. D’Alessandro), [email protected] (P. Hájek). 0022-1236/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jfa.2013.11.017

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replace continuous functions by the uniformly continuous ones (which is a natural condition that coincides with continuity in the finite dimensional setting). The natural problem of the proper generalization of the result for infinite dimensional spaces was posed by Shilov [32] (in the case of a Hilbert space). Aron [3] (see also Aron and Prolla [4]) observed that the uniform closure finite type, denoted by Pf (X), which consists of on BX of the space of all polynomials of the all polynomials admitting a formula P (x) = nj=1 φj , xnj , φj ∈ X ∗ , nj ∈ N, is precisely the space of all functions which are weakly uniformly continuous on BX (Theorem 2.7 below). Since there exist infinite dimensional Banach spaces such that all bounded polynomials are weakly uniformly continuous on BX (e.g. c0 , or more generally all Banach spaces not containing a copy of 1 and such that all bounded polynomials are weakly sequentially continuous on BX ), this result gives a very satisfactory solution to the problem. Unfortunately, most Banach spaces, including Lp , p ∈ [1, ∞), do not have the special property used in [4]. In this case, no characterization of the uniform limits of polynomials is known. But the problem has a more subtle formulation as well. Let us consider the algebras An (X) consisting of all polynomials which can be generated by finitely many algebraic operations of addition and multiplication, starting from polynomials on X of degree not exceeding n ∈ N. Of course, such polynomials can have arbitrarily high degree. The first mentioned result can now be formulated as stating that A1 (X) consists precisely of all functions which are weakly uniformly continuous on BX . It is clear that if n is the lowest degree such that there exists a polynomial P in P(n X) which is not weakly uniformly continuous then A1 (X) = A2 (X) = · · · = An−1 (X)  An (X). The problem of what happens from n on has been studied in several papers, notably [29,24,17] etc. The natural conjecture appears to be that once the chain of equalities has been broken, it is going to be broken at each subsequent step. The proof of this latter statement given in [24], for all classical Banach spaces, based on the theory of algebraic bases is unfortunately not entirely correct, as was pointed out to the authors by Michal Johanis. It is not clear to us if the theory of algebraic bases developed therein can be salvaged. Fortunately, the main statement of this theory, Lemma 1.7 below, can be proved using another approach. The complete proof will appear in [2]. Most of the results in this area which used [24] are therefore safe. The main result of this note, Theorem 2.5, contains all previously known results (all confirming the above conjecture) as special cases. Its proof is based on Lemma 1.7 in combination with some new results on the asymptotic behaviour of polynomials on infinite dimensional spaces. We then proceed by giving solutions to three other problems posed in the literature, which are concerning smooth functions rather than polynomials, but which belong to the same field of study of smooth mappings on a Banach space. The first result is a construction of a nonequivalent C k -smooth norm on every Banach space admitting a C k -smooth norm, answering a problem posed in several places in the literature, e.g. in [8]. We solve a question in [9] by proving that a real Banach space admitting a separating real analytic function whose holomorphic extension is Lipschitz in some strip around X admits a separating polynomial. Finally, we solve a problem posed by Benyamini and Lindenstrauss in [11], concerning the extensions of uniformly differentiable functions from the unit ball into a larger set, preserving the values in some neighbourhood of the origin. More precisely, we construct an example of a uniformly differentiable real valued function f on the unit ball of a certain Banach space X, such that there exists no uniformly differentiable function g on λBX , for any λ > 1, which coincides with f in some neighbourhood of the origin.

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1. Subsymmetric polynomials In this section we collect some background results and concepts concerning polynomials in Banach spaces, that will be used in the sequel. We refer to [18] for the standard notation and results concerning polynomials. For a polynomial P ∈ P(d X; Y ) we denote by Pˇ ∈ Ls (d X; Y ) the unique symmetric d-linear mapping such that Pˇ (d x) = P (x), x ∈ X. By PK (d X; Y ) we denote the space of all compact and d-homogeneous polynomials from X to Y . Most of the time we will be concerned with the restrictions of the polynomials whose domain is a Banach space X, to a suitable subspace Y → X with a Schauder basis {ej }∞ j =1 (resp. a finitely dimensional Banach space). In this case it is possible, and very useful, to rely on the concrete representation of (the restriction of) the polynomial using the monomial expansion in terms of the vector coordinates.  Let n ∈ N. For a multi-index α ∈ Nn0 we denote its order by |α| = nj=1 αj . We denote the set of multi-indices of order d ∈ N0 by   I(n, d) = α ∈ {0, . . . , d}n : |α| = d . In order to treat infinite dimensional Banach spaces, we extend the definition also to the case when n = ∞, setting   ∞  I(∞, d) = α ∈ {0, . . . , d}N : |α| = αj = d . j =1

 For n ∈ N we have |I(n, d)| = n+d−1 n−1 . Given x = (x1 , . . . , xn ) ∈ Kn and α = (α1 , . . . , αn ) ∈ I(n, d) we use the standard multi-index notation

α xα = xl l . αl =0

The case n = ∞ is similar and corresponds to multi-indices whose domain is N. More precisely, ∗ ∞ ∗ for a fixed Schauder basis {ej }∞ j =1 of X, with a dual basis {xj }j =1 ⊂ X , α ∈ I(∞, d) and ∞ x = j =1 xj ej ,

α

α xl∗ , x l . xα = xl l = αl =0

αl =0

Note that x α ∈ P(d Kn ) for any α ∈ I(n, d), resp. x α ∈ P(d X). Given α = (α1 , . . . , αn ) ∈ I(n, d), we denote α! = α1 ! × · · · × αn !. We also use the corresponding multinomial coefficient by

  d! d d d! = . = = α1 ! · · · αn ! α! α α1 , . . . , αn We put a partial ordering on multi-indices defined as follows. If α = (α1 , . . . , αn ) ∈ I(n, d), β = (β1 , . . . , βn ) ∈ I(n, p), p  d, and αj  βj holds for all j ∈ {1, . . . , n} then we say that α  β, and we also denote α − β = (α1 − β1 , . . . , αn − βn ) ∈ I(n, d − p).

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Let P ∈ P(d X; Y ), v ∈ X. The directional derivative ∂P P (x + λv) − P (x) (x) = lim λ→0 ∂v λ is easily shown to be a polynomial in x which satisfies the formula   ∂P (x) = d · Pˇ v, d−1 x ∈ P d−1 X; Y . ∂v By induction, for a fixed α = (α1 , . . .) ∈ I(∞, p), p  d, where αi = 0, i > k, and y1 , . . . , yk ∈ X, we get

∂ α1 y

  ∂pP d! (x) = Pˇ α1 y1 , . . . , αk yk , d−p x ∈ P d−p X; Y . α (d − p)! 1 . . . ∂ k yk

(1)

d Let X be a Banach space with a Schauder basis {ej }∞ j =1 , Y be a Banach space, P ∈ P( X; Y ). ρ There is a unique set of vectors yα ∈ Y , α = (α1 , . . . , αk ) ∈ I(∞, d), ρj ∈ N, 1  ρ1 < ρ2 < · · · < ρk ,

yαρ =

∂d P 1 (0), α α1 ! · · · αk ! ∂ 1 eρ1 . . . ∂ αk eρk

(2)

such that the formula  P

∞  j =1

 xj ej

=





xρα11 · · · xραkk yαρ

(3)

α∈I(∞,d) 1ρ1 <···<ρk

holds for every finitely supported vector x ∈ X. In the special case Y = R the coefficients are just ρ real numbers aα . Definition 1.1. A Schauder basis {ej }∞ if there j =1 of a Banach space X is called symmetric  exists K > 0 such that for any bijection σ : N → N, the formal linear operator Iσ ( ∞ a j =1 j ej ) = ∞ −1 j =1 aσ (j ) ej is an isomorphism of X such that Iσ

Iσ < K. A Schauder basis {ej }∞ j =1 of a Banach space X is called spreading invariant if there exists K> 0 such thatfor any increasing mapping σ : N → N, the formal linear oper∞ ator Iσ ( ∞ j =1 aj ej ) = j =1 aj eσ (j ) is an isomorphism into a subspace of X such that −1



Iσ < K. A spreading invariant and unconditional basis is called subsymmetric. We remark that a symmetric basis is automatically unconditional. A subset U of a Banach space X with a Schauder basis {ej }∞ j =1 is called symmetric (resp. spreading invariant) if for any bijection σ : N → N (resp. for any increasing mapping σ : N → N), Iσ (U ) ⊂ U .

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Definition 1.2. Let {ej }∞ j =1 be a Schauder basis of a Banach space X, U ⊂ X be symmetric (resp. spreading invariant), and f : U → Y be a function. If  ∞   ∞  ∞    f aj ej = f aj eσ (j ) , aj ej ∈ U, j =1

j =1

j =1

for any bijection σ : N → N (resp. for any increasing mapping σ : N → N) then we say that f is symmetric (resp. subsymmetric) on U . These notions will typically be applied to functions whose domain is a Banach space with a symmetric (resp. spreading invariant) basis, or a subspace of a space with a Schauder basis consisting of finitely supported vectors. We use the same terminology also for functions acting from X = Rn , with the fixed and linearly ordered linear basis {ej }nj=1 . In this case the notion of subsymmetric is reduced to the identity f (x) = f (y) being valid for every pair x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) of elements of Rn such that the sequences formed by all non-zero coordinates of x and y coincide (e.g. x = (2, 0, 0, 1.5, π, 0), y = (0, 2, 1.5, 0, 0, π)). The relation (4) below is the fundamental result of the theory of spreading models. Its proof is based on repeated use of the finite Ramsey theorem, and can be found in e.g. in [22, p. 294] (the main idea of the crucial step is exposed in the proof of a simpler Theorem 1.6 below). We prefer to omit the standard proof of the additional result (5) concerning subsymmetric polynomials, which can be obtained by working simultaneously with the original norm · and P in the aforementioned proof, and keeping in mind that d-homogeneous polynomials form a closed set in the topology of uniform convergence on the unit ball. Theorem 1.3 (Brunel, Sucheston). (See [10].) Let {εn }∞ n=1 be sequence of positive real numbers decreasing to zero, {N (k)}∞ be an increasing sequence of natural numbers, and {xn }∞ k=1 n=1 be a normalized basic sequence in a Banach space X. Then there exists a subsequence {yn }∞ n=1 of ∞ , such that for all and a Banach space (Y,

| ·

|) with a spreading invariant basis {e } {xn }∞ n n=1 n=1 k ∈ N and all scalars aj , j = 1, . . . , N(k), N (k) N (k)  N (k)                  (1 − εk ) aj ej    aj ynj   (1 + εk ) aj ej ,       j =1

j =1

(4)

j =1

whenever k  n1 < · · · < nN (k) . If P ∈ P(d X) then we may in addition assume that there is a subsymmetric polynomial R ∈ P(d Y ) such that for all k R

 N (k) 

 aj ej

− εk  P

j =1

whenever k  n1 < · · · < N (k),

 N (k) 

 aj ynj

j =1

N (k) j =1

R

 N (k) 

 aj ej

+ εk ,

(5)

j =1

aj ynj ∈ BX .

Definition 1.4. The Banach space Y constructed above is said to be a spreading model built on 1 ∞ k the sequence {xn }∞ n=1 . If εn = 2n , Nk = 2 we call {yn }n=1 a “characteristic subsequence” of ∞ {xn }n=1 .

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It is clear that in general a basic sequence may admit many non-isomorphic spreading models. We say that Y is a spreading model of X provided Y results as a spreading model built on some normalized basic sequence in X. For a given d ∈ N denote  J(d) = α = (α1 , . . . , αk ): k ∈ N, αj ∈ {1, . . . , d},

k 

 αj = d .

j =1

For completeness, we also set J(0) = {∅}. Given α = (α1 , . . . , αk ) ∈ J(d) we let  Pα

∞ 

 xj ej



=

j =1

xρα11 · · · xραkk ,

(6)

1ρ1 <···<ρk

∞

for all finitely supported j =1 xj ej ∈ c00 , and set P∅ = 1. Clearly, Pα is a subsymmetric polynomial. Polynomials which satisfy (6) are called standard. This terminology applies also to the case when X = Rn . These polynomials form a linear basis of the finite dimensional linear space of all d-homogeneous subsymmetric (and not necessarily bounded) polynomials on span{ej }. More precisely, we have the following well-known fact. n Fact 1.5. Let X be the linear span of a Schauder basis {ej }∞ j =1 (resp. X = R ). If a polynomial ρ P ∈ Pd (X) is subsymmetric then for fixed α = (α1 , . . . , αk ), constants yα do not depend on the choice of ρ = ρ1 < · · · < ρk . In particular, the following equality holds

 P

∞ 

 xj ej

j =1

for all finitely supported

=

d   k=0 α∈J(k)

∞

j =1 xj ej

 Pα

∞ 

 (7)

xj ej yα

j =1

∈ X (resp. for all x ∈ Rn ).

We will also rely on a finite dimensional version of the above result. Theorem 1.6. Let ε > 0, d, n ∈ N. Then there exists N = N (d, n, ε) such that for every n N P ∈ P(d N 1 ), P  1, there exists a subsequence {ekj }j =1 of the basic vectors {ek }k=1 , and n d a subsymmetric polynomial Q ∈ P( 1 ) such that P − Q < ε. ρ

Proof. By combining the formulas (1), (2) and (3), we see that sup |aα | < K P , for some K > 0, regardless of N and the form of P . Denote  Rα

∞  j =1

 xj ej

=



aαρ xρα11 · · · xραkk ,

α ∈ J(d).

(8)

1ρ1 <···<ρk

 Then clearly P = α∈J(d) Rα , and since this sum has a finite number of summands (independently of N and the form of P ), it suffices to proceed with the proof for each Rα separately, showing that in this case we arrive at Q = cα Pα . Suppose α = (α1 , . . . , αk ). Since α is fixed now, we have a one-to-one coding

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(ρ1 , . . . , ρk ) ↔ A,

A ⊂ {1, . . . , N },

1633

|A| = k.

We can rewrite (8) as  Rα

∞ 





=

xj ej

A={ρ1 ,...,ρk }

j =1

aαA xρα11 · · · xραkk .

(9)

We suppose that P ∈ P(d N 1 ), and we paint the k-element subsets A ⊂ {1, . . . , N } using colours from the finite set m ∈ {−M, −M + 1, . . . , M}, M = [K] δ + 1, according to the relation  A  a − mδ   δ. α

By the finite Ramsey theorem, if N is large enough, there is a monochromatic subset B ⊂ {1, . . . , N } of cardinality n. More precisely, every A ⊂ B, |A| = k has the same colour m. So letting cα = mδ and Q = cα Pα leads to a polynomial satisfying the estimate  

     Rα aj ej − Q aj ej   δ  j ∈B

j ∈B

provided we have chosen δ <

ε nn .

 A={ρ1 ,...,ρk }⊂B

  α  x 1 · · · x αk   δ n < δnn < ε, ρ1 ρk k

2

In this paper we are going to work with algebras A of polynomials on a Banach space X, i.e. subsets of P(X) that are closed with respect to addition, pointwise multiplication, and scalar multiplication. Given an algebra A ⊂ P(X), we say that the set B ⊂ A generates the algebra A, if for every p ∈ A there is a finite set {b1 , . . . , bl } ⊂ B and a polynomial P ∈ P(Rl ) such that p = P (b1 , . . . , bl ). nFor every Banach space X, we denote by An (X) the algebra generated by polynomials from j =0 Pj (X). Given α = (α1 , . . . , αk ) ∈ J(d), and N  k, we let  PαN

N 

 xj ej

=

j =1



xρα11 · · · xραkk

(10)

1ρ1 <···<ρk

and set P∅N = 1. For N  d, the polynomials PαN , for α ∈ J(d), form a linear basis of the space of subsymmetric d-homogeneous polynomials on RN , which will be denoted by Hd (RN ). As we know, if k  N then Hk (RN ) has a linear basis consisting of PαN , α ∈ J(k) (see (10)). In other words, P ∈ Hk (Rn ) has the unique standard form P (x1 , . . . , xN ) =



aα PαN (x1 , . . . , xN ),

aα ∈ R.

(11)

α∈J(k)

The spaces of subsymmetric polynomials Hk (Rk ) and Hk (RN ), N > k, are canonically isomorphic as their linear bases can be indexed the same set J(k). The set of polynomials k N ) generates the algebra of subsymmetric polynomials denoted by S (RN ). H (R l k l=0

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Lemma 1.7. (See [2].) For every n ∈ N, there exists an ε > 0 such that for every m  M(n), m

  p(x1 , . . . , xm ) − sn+1 (x1 , . . . , xm )  ε,

sup

i=1

|xi |1

for every p from the algebra Sn (Rm ) generated by subsymmetric polynomials of degree at most n. The above quantitative lemma implies the following fundamental criterion. Theorem 1.8. Let X be an infinite dimensional Banach space, and P ∈ P(n X) be a polynomial with the following property. For every N ∈ N, ε > 0 there exists a normalized finite basic sequence {ej }N j =1 such that   N   N       sup P aj ej − ajn   ε. N   |a |1 j =1

j =1

j

j =1

Then P ∈ / An−1 (X). Proof. By contradiction, suppose that for any ε > 0, there exists a finite set {P1 , . . . , Pk } ⊂ P(n−1 X) and a polynomial R ∈ P(Rk ) such that   sup P (x) − Q(x)  ε x∈BX

holds, where Q = R(P1 , . . . , Pk ). Choose the sequence {ej }N j =1 by assumption of the theorem. By Theorem 1.6, if N ∈ N is large enough, there is a subset A ⊂ {1, . . . , N}, |A| > M(n), and such that each Pj , j = 1, . . . , k, restricted to {ej }j ∈A is subsymmetric with an error term as small as we wish. This contradicts Lemma 1.7. 2 2. Wsc and polynomials into 1 Let X, Y be Banach spaces. Using the classical duality for the projective tensor product [22, p. 692], combined with the duality between the symmetric tensor product and the homogeneous scalar valued polynomials [18, p. 19], we obtain the following well-known duality relationship. n π,s

 ∗ n   X ⊗π Y = L X; Y ∗ = P n X; Y ∗ . π,s

(12)

 As special cases, we of course have ( nπ,s X)∗ = P(n X), (X ⊗π Y )∗ = L(X; Y ∗ ). Recall a result of Bessaga and Pelczynski [22, p. 206]. Let X be a Banach space, c0 → X ∗ . Then X contains a complemented copy of 1 (and hence X ∗ actually contains a complemented copy of ∞ ). Applying this result to the duality relation (12) we get the next (probably known) result. Theorem 2.1. Let X be a Banach space. The following are equivalent for n ∈ N. 1. PK (n X; 1 ) = P(n X; 1 ), 2. c0  → P(n X).

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 n Proof. Suppose 2 fails. Since ( nπ,s X)∗ = P(n X), 1 is complemented in by the π,s X Bessaga–Pelczynski theorem. Hence 1 is a range of a bounded linear operator from nπ,s X and 1 fails by the universality of the projective symmetric tensor product. On the other hand, if 1 fails, then there is a non-compact bounded linear operator T : nπ,s X → 1 . Setting B = T (Bnπ,s X ), we claim that B contains B1 (up to isomorphism). Indeed, since B is not relatively compact, B is not weakly compact [22, p. 277]. By the Eberlein–Smulyan theorem there exists a bounded sequence {xn } ⊆ B with no weakly convergent subsequences, which cannot be weakly Cauchy either (Schur). Thus, by Rosenthal’s 1 -theorem, {xn } admits a subsequence equivalent to the usual 1 -basis, which proves the claim. nFinally, using the lifting property of 1 [22, p. 238], 1 is a complemented subspace of π,s X, whence 2 fails by duality. 2 We will need two principles for passing to suitable sequences in the domain. The first one is based on an improvement of the classical result that 2 is a linear quotient of any Banach space containing a copy of 1 . Lemma 2.2. Let X be a Banach space, 1 → X, p  2. Then there exists T ∈ L(X; p ) and a basic sequence {fj } in X equivalent to 1 -basis such that T (fj ) = ej is the unit basis in p . Proof. It suffices to prove the result for p = 2, since then we can compose T with the formal identity Id : 2 → p , which is a bounded linear operator. Let L : 2 → L1 be an isomorphic embedding, {ej } be the basis of 2 . By Pelczynski–Hagler [27, p. 253], there is an isomorphic embedding M : L1 → X ∗ . So {yj = M ◦ L(ej )} is a weakly null sequence in X ∗ , which is equivalent to the 2 -basis. There is a normalized sequence {f˜j } ∈ X ∗∗ biorthogonal to M ◦ L(ej ). By Goldstine’s theorem we replace f˜j by fj ∈ BX so that fj , yk  = 0, k  j , fj , yj  = 1. Since {yj } is weakly null, we can pass to subsequences so that {fj , yj } is a biorthogonal system. Since M ∗ (X) ⊂ L∞ and {L(ej ), M ∗ (fj )} is a biorthogonal system in L1 , L∞ , by the DPP property of L1 , {M ∗ (fj )} does not contain a weakly Cauchy subsequence. By Rosenthal’s 1 -theorem, we may assume without loss of generality that it is an 1 -basis. By the lifting property of 1 , {fj } is an 1 -basis. Finally, R = L∗ ◦ M ∗ : X ∗∗ → 2 is a quotient mapping such that R(fj ) = ej . So T = RX : X → 2 is the desired operator. 2 In particular, let X be a Banach space, 1 → X. Then there is a P ∈ P(2 X; 1 ) such that it takes a sequence {fj } in X, equivalent to an 1 -basis, into {ej } a unit basis in the range 1 . Proposition 2.3. Let X be a Banach space, 1 → X, k ∈ N, k  2. Then there exists a polynomial P ∈ P(k X) and a basic sequence {fj } in X equivalent to an 1 -basis such that  P

∞  j =1

 aj fj

=

∞ 

ajk .

j =1

In particular, P is not weakly continuous at the origin and   Pwu k X = P k X .

(13)

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Proof. Let T and {fj }be as above and let {gj } be the sequence of the coordinate functionals k on k . Letting P (x) = ∞ j =1 (gj (T (x))) proves (13). Assume by contradiction that P is weakly continuous, so given ε > 0 there exist φ1 , . . . , φn ∈ X ∗ and δ > 0 such that |φj (x)| < δ, j = 1, . . . , n, implies |P (x)| < ε. We have that φj [fj ] ∈ ∞ . By a simple argument there exist pairwise distinct indices m, l, r such that     φj (fm ) − φj (fl ), φj (fm ) − φj (fr ) < δ,

j = 1, . . . , n.

So choosing ε > 0 small enough and letting x = fm − 12 fl − 12 fr clearly witnesses the contradiction. 2 We will need a modification of a well-known principle for dealing with non-weakly sequentially continuous polynomials of minimal degree, which has been used many times in the literature (see e.g. [12] for its most general formulation). In our case, we replace the non-wsc property by the non-compactness, and add the assumption 1 → X. Lemma 2.4. Let X, Y be Banach spaces, 1 → X, P(k X; Y ) = PK (k X; Y ) for all k < n, and ∞ P ∈ P(n X; Y ) \ PK (n X; Y ). Then there is a weakly null sequence {yk }∞ k=1 such that {P (yk )}k=1 is not relatively compact. Proof. By Rosenthal’s 1 -theorem there is a δ > 0 and a weakly Cauchy sequence {xk }∞ k=1 such that   P (xk ) − P (xl ) > δ,

k = l ∈ N.

(14)

By a simple application of the multilinearity of Pˇ , P (xk − xl ) = P (xk ) +

n−1   n j =1

j

 (−1)j Pˇ j xl , n−j xk + (−1)n P (xl ).

By assumption, all polynomials of degree less than n are compact, so for any fixed k, passing to a subset of indices Nk ⊂ Nk−1 , N0 = N, there exist the limits  j yk = lim Pˇ j xl , n−j xk , l∈Nk

j = 1, . . . , n − 1.

Let M be the diagonal set of Nk , k ∈ N. Next, fix for each k ∈ M, an mk such that for all j ∈ {1, . . . , n − 1}  j   y − Pˇ j xl , n−j xk  < k

δ , 20nn+1

l  mk , l ∈ M.

Then    n−1     δ n   j (−1)j yk + (−1)n P (xl )  < , P (xk − xl ) − P (xk ) + j  20  j =1

l  mk , l ∈ M.

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Whence,   n−1      n δ  n j j  (−1) yk  < , P (xk − xl ) − P (xk ) − (−1) P (xl ) − j  20 

l  mk , l ∈ M.

j =1

Thus     P (xk − xl ) − P (xp − xr )  P (xk ) − (−1)n P (xl ) − P (xp ) + (−1)n P (xr ) − δ , 10 whenever k, p ∈ N, l, r ∈ M, l  mk , and r  mp . Suppose that k, l, p ∈ N are given and denote z = (−1)n P (xk ) + P (xl ) − (−1)n P (xp ). Using (14), there is an rk,l,p ∈ N such that

P (xr ) − z  2δ for all r  rk,l,p . Whence   P (xk − xl ) − P (xp − xr )  δ − δ > δ , 2 10 4 whenever k, p ∈ N, l, r ∈ M, l  mk , and r  max{mp , rk,l,p }. Now it suffices to find lk ∈ M such that lk  max{mk , r1,l1 ,k , . . . , rk−1,lk−1 ,k } and put yk = xk − xlk . Then {yk } is weakly null and {P (yk )} is a 4δ -separated sequence. 2 Theorem 2.5. Let X be a Banach space, and m be the minimal integer such that there is a noncompact P ∈ P(m X; 1 ). Then n  m implies P(n X) ⊂ An−1 (X). Proof. If 1 → X then it suffices to combine Theorem 1.8 and Proposition 2.3. For the rest of ∞ the proof we assume that 1 → X. Denote {fj }∞ j =1 the canonical basis in 1 , P = (Pk )k=1 ∈ m m P( X; 1 ), Pk ∈ P( X). We claim that by performing some adjustments to P , we may assume in addition that there exists a weakly null normalized basic sequence {xn }∞ n=1 ⊆ X such that P (xj ) = fj for each j . To this end, note that by Lemma 2.4 there exists a weakly null sequence {yk }∞ k=1 in X such is not relatively compact, i.e. it contains a separated subsequence, which we that {P (yk )}∞ k=1 call again {P (yk )}∞ . By [1, p. 22], by passing to a subsequence we may assume that {yk } k=1 contains no weakly null is a normalized basic sequence. As 1 is a Schur space, {P (yk )}∞ k=1 contains a subsequence, again {P (y subsequences. By Rosenthal’s 1 -theorem, {P (yk )}∞ k )}, k=1 equivalent to the 1 -basis. By a well-known result (according to Bill Johnson, who has pointed out to us some very closely related other results), every sequence in 1 , which is equivalent to the 1 -basis, contains a further subsequence which spans a complemented subspace. Since we were unable to find this result explicitly in the literature, let us indicate the idea of proof. Supposing that {zk } is the 1 -basic sequence in 1 , we may by passing to a subsequence assume that zk is pointwise  convergent to u0 ∈ 1 , and there exists a sequence of disjoint block vectors {uk } such that k |zk − u0 − uk | < ∞. The case when u0 = 0 is well-known [22, Proposition 4.45], so let us assume the contrary. Moreover, we may assume that the norms of u0 restricted to the supports of uk form a fast decreasing sequence. Then by the classical results [22, Thereom 4.23, Proposition 4.45], we have that the sequence {uk }∞ k=0 is equivalent to the 1 -basis, which is moreover complemented in 1 . Hence, u0 , z1 − u0 , z2 − u0 , . . . is also equivalent to a complemented 1 -basis in 1 . To finish, it suffices to find a suitable projection in this latter space, which takes (a0 , a1 , . . .) → ( ∞ k=1 ak , a1 , a2 , . . .).

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Hence there exists a weakly null and normalized basic sequence {xk } = {ynk } ⊆ {yn } such that P (xk ) = gk , where {gk } is equivalent to an 1 -basis which spans a complemented subspace in 1 . Hence composing P with the appropriate projection in 1 , we substitute 1 with its complemented subspace [gk ]∞ k=1 , and the claim follows. ∗ Let {φj }∞ be a bounded biorthogonal sequence to {xj }∞ j =1 j =1 in X . We are mostly going to be interested in the behaviour of P restricted to Y = span{xj : j ∈ N} → X. For the sake of convenience, set Y{j : j k+1} := span{xj : j  k + 1}. Note that we have P (λxj ) = λm fj . Formula (3) for the restriction of P to Y can be rewritten, by collecting the appropriate finitely  many terms, into the following formula, which holds for all finitely supported vectors x = aj xj ∈ Y .  Pk

∞ 

 aj xj

j =1





=

q α,q,r (a1 , . . . , ak−1 )α ak Sk

∞ 

 aj xj ,

(15)

j =k+1

p+q+r=m, α∈I(k−1,p)

α,q,r

Sk ∈ P(r Y{j : j k+1} ). Note that by the minimality assumption on m, for a fixed 0 = β = (β1 , . . .) ∈ I(∞, t), t  p < m where βi = 0, i > k − 1, we have that ∂t P= ∂ β1 x1 . . . ∂ βk−1 xk−1

∂t Pj ∂ β1 x1 . . . ∂ βk−1 xk−1

 : X → 1

is a compact (m − t)-homogeneous polynomial with range in 1 . So, for a fixed β of the aforementioned type,     ∂t  = 0. lim  P j   j →∞ ∂ β1 x1 . . . ∂ βk−1 xk−1 For y =

k−1 i=1

ai xi ∈ [x1 , . . . , xk−1 ] and l > k − 1,

  ∞  ∂t Pl y + aj xj = ∂ β1 x1 . . . ∂ βk−1 xk−1 j =l

 p+q+r=m αβ α∈I(k−1,p)

α! q α,q,r (ai )α−β al Sl (α − β)!



∞ 

 aj xj .

j =l+1

We claim that for a fixed 0 = β = (β1 , . . .) ∈ I(∞, t), t  m where βi = 0, i > k − 1, q, r such that t + q + r = m, we have that  β,q,r   lim Sl Y

{j : j l+1}

l→∞

= 0.

(16)

For the proof of the claim by contradiction, choose a maximal β ∈ I(k − 1, t) which fails (16). Hence, for any (if it exists) α ∈ I(k − 1, p), p > t, q, r such that p + q + r = m,  α,q,r   lim Sl Y

l→∞

{j : j l+1}

= 0.

(17)

Passing to a suitable subsequence of l → ∞ (for simplicity assuming it is still indexed by N) we conclude that there exists a normalized sequence of vl ∈ Y{j : j l+1} such that α,q,r bα,q,r = liml→∞ Sl (vl ) exist for all α, q, r, and there is at least one non-zero term (with

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α = β) among them. Moreover, if α ∈ I(k − 1, p) where p > t then bα,q,r = 0. That means that  a for a suitably chosen y = k−1 i=1 i xi ∈ [x1 , . . . , xk−1 ], and al ∈ R, we have  ∂t q Pl (y + al xl + vl ) = β!al bβ,q,r = 0, β β k−1 1 l→∞ ∂ x1 . . . ∂ xk−1 q+r=m−t lim

which contradicts the minimality of m. Fix an arbitrary sequence δk  0. By passing to a fast enough growing subsequence of {xj } q,r 0,q,r we can disregard in (15) all terms with p  1, so that (using the short notation Sk = Sk )   ∞   ∞      q q,r    sup a x a S a x − P   δk .  k j j j j k k ∞  

aj xj 1 j =1

j =1

(18)

j =k+1

q+r=m

∞ k Let {εnk }∞ n=1 , εn  0 be decreasing sequences of real numbers, {Nk (j )}j =1 be an increasing sequence of natural numbers. To start with, by applying Theorem 1.3, we may assume that {xn }∞ n=1 is a characteristic sequence of its spreading model E with a subsymmetric basis {en }∞ n=1 . By a repeated application of Theorem 1.3 there are nested subsequences N ⊃ M1 ⊃ M2 ⊃ · · · of index sets so that the following holds. For a subsequence {xn }n∈Mk of {xn }∞ n=1 q,r there is a subsymmetric polynomial Rk ∈ P(r E), r, q, such that for all scalars aj , j = 1, . . . , Nk (K),

q,r Rk

 N (K) k 

 aj ej

k − εK

q,r  Sk

 N (K) k 

j =1

 q,r  Rk

aj xnj

j =1

provided K  n1 < · · · < nNk (K) , nj ∈ Mk , and

 q,r Rk

∞ 

 aj ej

 aj ej

k , + εK

(19)

j =1

Nk (K)



=

j =1

 N (K) k 

j =1

aj xnj  1. By (7), 

aαq,r,k Pα

∞ 

 aj ej

(20)

j =1

α∈J(r)

for all finitely supported vectors. ∞ By passing to a suitable diagonal sequence M = {mi }∞ i=1 of the system {Mk }k=1 , and keeping q,r in mind that the set {Rk }k,q,r in P(E) is uniformly bounded, we may also assume that there exist finite limits bαq,r = lim aαq,r,k , k→∞

k ∈ M.

(21)

We consider the subsymmetric polynomial W q,r ∈ P(r E),  W q,r

∞  j =1

 aj ej

=

 α∈J(r)

 bαq,r Pα

∞  j =1

 aj ej .

(22)

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q,r We claim that W = 0, unless r = 0. Assuming the contrary, there is a finitely supported vector v = Ti=1 vi ei such that



W q,r (v) = δ = 0.

q+r=m, r1

We may assume without loss of generality that δ > 0. Hence, for a sliding finitely supported block vector wj = Ti=1 vi+j xmi+j , we get by (19), (20), and (21), 

lim inf j

q+r=m, r1

δ q,r Sl (wj ) > , 2

(23)

holds for all l ∈ M large enough. But this contradicts again the minimality assumption on m. Indeed, we denote by U a w ∗ -cluster point of {P (x + wk ): k ∈ N} in the dual Banach space Pm (X; 1 ). In particular, for every x there is a subsequence L ⊂ N such that U (x) =

lim

j →∞, k∈L

P (x + wj ).

Let U = U 0 + U 1 + · · · + U m = (Uk0 + Uk1 + · · · + Ukm )∞ k=1 be the unique splitting of U into a sum of j -homogeneous summands U j . Then by (23) m−1 



Uki (ak xk ) 

i=0

q+r=m, r1

δ q ak W q,r (v)  , 2

a contradiction with the minimality of m. This verifies the claim that W q,r = 0, unless r = 0. Combining all the previous results we conclude that there is an infinite increasing sequence M ⊂ N, and c = 0, such that the following holds. For any ρ > 0, and N ∈ N there is finite set {t1 , . . . , tN } ⊂ M such that   N        aj xti − cakm  < ρ, Pk  

k ∈ {t1 , . . . , tN }.

i=1

It is now clear that the polynomial Q ∈ P(m+l X), l  0, defined as Q(x) =

∞ 

φjl (x)Pj (x),

j =1

satisfies the condition laid out in Theorem 1.8, whence Q ∈ P(m+l X) \ Am+l−1 (X).

2

Corollary 2.6. (See [24].) Let X be a Banach space admitting a non-compact linear operator T ∈ L(X; p ), p ∈ [1, ∞). Then letting n = p we obtain An (X)  An+1 (X)  · · · .

(24)

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Proof. By standard argument, we may assume that T (BX ) contains the unit vectors in p . It then suffices to compose T with the polynomial P ∈ P(n p ; 1 ), given by formula (xj ) → (xjn ) to obtain a non-compact n-homogeneous polynomial from X into 1 . It remains to apply Theorem 2.5. 2 If p > 1, 2 is isomorphic to a complemented subspace of Lp ([0, 1]) [22, p. 210]. If p = 1, 1 is isomorphic to a (complemented) subspace of L1 ([0, 1]) [22, Theorem 4.57]. Also, if K is a non-scattered compact then by the classical result C(K) contains a copy of 1 . Hence if X = Lp ([0, 1]), 1  p  ∞, or X = C(K) where K is a non-scattered compact then A1 (X)  A2 (X)  · · · . The above Theorem 2.5, together with the positive results of [3–5] below, implies all known results in this area. Theorem 2.7. (See [3,4].) Let X, Y be Banach spaces, then Pf (X; Y ) = Cwu (BX ; Y ). Corollary 2.8. (See [5].) Let X, Y be Banach spaces, and suppose that X does not contain a subspace isomorphic to 1 . Then Pwu (n X; Y ) = Pwsc (n X; Y ). The next result was first formulated in [24]. Theorem 2.9. Let X be a Banach space, 1 → X. Then A1 (X)  A2 (X)  · · · . Proof. Combine Proposition 2.3 and Theorem 1.8.

2

Corollary 2.10. Given 1  p < ∞, we have the following: A1 (p ) = · · · = An−1 (p )  An (p )  An+1 (p )  · · · where n − 1 < p  n. Proof. By [5] we know that Pn (p ) = Pnwu (p ) whenever n < p. So using Theorem 2.7 we obtain that An−1 (p ) = A1 (p ). The rest follows readily from Corollary 2.6. 2 Corollary 2.11. Let X be a Banach space, q > 1, for n > p we have

1 p

+

1 q

= 1. Assume that X ∗ has type q. Then

A1 (X)  An (X)  An+1 (X)  · · · . ∗ Proof. By [23] there is a normalized basic sequence {yk }∞ k=1 in X which has the upper ∗ q-estimate. Thus, T : q → X , T (ek ) = yk is a non-compact bounded linear operator. Since T is weakly compact, T ∗ : X → p , is a non-compact operator. An appeal to Lemma 2.6 finishes the argument. 2

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Corollary 2.12. (See [17].) Let X be a Banach space with an unconditional FDD, 1 → X, and suppose that n is the least integer such that there exists a P ∈ P(n X) which is not weakly sequentially continuous. Then A1 (X) = · · · = An−1 (X)  An (X)  An+1 (X)  · · · . Proof. It was shown in [17], by using the averaging technique from [6] as in [16], that under these assumptions c0 → P(n X). 2 The main remaining problem is of course the following. Problem 2.13. Let X be a separable Banach space. Is there an n ∈ N such that A1 (X) = · · · = An−1 (X)  An (X)  An+1 (X)  · · · ? Note that if the dual X ∗ contains a subspace isomorphic to c0 or a superreflexive space, then we can conclude that (24) holds for some n. Indeed, in this case either 1 → X or, by using the James–Gurarii theorem [22, p. 450] X admits a non-compact linear operator into some p . This leaves us with two possibilities. If X fails (24), for any n ∈ N, then either X ∗ is 1 -saturated or it contains a Tsirelson-like subspace Y , in the sense that Y contains no copy of 1 , c0 , or a superreflexive space. 3. Some results on smooth functions In this section we solve several open problems from the literature regarding the behaviour of smooth functions on Banach spaces. We begin with a problem posed in various papers, e.g. in [8] or [7], concerning the existence of a non-complete C k -smooth renorming of a Banach space which admits a C k -smooth equivalent norm, where k  2. The non-complete C k -smooth renorming plays an important role in some applications regarding the so-called smooth negligibility and the existence of C k -smooth diffeomorphisms between certain subsets of the given Banach space X, see e.g. [19]. Our result can be used to simplify some parts of the theory of these mappings, in particular the techniques which bypass the use of the non-complete norm, used in [7], are no longer needed. We begin with an auxiliary result. Theorem 3.1. Let X be a Banach space with w ∗ -sequentially compact dual ball. If c0 ∼ = Y → X then Y contains a further subspace c0 ∼ = Z → Y such that Z is complemented in X. Proof. If c0 → X then X ∗ has a quotient 1 . By the lifting property, we also have 1 → X ∗ is a complemented subspace, and moreover, the basis {ej } of c0 and {fj } of 1 in X ∗ form a biorthogonal system. Since BX∗ is w ∗ -sequentially compact, by passing to a subsequence we get that fj → f in w ∗ -topology. So {g2j } = {f2j − f2j +1 } is w ∗ -null, and also equivalent to an 1 -basis,  which is still biorthogonal to {e2j }, again a c0 -basis sequence. Thus T : X → X, T (x) = g2j (x)e2j is a projection, and c0 is a complemented subspace of X. 2 In fact, a more general version of the above result was shown by Schlumprecht in his PhD thesis [31]. The condition on X is quite common, e.g. all weak Asplund spaces, or WLD spaces have it [15,20,22].

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Theorem 3.2. Let X be an infinite dimensional Banach space admitting a C k -smooth norm, k  2. Then X admits a decomposition X = Y ⊕Z where Y is infinite dimensional and separable. In particular, X admits a non-complete C k -smooth renorming. Proof. By Corollary 3.3. in [14] we have that either c0 → X or X is superreflexive. Either way, using the previous Theorem 3.1 (or the existence of PRI on superreflexive spaces), X = Y ⊕ Z where Y is infinite dimensional and separable. But since every separable Banach space injects into c0 , it admits a non-complete C ∞ -smooth norm, it follows that X admits a C k -smooth non-complete norm. 2 We point out that for k = 1 the existence of a (non-equivalent) C 1 -smooth norm on a given Banach space (or even any Asplund space) X remains open. The theorem below solves a problem posed in [9], concerning an assumption used by these authors in the proof of their main result. Before we pass the description of our result, let us recall that for every real Banach space X one may construct its complexified version X C , which is (as a real Banach space) isomorphic to X ⊕ X. The complex norm on X C is not uniquely determined, but this fact plays no role in our argument. We refer to the paper [28] for details. C 1 -smooth

Theorem 3.3. Let X be a real Banach space which admits a real analytic separating function whose complex extension exists and is Lipschitz on some strip around X, i.e. on X + 2rBXC ⊂ X C , for some r > 0. Then X is superreflexive and admits a separating polynomial. Proof. By contradiction. Let f : BX → R be a separating real analytic and Lipschitz function with f (0) = 0, df (0) = 0 and infSX f > 1, and such that the complex extension f˜ : BX + rBXC → C exists and is K-Lipschitz, r > 0. Denote S = BX + rBXC ⊂ X C . This implies that f˜ is bounded by K + r on S. By the Cauchy formula [30, Theorem 10.28] (for the second derivative of f˜) 2 d f˜(a)[h] = 2πi



2

f˜(a + ζ h) dζ ζ3

(25)

γ

holds for every a, h ∈ BX , and the path γ (t) = reit , t ∈ [0, 2π ]. Noting that the denominator in the Cauchy formula is in absolute value r 3 , we obtain that d 2 f (a) is uniformly bounded on BX . Hence df is Lipschitz on BX . By a result of Fabian, Whitfield and Zizler in [21], Theorem 3.2 in [14] X is superreflexive. By a result of Deville, Theorem 4.1 in [14] X has a separating polynomial. 2 In the last part of this section we give a solution to an extension problem, posed in the monograph of Benyamini and Lindenstrauss [11, p. 278], concerning uniformly differentiable functions on the unit ball of a Banach space X. Suppose that f : BX → R is a uniformly differentiable function in the interior of the unit ball BX . Is there a uniformly smooth extension of f whose domain is the whole X, or at least some neighbourhood of BX ? A weaker version of this problem (if we expect a positive solution, i.e. the existence of some extension) would be to require that the extension coincides with f at least in some open neighbourhood of the origin. We will show that even the weaker version of the problem has a negative solution. Our solution

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is based on the application of the theory of W-class of Banach spaces, which was developed in a series of papers [25,26,13,12] (this class was denoted by C-class in the first three papers). Definition 3.4. We say that a Banach space X is a Wλ -space, λ ∈ (0, 1], if every uniformly differentiable function f : BX → R takes weakly Cauchy sequences in λBX to convergent sequences. We say that X is a W-space if it is a Wλ -space for some λ > 0. It was shown in [12] that every C(K) space is a W1 -space. In particular, c0 in the supremum norm is also a W1 -space (this was shown in [25]). Being a W-space is invariant under isomorphism, but the precise value of λ may change. Proposition 3.5. For every m ∈ N, m  2, there is an equivalent renorming of c0 such that (c0 ,

· m ) belongs to W 1 -class, but it does not belong to W 1 -class. m

m−1

Proof. The renorming · m of c0 is determined by its closed unit ball Bm ⊂ c0 ,   Bm = conv {±mej }∞ j =1 ∪ Bc0 . Clearly, Bc0 ⊂ Bm ⊂ mBc0 . Note that if x = (xj ) ∈ Bm then card{j : |xj |  1 + x=

n  k=1

 ak mek + a0

∞  j =1

 bj ej

=

∞  j =1

xj ej ,

(26)

1 2 m}  m .

where

Indeed, suppose

n 

ak = 1 and ak  0.

k=0

Letting A = {k: ak  m12 }, clearly card(A)  m2 . Now |xj | = |a0 bj + aj m|  a0 + maj < 1 ∞ 1 + m1 , unless j ∈ A. Choose φm : R → R+ 0 a C -smooth even convex function, φm [−1 − m , 1  are -Lipschitz. Let now 1 + m1 ] = 0, φm (t) > 0, t > 1 + m1 , and such that both φm , φm m2 2m+1 ∞ Φm (x) = j =1 φm (xj ). It is clear from the previous discussion that Φm depends on at most m2 -coordinates in a neighbourhood of any interior point in Bm . Hence it is a uniformly differ1 entiable symmetric function such that both Φm , dΦm are 2m+1 -Lipschitz. But Φm (tej ) > 0 for 1 1 every t > 1 + m , j ∈ N, hence Φm restricted to m−1 Bm does not take weakly null sequences into null sequences. 2 Example 3.6. There is a Banach space X and a uniformly differentiable function f : BX → R which cannot be extended to a uniformly differentiable function on any λBX , λ > 1, preserving its original values in some neighbourhood of 0.   Proof. Let X = 2 ∞ · m ), Pm : X → (c0 , · m ) be the canonical projections onto m=2 (c0 ,  the direct summands. Let f (x) = ∞ m=2 Φm ◦ Pm (x). The functions f and df are 1-Lipschitz, so f is uniformly differentiable (even with a Lipschitz derivative). It is also clear that (26) implies 1 1 that Φm cannot be extended to (1 + m−1 )Bm , preserving its values on m−1 Bm . Since m can be chosen arbitrary large, the result follows. 2

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Acknowledgments We would like to thank the referee for a very careful review, and for identifying some weak points in our arguments. We also thank our colleague Michal Johanis for meticulous proofreading of the manuscript, as well as finding out that one of our principal references [24] is flawed. Finally, we thank Bill Johnson for some useful remarks. References [1] F. Albiac, N.J. Kalton, Topics in Banach Space Theory, Grad. Texts in Math., vol. 233, Springer, 2006. [2] S. D’Alessandro, P. Hájek, M. Johanis, Corrigendum to the paper “Polynomial algebras on classical Banach spaces”, Israel J. Math. 106 (1998) 209–220, in press. [3] R.M. Aron, Approximation of differentiable functions on a Banach space, in: M.C. Matos (Ed.), Infinite Dimensional Holomorphy and Applications, in: North-Holland Math. Stud., vol. 12, 1977, pp. 1–17. [4] R.M. Aron, J.B. Prolla, Polynomial approximation of differentiable functions on Banach spaces, J. Reine Angew. Math. 313 (1980) 195–216. [5] R.M. Aron, C. Hervés, M. Valdivia, Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52 (1983) 189–204. [6] R.M. Aron, M. Lacruz, R. Ryan, A. Tonge, The generalized Rademacher functions, Note Mat. 12 (1992) 15–25. [7] D. Azagra, Diffeomorphisms between spheres and hyperplanes in infinite-dimensional Banach spaces, Studia Math. 125 (1997) 179–186. [8] D. Azagra, T. Dobrowolski, Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications, Math. Ann. 312 (1998) 445–463. [9] D. Azagra, R. Fry, L. Keener, Real analytic approximation of Lipschitz functions on Hilbert space and other Banach spaces, J. Funct. Anal. 262 (2012) 124–166. [10] B. Beauzamy, J.-T. Lapresté, Modeles étalés des espaces de Banach, Travaux en Cours, Hermann, Paris, 1984. [11] Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 1, Amer. Math. Soc. Colloq. Publ., vol. 48, 2000. [12] Y.S. Choi, P. Hájek, H.J. Lee, Extensions of smooth mappings into biduals and weak continuity, Adv. Math. 234 (2013) 453–487. [13] R. Deville, P. Hájek, Smooth noncompact operators from C(K), K scattered, Israel J. Math. 162 (2007) 29–56. [14] R. Deville, G. Godefroy, V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monogr. Surv. Pure Appl. Math., vol. 64, Longman, London, 1993. [15] J. Diestel, Sequences and Series in Banach Spaces, Grad. Texts in Math., vol. 92, Springer-Verlag, 1984. [16] V. Dimant, S. Dineen, Banach subspaces of spaces of holomorphic functions and related topics, Math. Scand. 83 (1998) 142–160. [17] V. Dimant, R. Gonzalo, Block diagonal polynomials, Trans. Amer. Math. Soc. 353 (2000) 733–747. [18] S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Springer Monogr. Math., Springer-Verlag, London, 1999. [19] T. Dobrowolski, Smooth and R-analytic negligibility of subsets and extension of homeomorphism in Banach spaces, Studia Math. 65 (1979) 115–139. [20] M. Fabian, Differentiability of Convex Functions and Topology—Weak Asplund Spaces, John Wiley and Sons, 1997. [21] M. Fabian, J.H.M. Whitfield, V. Zizler, Norms with locally Lipschitzian derivatives, Israel J. Math. 44 (1983) 262–276. [22] M. Fabian, P. Habala, P. Hájek, V. Montesinos, V. Zizler, Banach Space Theory: The Basis for Linear and Nonlinear Analysis, CMS Books Math., Springer-Verlag, New York, 2011. [23] J. Farmer, W.B. Johnson, Polynomial Schur and polynomial Dunford–Pettis properties, Contemp. Math. 144 (1993) 95–105. [24] P. Hájek, Polynomial algebras on classical Banach spaces, Israel J. Math. 106 (1998) 209–220. [25] P. Hájek, Smooth functions on c0 , Israel J. Math. 104 (1998) 89–96. [26] P. Hájek, Smooth functions on C(K), Israel J. Math. 107 (1998) 237–252. [27] P. Hájek, V. Montesinos, J. Vanderwerff, V. Zizler, Biorthogonal Systems in Banach Spaces, CMS Books Math., Canadian Mathematical Society, Springer-Verlag, 2007.

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