POLYNOMIAL CONTINUITY ON l 1 MANUEL GONZÁLEZ, JOAQUÍN M. GUTIÉRREZ, AND JOSÉ G. LLAVONA. (Communicated by Palle E. T. Jorgensen)

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 5, May 1997, Pages S (97) POLYNOMIAL CONTINUITY ON l 1 MANUEL GONZÁLEZ, JOAQUÍN M GUTIÉRREZ, AND JOSÉ G LLAVONA (Communicated by Palle E T Jorgensen) Abstract A mapping between Banach spaces is said to be polynomially continuous if its restriction to any bounded set is uniformly continuous for the wea polynomial topology A Banach space X has property (RP) if given two bounded sequences (u j ), (v j ) X, wehavethatq(u j ) Q(v j ) 0for every polynomial Q on X whenever P (u j v j ) 0 for every polynomial P on X; ie, the restriction of every polynomial on X to each bounded set is uniformly sequentially continuous for the wea polynomial topology We show that property (RP) does not imply that every scalar valued polynomial on X must be polynomially continuous Throughout, X and Y are Banach spaces, X the dual of X, B X its closed unit ball, S X its unit sphere, and N the set of natural numbers Given N, we denote by P( X, Y )thespaceofall-homogeneous (continuous) polynomials from X into Y ; L s ( X, Y ) is the space of all (continuous) symmetric -linear mappings from X := X () X into Y Whenever Y is omitted, it is understood to be the scalar field K (real R or complex C) We identify P( 0 X)=K, and denote P(X) := =0 P( X) For the general theory of polynomials on Banach spaces, we refer to [6] As usual, e n stands for the sequence (0,,0,1,0,) with 1 in the nth position To each polynomial P P( X, Y ) we can associate a unique symmetric -linear mapping ˆP L s ( X, Y )sothatp(x)= ˆP(x,,x) for all x X, and a (bounded linear) operator T P : X L s ( 1 X, Y )givenby T P (x)(x 1,,x 1 )= ˆP(x, x 1,,x 1 ) Following [1], we say that a mapping f : X Y is polynomially continuous (P -continuous, for short) if, for every ɛ>0 and bounded B X, there are a finite set {P 1,,P n } P(X)andδ>0sothat f(x) f(y) <ɛwhenever x, y B satisfy P j (x y) <δ(1 j n) Clearly, the definition may be restated assuming that the polynomials P 1,,P n are homogeneous Suppose we require the polynomials {P 1,,P n } P(X) in the above definition to be of degree one, ie, to be continuous linear forms on X Then we obtain that f is wealy uniformly continuous on bounded subsets, a notion that has been studied Received by the editors October 30, Mathematics Subject Classification Primary 46E15; Secondary 46B20 Key words and phrases Polynomials on Banach spaces, wea polynomial topology, polynomials on l 1 The first author was supported in part by DGICYT Project PB (Spain), and the second and third authors by DGICYT Project PB (Spain) 1349 c 1997 American Mathematical Society

2 1350 MANUEL GONZÁLEZ, JOAQUÍN M GUTIÉRREZ, AND JOSÉ G LLAVONA by many authors (see [1]) Since an operator is compact if and only if it is wealy (uniformly) continuous on bounded sets [2, Proposition 25], every compact operator is P -continuous If a polynomial is wealy (uniformly) continuous on bounded sets (such as every scalar valued polynomial on c 0 ), then it is clearly P -continuous We shall need the following result: Proposition 1 A polynomial P is P -continuous if and only if so is the associated operator T P Proof Suppose P P( X, Y )isp-continuous Given ɛ>0, we can find δ>0and {P 1,,P n } P(X)sothat P(x) P(y) <ɛwhenever P j (x y) <δfor all 1 j n and x, y B X Assume x, y satisfy the above conditions, and z 1,,z 1 B X The polarization formula [6, Theorem 110] yields: (T P (x) T P (y)) (z 1,,z 1 ) = ˆP(x, z 1,,z 1 ) ˆP(y, z 1,,z 1 ) = ( ) ɛ1 x+ɛ 2 z 1 + +ɛ z 1!2 ɛ 1 ɛ [P ɛ j=±1 ( )] ɛ1 y+ɛ 2 z 1 + +ɛ z 1 P Assuming that every P j is homogeneous, we have ( P ɛ1 x + ɛ 2 z 1 + +ɛ z 1 j ɛ ) 1y+ɛ 2 z 1 + +ɛ z 1 < P j (ɛ 1 x ɛ 1 y) = P j (x y) < δ for 1 j n, andso T P (x) T P (y) ɛ! Conversely, let T P be P-continuous For 0 < ɛ < 1, there is δ > 0 and {P 1,,P n } P(X)sothat T P (x) T P (y) <ɛ, whenever P j (x y) <δ for any 1 j n and x, y B X Forsuchx, y we have P (x) P (y) ˆP(x,,x) ˆP(x,y,x,,x) + ˆP(x,y,x,,x) ˆP(x,y,y,x,,x) + + ˆP(x,y,,y) ˆP(y,,y) = (T P (x) T P (y))(x,,x) + (T P (x) T P (y))(x,y,x,,x) + + (T P (x) T P (y))(y,,y) < ɛ, and the proof is complete We say that a net (x α ) X converges to x in the wea polynomial topology (pw-topology, for short) [3, 6] if for every P P(X)wehaveP(x α ) P(x) It is clear that a mapping f : X Y is pw-continuous on bounded sets if and only if for every x X, ɛ>0 and bounded B X with x B, there are δ>0and

3 POLYNOMIAL CONTINUITY ON l {P 1,,P n } P(X)sothatwehave f(x) f(y) <ɛwhenever P j (x y) <δ for 1 j n and y B Obviously, an operator is P -continuous if and only if it is pw-continuous on bounded sets We now relate the P -continuity with property (RP) of Aron, Choi and Llavona [1] We say that X has property (RP) if given two bounded sequences (u j )and(v j ) in X, wehavethatq(u j ) Q(v j ) 0 for every Q P(X) whenever P (u j v j ) 0 for every P P(X) Every superreflexive space and every space with the DPP not containing l 1 have property (RP) [1] Clearly, if every scalar valued (continuous) polynomial on X is P -continuous, then X has property (RP) It is proved in [1] that C[0, 1], L 1 [0, 1] and L [0, 1] do not satisfy property (RP), and that there are 3-homogeneous polynomials on the spaces C[0, 1] and L [0, 1] which are not P -continuous Similarly, there is a non-p -continuous 2-homogeneous polynomial on L 1 [0, 1] It is natural to as whether property (RP) implies that every scalar valued polynomial is P -continuous We show that the answer is no by giving examples of polynomials on l 1 which are not P -continuous We first need to construct a pw-null net in the sphere of l 1 We need a previous lemma Lemma 2 Let U be a wea zero neighbourhood in l 1 Then, for each m N we can find x =(x n ) S l1 U and r>mso that x n =0whenever n<mand n>r Proof We can find ξ 1,,ξ B l and ɛ>0 such that U {x l 1 : ξ j (x) <ɛfor 1 j } Let ξ j = ( ) ξj n There is an infinite set A N so that n=1 ξ p j ξq j < 2ɛ whenever 1 j and p, q A Fix p, q A (m p<q), and set x := (e p e q )/2 and r=qthen ξ j (x) = ξ p j ξq j /2<ɛfor 1 j, and the proof is complete The following two results use the idea of [4] Lemma 3 Let F be a finite family of continuous symmetric multilinear forms on l 1, ɛ>0and N 1 Then there exist x 1,,x N S l1, with disjoint supports, such that F (x i1,,x im ) <ɛwhenever F F is an m-form and i 1,,i m are distinct indices between 1 and N Proof Since each F Fis symmetric, it is enough to obtain the estimate when i 1 < <i m By Lemma 2, we can find n 1 N and x 1 S l1, having all but the first n 1 coordinates equal to zero, so that F (x 1 ) < ɛ for all F F l 1 Again by Lemma 2, we can choose n 2 N and x 2 S l1 having disjoint support with x 1 and all but the first n 2 coordinates equal to zero, so that F (x 2 ) <ɛfor all F F l 1, and F (x 1,x 2 ) <ɛfor all F F L s ( 2 l 1 ) In this way, we obtain x j s with disjoint supports, so that F (x i1,,x im ) <ɛfor all F F L s ( m l 1 )andall i 1 < <i m Theorem 4 There is a pw-null net in S l1 Proof It is enough to show that for every finite family F P(l 1 )andɛ>0, there is an x S l1 so that P (x) <ɛfor all P F Fix N large, choose x 1,,x N S l1 with disjoint supports satisfying the conditions of Lemma 3 for the family { ˆP : P F}of symmetric multilinear forms, and set x := 1 N (x 1 + +x N ) S l1

4 1352 MANUEL GONZÁLEZ, JOAQUÍN M GUTIÉRREZ, AND JOSÉ G LLAVONA If P F P( m l 1 ), we write P (x) = 1 N m N i 1,,i m=1 ˆP (x i1,,x im )=Σ 1 +Σ 2, where Σ 1 is the sum over m-tuples of distinct indices, and Σ 2 is the sum over the remaining indices By Lemma 3, Σ 1 <ɛ/2 Since there are N m N(N 1) (N m +1) summands in Σ 2,weobtain Σ 2 for N large enough [ 1 ( 1 1 N ) ( 1 m 1 )] ˆP < ɛ N 2, As a consequence, if X contains a copy of l 1, then the unit sphere of X contains a pw-null net as well We now give the main result Theorem 5 For every N ( 2), there is a -homogeneous scalar valued polynomial on l 1 which is not P -continuous Proof Suppose first that =2andl 1 is constructed over the real numbers We need a sequence (x j ) l,equivalenttothel 1 -basis, such that x j i = xi j for all i, j N, wherex i j := x j(e i ) (then we say that the sequence is symmetric) We select a Rademacher-lie sequence (y j ) l, taing y 1 := (1, 1, 1, 1,), and letting y j be the sequence consisting of infinitely many times the following bloc of 2 j integers: 1, (2j 1 ),1, 1, (2j 1 ), 1 Clearly, (y j ) is 1-equivalent to the unit vector basis of l 1 ; ie, for every finite set of real numbers α 1,,α n,wehave (1) α j = α j y j j=1 j=1 If we tae x 1 := y 1 and, for j>1, modify the first j 1 coordinates of y j in the obvious way, then we get a symmetric sequence (x j ) which is still 1-equivalent to the unit vector basis of l 1 Now, define an operator T : l 1 l by T (e j ):=x j SinceTis an embedding, we conclude from Theorem 4 that it is not P -continuous Therefore, by Proposition 1, the 2-homogeneous polynomial P : l 1 R given by P (y) =(T(y))(y) for y l 1 is not P -continuous The same sequence can be used in the complex case, since, for every finite set of complex numbers α 1,,α n, we have (see [5, XI, Proposition 4]) (2) α j 4 α j y j j=1 The sequence (y j ) will be used in the case >2 as well Letting { y i2+ +i j if j min {i 2,,i }, A j (e i2,,e i ):= if i r =min{i 2,,i }<j, j=1 y j+i2+ +ir 1+ir+1+ +i i r

5 POLYNOMIAL CONTINUITY ON l we obtain A j L s ( 1 l 1 ) Moreover, the sequence (A j ) is equivalent to the unit vector basis of l 1 Indeed, given real numbers α 1,,α n, using (1), choose i 2,,i N such that n min{i 2,,i } and α j = α j y i2+ +i j j=1 j=1 = α j A j j=1 Note that the sequence (A j ) is symmetric in the sense that A i1 (e i2,,e i )isinvariant under permutation of the indices i 1,,i In the complex case we proceed similarly, using (2) in place of (1) In both cases we define T : l 1 L s ( 1 l 1 )by T(e j )=A j Then the polynomial P P( l 1 )givenby P(α):= α i1 α i A i1 (e i2,,e i ), for α =(α j ) j=1 l 1, i 1,,i =1 is not P -continuous, since the associated operator T is an isomorphism The authors wish to than J A Jaramillo, M Lacruz and S Troyansi for helpful discussions References [1] R M Aron, Y S Choi and J G Llavona, Estimates by polynomials, Bull Austral Math Soc 52 (1995), CMP 96:03 [2] R M Aron and J B Prolla, Polynomial approximation of differentiable functions on Banach spaces, J Reine Angew Math 313 (1980), MR 81c:41078 [3] T K Carne, B Cole and T W Gamelin, A uniform algebra of analytic functions on a Banach space, Trans Amer Math Soc 314 (1989), MR 90i:46098 [4] A M Davie and T W Gamelin, A theorem on polynomial-star approximation, Proc Amer Math Soc 106 (1989), MR 89:46023 [5] J Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Math 92, Springer, Berlin 1984 MR 85i:46020 [6] J Mujica, Complex Analysis in Banach Spaces, Math Studies 120, North-Holland, Amsterdam 1986 MR 88d:46084 Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cantabria, Santander, Spain address: gonzalem@ccaix3unicanes Departamento de Matemáticas, ETS de Ingenieros Industriales, Universidad Politécnica de Madrid, C José Gutiérrez Abascal 2, Madrid, Spain address: c @ccupmupmes Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, Madrid, Spain address: llavona@eucmaxsimucmes