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Bases in Spaces of Multilinear Forms over Banach Spaces
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
200, 548]566 Ž1996.
0224
Bases in Spaces of Multilinear Forms over Banach Spaces V. Dimant and I. Zalduendo Departamento de Economıa Uni¨ ersidad de San Andres, ´ y Matematica, ´ ´ CC 1983-Correo Central, (1000) Buenos Aires, Argentina Submitted by Richard M. Aron Received October 13, 1994
INTRODUCTION In this article we study the existence of monomial bases in spaces of multilinear forms over Banach spaces. During the past years it has become increasingly apparent that there are close relationships between the existence of monomial bases in spaces of polynomials, the reflexivity of such spaces, and compactness or weak sequential continuity Žcomplete continuity. of polynomials. Thus, for example, for reflexive Banach spaces with bases E and F, Holub w17x has proved that reflexivity of L 2 Ž E = F . is equivalent to compactness of all linear maps from E to F U . Ryan w23x shows that when E and F have shrinking bases, the same condition Žcompactness of linear operators E ª F U . is equivalent to the existence of a monomial basis in L 2 Ž E = F .. Alencar w2x proved that for reflexive Banach spaces with basis, the existence of monomial basis in the space of homogeneous polynomials is equivalent to reflexivity of this space. Also, the non-existence of monomial basis has been related in some cases w3, 7, 13x to the containment of a copy of l` . We extend a result in w14x by applying the Gonzalo]Jaramillo indexes and the estimates found in w24x concerning multilinear forms, and we use this to give conditions for the existence of monomial basis in L m Ž E1 = ??? = Em ., the space of m-linear forms over a product of Banach spaces. We begin by giving, in Section 1, a number of conditions including compactness and weak sequential continuity, which are equivalent to the existence of monomial bases. We apply these equivalences and the Gonzalo]Jaramillo indexes to the problem of existence of monomial bases in spaces of multilinear forms over spaces with upper or lower p-bounds for 548 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
549
BASES IN SPACES OF MULTILINEAR FORMS
sequences in Section 2, and in spaces of homogeneous polynomials in Section 3. Some examples, applications and open problems are given in Section 4.
1. MULTILINEAR FORMS AND MONOMIAL SCHAUDER BASES Throughout, E1 , . . . , Em will be Banach spaces over the complex field, with Schauder bases e1n4nG 1 , . . . , e nm 4nG 1 , respectively. For each i s 1, . . . , m and for each n g N, e niU will denote the coordinate functional. We will denote by L m Ž E1 = ??? = Em , F . the space of all continuous m-linear mappings from E1 = ??? = Em into F, and by L m Ž E1 = ??? = Em . we mean L m Ž E1 = ??? = Em , C.. We say that A g L m Ž E1 = ??? = Em , F . is compact if A maps the product of unit balls BE1 = ??? = BE m into a relatively compact subset of F, and that A is weakly sequentially continuous if A maps weakly convergent sequences in E1 = ??? = Em into norm convergent sequences in F. The set of all compact m-linear mappings from E1 = ??? = Em into F will be denoted by K m Ž E1 = ??? = Em , F . and the set of all weakly sequentially continuous m-linear mappings will be m Ž denoted by Lwsc E1 = ??? = Em , F .. The notation E1 = ??? = Eˆj = ??? = Em means the Ž m y 1.-product of the spaces E1 , . . . , Em , excluding the space Ej . The notation AŽ x 1 , . . . , ˆ x j , . . . , x m . for an Ž m y 1.-linear form A is interpreted analogously. Let A g L m Ž E1 = ??? = Em ., n g N, and 1 F k F m, we define 5 A 5 kn s sup < A Ž x 1 , . . . , x m . < : 5 x i 5 s 1
½
k for all i and x k g e nk , e nq1 ,...
5
and 5 A 5 n s sup5 A 5 kn : k s 1, . . . , m4 . Also, if T g L Ž Ei , F ., we write 5 T 5 n s sup 5 T Ž x . 5 : 5 x 5 s 1,
½
i x g e ni , e nq1 ,...
5.
For each Ž i1 , . . . , i m . g N m we define Bi1 , . . . , i m g L m Ž E1 = ??? = Em . by Bi1 , . . . , i m s e1i1U ??? e immU . We consider in N m the square ordering, described inductively by Ryan in w22x Žsee also w11x.: for m s 2, Ž1, 1., Ž1, 2., Ž2, 2., Ž2, 1., Ž1, 3., Ž2, 3., Ž3, 3., Ž3, 2., . . . ; and given the ordering s1 , s2 , . . . of N my 1 , the order in N m is Ž s1 , 1., Ž s1 , 2., Ž s2 , 2., Ž s2 , 1., Ž s1 , 3., . . . . Thus the notation Bi , . . . , i 4 means 1 m
550
DIMANT AND ZALDUENDO
the sequence ordered in that way, and ÝŽ i1 , . . . , i m .g N m means a series with this ordering. On the other hand, Ý`i1s1 ??? Ý`i m s1 refers to an iterated sum. In w4x E is said to have the LBP property Žfor Littlewood, Bogdanowicz, and Pelczynski. if certain analytic functions can be uniformly approximated on bounded sets by polynomials of finite type. Accordingly, we say that E1 = ??? = Em has the m-LBP property if each A g L m Ž E1 = ??? = Em . is approximable Žuniformly on the product of unit balls. by finite sums of products of linear functionals. We come now to the main theorem of this section. THEOREM 1. Let E1 , . . . , Em be Banach spaces with Schauder bases e1n4 , . . . , e nm 4 . Then the following are equi¨ alent: Ži. For all A g L m Ž E1 = ??? = Em ., 5 A 5 n ª 0. Žii. For all A g L m Ž E1 = ??? = Em ., A s Ý`i s1 ??? Ý`i s1 1 m 1 AŽ e i1, . . . , e imm . Bi1 , . . . , i m Ž uniformly on the product of unit balls.. Žiii. For each i s 1, . . . , m, e ni 4n G 1 is shrinking and E1 = ??? = Em has the m-LBP property. Živ. Bi , . . . , i 4 is a Schauder basis of L m Ž E1 = ??? = Em .. 1 m Žv. The basis e1n4nG 1 is shrinking, for all C g L my 1 Ž E2 = ??? = Em .5 C 5 n ª 0 and L Ž E1 , L my1 Ž E2 = ??? = Em .. s KŽ E1 , L my1 Ž E2 = ??? = Em ... Žvi. For each i, j s 1, . . . , m, e ni 4nG 1 is shrinking and L my 1 E1 = ??? = Eˆj = ??? = Em , EUj
ž
/
s K my 1 E1 = ??? = Eˆj = ??? = Em , EUj .
ž
/
Žvii. For each i, j s 1, . . . , m, e ni 4nG 1 is shrinking and L my 1 E1 = ??? = Eˆj = ??? = Em , EUj
ž
/
my 1 s Lwsc E1 = ??? = Eˆj = ??? = Em , EUj .
ž
/
Žviii. For each i s 1, . . . , m, e ni 4nG 1 is shrinking and there is a j g my1 Ž 1, . . . , m4 for which L my 1 Ž E1 = ??? = Eˆj = ??? = Em , EUj . s Lwsc E1 U = ??? = Eˆj = ??? = Em , Ej .. Proof. We first see the equivalence between Ži., Žii., and Žiii.. Ži. « Žii. We use induction on m. For m s 1, the coordinate functionals are a basis of EU if the basis of E is shrinking; so this is clear. For
551
BASES IN SPACES OF MULTILINEAR FORMS
m G 2, let Ž x 1, . . . , x m . g E1 = ??? = Em , 5 x i 5 s 1 for all i and let « ) 0. Note that the equality in Žii. is true pointwise: AŽ x 1 , . . . , x m . s A
`
ž
i1s1
`
s
e1i1U Ž x 1 . e 1i1 , . . . ,
Ý
i m s1
e immU Ž x m . e imm
/
`
???
Ý
`
Ý
Ý A Ž e1i , . . . , eim . Bi , . . . , i Ž x 1 , . . . , x m .
i1s1
i m s1
N1
Nm
m
1
m
1
so we have AŽ x , . . . , x . y
Ý
???
???
Ý A Ž e1i , . . . , eim . Bi , . . . , i
m
1
i1s1 N1
F
Ý
i1s1
ž
`
Ý
i 2s1
N1
Ý
Ý
i1s1
m
1
i m s1
m
1
`
Ý
???
???
Ý A Ž e1i , . . . , eim . Bi , . . . , i
ž
` i 2s1
ž
`
m
1
m
1
1
m
Nm
???
i1sN1q1
/
Ý A Ž e1i , . . . , eim . Bi , . . . , i Ž x 1 , . . . , x m .
i m s1
m
1
`
Ý
m
1
` i m s1
i 2s1
/
Ý A Ž e1i , . . . , eim . Bi , . . . , i Ž x 1 , . . . , x m .
i m s1
N2
Ý
m
Ý A Ž e1i , . . . , eim . Bi , . . . , i Ž x 1 , . . . , x m .
`
y
q A
1
Ý
Ý
m
1
Nm
???
i1sN1q1 i 2s1
s
m
1
i m s1
i 2s1
q
m
1
`
N2
y
Ý A Ž e1i , . . . , e im . Bi , . . . , i Ž x 1 , . . . , x m .
i m s1
x 1i1 e 1i1 , x 2 , . . . , x m
/
1
m
.
The second term of the last sum is at most 5 A 5 N1Ž1 q K . Žwhere K is the basis constant for e1n4. so we can choose N1 so that this term is less than «r2. For this N1 we can choose N2 , . . . , Nm so that the first term of the last sum is less than «r2, since if Ži. is valid for m-linear forms then it is valid for Ž m y 1.-linear forms and we can use the inductive hypothesis. Žii. « Žiii. Clearly Žii. implies that E1 = ??? = Em has the m-LBP property. In order to see that e nj 4n G 1 is shrinking, let w i g EiU and
552
DIMANT AND ZALDUENDO
A s w 1 ??? wm g L m Ž E1 = ??? = Em .. Suppose that for each i s 1, . . . , m, j x i g Ei , 5 x i 5 s 1, and x j g e nj , e nq1 , . . . , so we have < AŽ x 1 , . . . , x m . < s
`
Ý
i1s1
s
`
Ý
`
???
Ý
i m s1 `
???
i1s1
w 1 Ž e i1 . ??? wm Ž e i m . Bi1 , . . . , i mŽ x 1 , . . . , x m . `
???
Ý
Ý
i jsn
i m s1
w 1 Ž e i1 . ???
wm Ž e i m . Bi1 , . . . , i mŽ x 1 , . . . , x m . F
`
Ý
???
i1s1
`
`
???
Ý
Ý
i jsn
i m s1
w 1 Ž e i1 . ??? wm Ž e i m . Bi1 , . . . , i m .
Since the last of the above terms tends to 0 as n grows, it follows that 5 A 5 nj s 5 w j 5 n ª 0. Thus e nj 4n G 1 is shrinking. Žiii. « Ži. Let A g L m Ž E1 = . . . = Em .; let « ) 0. By Žiii. there exist j w 1 , . . . , w Nj g EUj such that N
Ý wi1
Ay
??? w im -
is1
« 2
.
In order to prove that 5 A 5 kn ª 0, let M s max5 w ij 5 : i s 1, . . . , N; j s 1, . . . , m4 and choose n 0 g N such that 5 w ik 5 n - «r2 NM my 1 for i s 1, . . . , N and n G n 0 . Suppose that x j g Ej , 5 x j 5 s 1 for j s 1, . . . , m and k x k g e nk , e nq1 , . . . , so we have < AŽ x 1 , . . . , x m . < F
N
ž
Ý wi1
Ay
/
??? w im Ž x 1 , . . . , x m .
is1 N
q
Ý wi1
??? w im Ž x 1 , . . . , x m .
is1
-
« 2
N
Ý < wi1 Ž x 1 . < ??? < wim Ž x m . <
q
is1
« 2 N
qÝ is1
« 2 NM my 1
M my 1 s « .
553
BASES IN SPACES OF MULTILINEAR FORMS
Thus, 5 A 5 kn - « if n G n 0 . Since k is arbitrary, we have shown that 5 A 5 n ª 0, and this completes the equivalence between the three first items. Now we prove Žii. « Živ. « Ži.. Assume that Žii. holds so Bi1 , . . . , i m 4 spans a dense subspace in L m Ž E1 = ??? = Em .. But Bi1 , . . . , i m are the coordinate functionals corresponding to the canonical basis of the projecˆ ??? m ˆ Em Žsee w22x., so they form a basic setive tensor product E1 m Ž . quence. Then iv is valid. Assume, now, that Živ. holds. Let A g L m Ž E1 = ??? = Em . and suppose k that x j g Ej , 5 x j 5 s 1 for j s 1, . . . , m and x k g e nk , e nq1 , . . . . Let m D k s Ž i1 , . . . , i m . g N : i k G n4 . By the square ordering of N m , D k ; D s Ž i1 , . . . , i m . g N m : Ž i1 , . . . , i m . G Ž1, 1, . . . , 1, n.4 . Note also that if Ž i1 , . . . , i m . f D k then Bi , . . . , i Ž x 1, . . . , x m . s 0. Hence, 1 m AŽ x 1 , . . . , x m . s
Ý Ž i 1 , . . . , i m .gN m
s
Ý Ž i 1 , . . . , i m .gD
s
A Ž e1i1 , . . . , e imm . Bi1 , . . . , i mŽ x 1 , . . . , x m .
A Ž e1i1 , . . . , e imm . Bi1 , . . . , i mŽ x 1 , . . . , x m .
Ý Ž i 1 , . . . , i m .G Ž1, 1, . . . , 1, n .
A Ž e1i1 , . . . , e imm . Bi1 , . . . , i mŽ x 1 , . . . , x m . .
Therefore, < AŽ x 1 , . . . , x m . < F
Ý Ž i 1 , . . . , i m .G Ž1, 1, . . . , 1, n .
A Ž e1i1 , . . . , e imm . Bi1 , . . . , i m ª 0.
We conclude that 5 A 5 kn ª 0 for each k so 5 A 5 n ª 0 and Ži. holds. We remark that it is clear that Ži. implies that all the bases e ni 4nG 1 are shrinking and also 5 C 5 n ª 0 if C g L my 1 Ž E2 = ??? = Em ., so we should prove only the second part of items between Žv. and Žviii.. Let us see that Žv. holds. Let T g L Ž E1 , L my 1 Ž E2 = ??? = Em .. and let AT be the associated m-linear mapping in L m Ž E1 = ??? = Em .. By Žii., AT s
`
Ý
???
i1s1
`
Ý
i m s1
AT Ž e1i1 , . . . , e imm . Bi1 , . . . , i m ,
so we have T Ž x1 . s
`
Ý
i1s1
???
`
Ý T Ž e1i .Ž ei2 , . . . , eim . e1i U Ž x 1 . ei2U
i m s1
1
2
m
1
2
??? e immU
uniformly for x g BE1. Thus T is a limit of finite rank operators and, therefore, compact. 1
554
DIMANT AND ZALDUENDO
We now prove that Žv. « Ži.. Let A g L m Ž E1 = ??? = Em . and let T be the associated linear mapping in L Ž E1 , L my 1 Ž E2 = ??? = Em ... Note that for all n g N,
a n s sup 5 A 5 kn : k s 2, . . . , m4 s sup 5 T Ž x 1 . 5 n : 5 x 1 5 s 1 4 so we will prove that a n ª 0 and that 5 A 5 1n ª 0. Let « ) 0. Since T is compact, there exist x 11, . . . , x 1N in the unit ball of E1 such that T Ž x 1 g E1 : 5 x 1 5 s 14. ; D 1 F jF N B ŽTx 1j , «r2.. By Žv., for each x 1j there is an n 0 Ž x 1j . g N such that if n G n 0 Ž x 1j ., 5 Tx 1j 5 n - «r2. Then for n G n 0 s max n 0 Ž x 1j .4 we have that 5 T Ž x 1 .5 n - « for all x 1 with unit norm. Thus, a n ª 0. It remains to prove that 5 A 5 1n ª 0. For this, note that 5 A 5 1n s 5 T 5 n . Let « ) 0 and take x 1n g E1 such that 5 x 1n 5 s 1, x 1n g w e1n , e1nq1 , . . . x, and 5 T Ž x 1n .5 ) 5 T 5 n y « for each n. Since the basis of E1 is shrinking, we have that x 1n4 converges weakly to 0, so by the compactness of T, T Ž x 1n . ª 0. Therefore, 5 T 5 n ª 0. The proof of Žii. « Žvi. « Ži. is similar to the proof of Žii. « Žv. « Ži.. Finally, we prove Žii. « Žvii. « Žviii. « Ži.. Suppose that Žii. holds. Let F g L my 1 Ž E1 = ??? = Eˆj = ??? = Em , EUj ., for arbitrary j s 1, . . . , m and take for each i s 1, . . . , m, i / j, a sequence x ni 4n G 1 converging weakly to x i g Ei . We may suppose that all these sequences and their limits have norm at most 1. By Žii.,
F Ž y1 , . . . , ˆ y j, . . . , y m . s
`
Ý
`
???
Ý
i1s1
i m s1
F e 1i1 , . . . , ˆ e ijj , . . . , e imm
ž
j ij
/že /e
1U i1
Ž y 1 . ???
e immU Ž y m . e ijjU uniformly in BE1 = ??? = BˆE j = ??? = BE m . Let « ) 0 and let N1 , . . . , Nm such that N1
FŽ y , . . . , ˆ y ,..., y j
1
m
.y Ý
???
i1s1
Nm
Ý
i ms1
F e1i1 , . . . , ˆ e ijj , . . . , e imm
ž
j ij
/že /e
1U i1
Ž y 1 . ???
e immU Ž y m . e ijjU -
« 3
555
BASES IN SPACES OF MULTILINEAR FORMS
w
for all Ž y 1 , . . . , ˆ y j, . . . , y m . g BE1 = ??? = BˆE j = ??? = BE m . Since x ni ª x i, we can choose n 0 g N such that for all n G n 0 N1
Nm
Ý
???
i1s1
Ý
i m s1
N1
y
Ý
ž
j ij
/že /e
1U i1
Ž x 1n . ??? eimU Ž x nm . eijU
Nm
???
i1s1
-
F e1i1 , . . . , ˆ e ijj , . . . , e imm
Ý
i m s1
« 3
F e 1i1 , . . . , ˆ e ijj , . . . , e imm
ž
j ij
/že /e
m
1U i1
j
Ž x 1 . ??? e immU Ž x m . e ijjU
.
So we have that 5 F Ž x 1n , . . . , x nm . y F Ž x 1, . . . , x m .5 - « for n G n 0 , and F is weakly sequentially continuous. It is clear that Žvii. « Žviii.. We now prove that Žviii. « Ži.. Let A g mŽ L E1 = ??? = Em . and let F be the corresponding map in L my 1 Ž E1 = ??? = Eˆj = ??? = Em , EUj .. Suppose that for some k, 5 A 5 nk ¢ 0, and take « ) 0 and norm-one sequences x ni 4n g Ei for i s 1, . . . , m, such that w x nk ª 0 and < AŽ x 1n , . . . , x nm .< ) « . We may assume that x ni 4n G 1 is weakly Cauchy Žfor all i / k ., because in a Banach space with a shrinking basis e n4 , every bounded sequence x n4 has a weakly Cauchy subsequence. Indeed, eUn Ž x k .4k G 1 is Cauchy for all n, so the result is a consequence of w6, Lemma 2.7x. We must consider two cases to obtain a contradiction: Ža. k / j. Since F is weakly sequentially continuous we may use w6, Lemma 2.4x Žmodified for m-linear forms from a product of different spaces. to see that 5 F Ž x 1n , . . . , ˆ x nj , . . . , x nk , . . . , x nm . 5 ª 0. Therefore < A Ž x 1n , . . . , x nm . < s < F Ž x 1n , . . . , ˆ x nj , . . . , x nk , . . . , x nm . Ž x nj . < F 5 F Ž x 1n , . . . , ˆ x nj , . . . , x nk , . . . , x nm . 5 ª 0 and this is a contradiction. Žb. k s j. By w6, Corollary 2.5x, F maps weakly Cauchy sequences into norm Cauchy sequences. Thus F Ž x 1n , . . . , ˆ x nk , . . . , x nm . is a norm Cauchy
556
DIMANT AND ZALDUENDO
sequence in EkU , so it converges to z g EkU , and 5 z 5 n ª 0 because the basis of Ek is shrinking. Hence, < A Ž x 1n , . . . , x nm . < s < F Ž x 1n , . . . , ˆ x nk , . . . , x nm . Ž x nk . < F 5 F Ž x 1n , . . . , ˆ x nk , . . . , x nm . 5 n F 5 F Ž x 1n , . . . , ˆ x nk , . . . , x nm . y z 5 n q 5 z 5 n F 5 F Ž x 1n , . . . , ˆ x nk , . . . , x nm . y z 5 q 5 z 5 n ª 0. Again, we have a contradiction, and this completes the proof. We end this section with two corollaries and a few remarks. COROLLARY 1. If the Banach spaces E1 , . . . , Em all ha¨ e unconditional shrinking bases, then either Ža. Žb.
monomials form a basis of L m Ž E1 = ??? = Em ., or L m Ž E1 = ??? = Em . contains a copy of l` .
Proof. If Ža. does not hold, by Žv. of Theorem 1 either there is a non-compact T g L Ž E1 , L my 1 Ž E2 = ??? = Em .., or a C g L my 1 Ž E2 = ??? = Em . with 5 C 5 n not tending to zero. In the first case, L Ž E1 , L my 1 Ž E2 = ??? = Em .. contains l` by a result of Diestel and Morrison w8, Theorem 2x. In the second, by Ži. of Theorem 1 monomials do not form a Schauder basis of L my 1 Ž E2 = ??? = Em .; the result follows by induction. COROLLARY 2. If E1 , . . . , Em are reflexi¨ e Banach spaces with Schauder bases, then L m Ž E1 = ??? = Em . is reflexi¨ e if and only if it has monomial basis. Proof. Holub w17x proved that if E and F are reflexive Banach spaces with bases, then L Ž E, F . is reflexive if and only if L Ž E, F . s KŽ E, F .. Holub’s result and Theorem 1 yield the proof. Remark. Give E1 , . . . , Em such that L m Ž E1 = ??? = Em . has a monomial basis, one may wonder if it has a non-monomial basis as well. Another result of Holub w17x is that if X and Y are reflexive Banach spaces with basis and their projective tensor product is not reflexive, then it has a non-tensor product basis. If any of the spaces E1 , . . . , Em is not reflexive, neither is L m Ž E1 = ??? = Em . for this contains copies of EiX . By a result of Zippin w25x not all bases of L m Ž E1 = ??? = Em . can be boundedly complete; thus there is a non-monomial basis Žall monomial bases, being dual, are boundedly complete..
BASES IN SPACES OF MULTILINEAR FORMS
557
2. m-LINEAR FORMS OVER SPACES WITH UPPER OR LOWER p-BOUNDS In this section we apply Theorem 1 to spaces with bounds for sequences. We begin by recalling the necessary definitions of lower and upper indexes of a Banach space E, introduced by Gonzalo and Jaramillo in w14x. A sequence x i 4 in E is said to have an upper p-estimate Ž1 F p F `. if there is a positive constant C such that for every n-tuple of scalars a1 , . . . , a n , n
Ý
1rp
n
ai x i F C
is1
ž
Ý < ai < p is1
/
This is equivalent w15x to the sequence being weakly q-summable where 1rp q 1rq s 1. A Banach space is said to have property S p if every weakly null semi-normalized basic sequence in E has a subsequence with an upper p-estimate. Clearly property S p implies property Sr for all r F p. The lower index of E is then defined as l Ž E . s sup p G 1 : E has property S p 4 . In an analogous manner, but using lower q-estimates one obtains the definition of property Tq Žwith Tq implying Tr for all larger r ., and u Ž E . s inf q G 1 : E has property Tq 4 . We refer to w14x for more on the Gonzalo]Jaramillo indexes, but we wish to mention that for 1 - p - `, l Ž l p . s uŽ l p . s p. It is well known that monomials form a Schauder basis in the space of k-homogeneous polynomials over l p if k - p, and that when k G p, P k Ž l p . contains a copy of l` . Theorems 2 and 3 generalize these facts. First, we require a lemma. The proof is modeled on the proof of Theorem 2.4 of w14x. LEMMA.
Let E1 , . . . , EN , and F be Banach spaces such that 1 l Ž E1 .
q ??? q
1 l Ž EN .
-
1 uŽ F .
.
Then e¨ ery A g L N Ž E1 = ??? = EN , F . is weakly sequentially continuous.
558
DIMANT AND ZALDUENDO
Proof. Choose p1 , . . . , pN , q so that Ei has property S p i , F has property Tq , and 1 p1
q ??? q
1 pN
-
1 q
.
Take x ni 4 ; Ei converging weakly but not in norm to 0 Žif convergence is also strong for some i, there is nothing to prove.. We may extract semi-normalized basic subsequences x fi i Ž k .4 with upper pi-estimate. Thus for every i s 1, . . . , N, the map l p i ª Ei mapping e k to x fi i Ž k . is continuous. Given A g L N Ž E1 = ??? = EN , F . define F g L N Ž l p 1 = ??? = l p N , F . by F Ž e i1 , . . . , e i N . s A Ž x 1f 1 Ž i1 . , . . . , x fNN Ž i N . . . We shall see that y k s AŽ x 1f 1 Ž k . , . . . , x fNN Ž k . . is a sequence with an upper Ž1rp1 q ??? q1rpN .y1 -estimate. For this, it will be sufficient w15x to verify that Ž y k . is weakly s-summable, where ss 1y
ž
1 p1
q ??? q
1 pN
y1
/
.
But given any g g F U , g ( F g L N Ž l p 1 = ??? = l p N ., and 1 p1
q ??? q
1 pN
-
1 q
F 1,
so by w24x, we obtain that g Ž y k . is s-summable. Since q - Ž1rp1 q ??? q1rpN .y1 , no subsequence of Ž y k . can have a lower q-estimate. But if some subsequence of Ž y k . were bounded from below in norm, since F has property Tq we could extract a subsequence with a lower q-estimate. Thus y k tend to 0 in norm. Since this can be done with any subsequence of x ni 4 , we obtain that A is weakly sequentially continuous at 0. Suppose now that x ni 4 tends weakly to x i. We will see by induction on N that AŽ x 1n , . . . , x nN . converges to AŽ x 1, . . . , x N . in norm. For N s 1 this is a consequence of the weak sequential continuity at 0 and the linearity of A. Note that our hypothesis Ž1rp1 q ??? q1rpN - 1rq . and the first part of the proof allows the weak sequential continuity of the k-linear functions Ž k - N . obtained by fixing some variables in A. Now AŽ x 1n y x 1, . . . , x nN y x N . may be expanded to a sum of 2 N terms Ž2 Ny 1 added and 2 Ny1
559
BASES IN SPACES OF MULTILINEAR FORMS
subtracted .. By adding the subtracting AŽ x 1, . . . , x N . 2 Ny 2 times we obtain A Ž x 1n y x 1 , . . . , x nN y x N . s A Ž x 1n , . . . , x nN . y A Ž x 1 , . . . , x N . q Ý Ž A Ž . . . , x i , . . . , x nj , . . . . y A Ž x 1 , . . . , x N . . 1
q Ý Ž A Ž x 1 , . . . , x N . y A Ž . . . , x i , . . . , x nj , . . . . . , 2
where if N is odd we have 2 Ny 2 terms in Ý1 and 2 Ny 2 in Ý2 , whereas if N is even we have 2 Ny 2 y 1 in Ý1 and 2 Ny 2 q 1 in Ý2 . In any case, 5 A Ž x 1n , . . . , x nN . y A Ž x 1 , . . . , x N . 5 F 5 A Ž x 1n y x 1 , . . . , x nN y x N . 5 q Ý 5 A Ž . . . , x i , . . . , x nj , . . . . y A Ž x 1 , . . . , x N . 5 which tends to 0 by our inductive hypothesis and the weak sequential continuity of A at 0. THEOREM 2. Let E1 , . . . , Em be Banach spaces with shrinking bases e ni 4 and such that for some j 1 l Ž E1 .
q ??? q
ˆ1 l Ž Ej .
q ??? q
1 l Ž Em .
-
1 u Ž EUj .
.
Then the monomials Bi1 , . . . , i m s e1i1U ??? e immU form a Schauder basis of L m Ž E1 = ??? = Em .. Proof. Just apply the lemma and equivalences Živ. and Žviii. of Theorem 1. COROLLARY. Let E1 , . . . , Em be reflexi¨ e Banach spaces with Schauder bases and such that for some j 1 l Ž E1 .
q ??? q
ˆ1 l Ž Ej .
q ??? q
Then L m Ž E1 = ??? = Em . is reflexi¨ e.
1 l Ž Em .
-
1 u Ž EUj .
.
560
DIMANT AND ZALDUENDO
Proof. The proof is as that of Corollary 1 of Section 1. THEOREM 3. Let E1 , . . . , Em be Banach spaces with unconditional shrinking bases, and such that for all j 1
q ??? q
u Ž E1 .
ˆ1
q ??? q
u Ž Ej .
1
)
u Ž Em .
1 l Ž EUj .
.
Then L m Ž E1 = ??? = Em . contains a copy of l` . Proof. None of the Ei ’s contain a copy of l 1 , for all have shrinking bases. Thus they either contain copies of c 0 or are reflexive. Those which contain copies of c 0 have an infinite upper Gonzalo]Jaramillo index. Therefore the inequality in our hypothesis forces at least one of the Ei ’s to be reflexive. Say Em is reflexive. We have 1
q ??? q
u Ž E1 .
1 u Ž Emy1 .
)
1 U l Ž Em .
.
U Choose p, q1 , . . . , qmy1 such that Ei has property Tq i for i F m y 1, Em has property S p , and 1rq1 q ??? q1rqmy1 s 1rp. Given the unconditional shrinking bases Ž e ij . i of Ej , by normalizing when necessary, we obtain weakly null seminormalized basic sequences from which we extract subsequences Žwith the same subindexes., such that Ž e nj i . i has a lower q j-estimate. Since Em is reflexive, Ž e imU . i is shrinking; thus in the same manner as above we extract a subsequence Ž e nmiU . i with an upper p-estimate. For each i g N define Ti g L m Ž E1 = ??? = Em . by setting
Ti Ž x 1 , . . . , x m . s x 1n i ??? x nmi . In other words, Ti s e1nUi ??? e nmiU . Given scalars a1 , . . . , a n , n
Ý aiTi is1
n
s sup j5
Ý ai x 1n
5 x F1 is1
??? x nmi
i
n
s sup j5
5 x F1
Ý ai x 1n
??? x nmy1 e nmiU i
i
is1 1rp
n
F D sup 5 x j 5F1
ž
F sup < a i < D sup i
5 x j 5F1
/ 1rp
n
ž
/
561
BASES IN SPACES OF MULTILINEAR FORMS
F 5 a 5 ` D sup
5 x j 5F1
1rq 1
n
ž
Ý < x 1n < q1 i
is1
/
1rq my 1
n
???
ž
Ý < x nmy1 < q my 1 i
is1
n
F C1 ??? Cmy1 D 5 a 5 ` sup
Ý
j5
5 x F1
/
n
x 1n i e 1n i ???
is1
Ý x nmy1 enmy1 i
i
is1
F KC1 ??? Cmy1 D 5 a 5 ` , where K is the maximum unconditional basis constant of the bases of the Ei ’s. Thus n
Ý aiTi
F C 5 a5 ` .
is1
On the other hand, for each k g N, < ak < s
n
ž
/
Ý aiTi Ž e1n , . . . , enm . is1 n
F
Ý aiTi
k
k
5 e 1n 5 ??? 5 e nm 5 , k
k
is1
so 5 a 5 ` s sup < a k < F sup 5 e1n 5 ??? 5 e nm 5 k k k
k
n
Ý ai Ti
.
is1
Therefore there are constants C, CX such that CX 5 a5 ` F
n
Ý aiTi
F C 5 a5 ` ,
is1
and the closed subspace of L m Ž E1 = ??? = Em . spanned by the Ti ’s is isomorphic to c0 . Being a dual space, L m Ž E1 = ??? = Em . must also contain a copy of l`. Note that the proof of the theorem remains valid as long as one has the inequality for a given j, with a reflexive Ej . Also, it is not necessary for the basis of this Ej to be unconditional. An example in which the inequality of Theorem 3 is verified by some but not all j’s is provided by L 2 Ž c 0 = l 2 .. This space has a basis by Theorem 2.
562
DIMANT AND ZALDUENDO
3. POLYNOMIALS OVER SPACES WITH UPPER OR LOWER p-BOUNDS Here we apply the results of the previous sections to spaces of polynomials over a Banach space E. The set D s Ž i1 , . . . , i m . g N m : i1 G i 2 G ??? G i m 4 is given the ordering induced by the square ordering of N m . For each Ž i1 , . . . , i m . g D we will note by Pi1 , . . . , i m the m-homogeneous polynomial Pi1 , . . . , i mŽ x . s Bi1 , . . . , i mŽ x, . . . , x . and by the notation Pi1 , . . . , i m 4 we mean the sequence Pi1 , . . . , i m 4Ž i1 , . . . , i m .g D ordered as above. We then have the following proposition. PROPOSITION. Let E be a Banach space with Schauder basis such that Bi , . . . , i 4 is a basis of L m Ž E .. Then Pi , . . . , i 4 is a basis of P m Ž E .. 1 m 1 m Proof. Clearly, if Bi1 , . . . , i m 4 is a basis of L m Ž E . then Pi1 , . . . , i m 4 spans a dense subspace in P m Ž E .. Since Pi1 , . . . , i m are the coordinate functionals ˆ m, s E Žthe m-fold symmetric projective tensor associated with the basis of m product., they form a basic sequence. Consequently, Pi1 , . . . , i m 4 is a basis of P m Ž E .. As simple consequences of this proposition, we have the following analogues of Theorem 2 and its corollary. THEOREM 4.
Let E be a Banach space with shrinking basis such that
Ž m y 1 . u Ž EU . - l Ž E . . Then Pi1 , . . . , i m 4 is a basis of P m Ž E .. COROLLARY.
Let E be a reflexi¨ e Banach space with a basis such that
Ž m y 1 . u Ž EU . - l Ž E . . Then P m Ž E . is reflexi¨ e. Theorem 3 can also be proved for spaces of homogeneous polynomials. As we have mentioned above, when m ) p the space of m-homogeneous polynomials P m Ž l p . contains a copy of l` Žsee, for example, w1, 7, 13x.. The following theorem generalizes that result. THEOREM 5. such that
Let E be a Banach space with unconditional shrinking basis
Ž m y 1 . l Ž EU . ) u Ž E . . Then P m Ž E . contains a copy of l` .
BASES IN SPACES OF MULTILINEAR FORMS
563
Proof. If we put E1 s E2 s ??? s Em s E and q1 s ??? s qmy1 s q in the proof of Theorem 3 we obtain Ti s eUn i ??? eUn i g Lsm Ž E ., for the Ti ’s are symmetric. Thus c 0 ; w Ti x ; Lsm Ž E . ( P m Ž E .. Being a dual, P m Ž E . contains l`.
4. EXAMPLES AND APPLICATIONS In this section we apply our results to several concrete Banach space examples. In several cases, and particularly for the polynomial examples, these results are known, with different proofs. One should consult, for example, w5, 2, 3, 7, 12, 13x. The Gonzalo]Jaramillo indexes used can be found in w14x. EXAMPLE 1. l p spaces Ž1 - p - `.. In this case one has l Ž l p . s uŽ l p . s p. Thus the following conditions are all equivalent: Ža. 1rp1 q ??? q1rpm - 1. Žb. L m Ž l p 1 = ??? = l p m . is reflexive. Žc. L m Ž l p 1 = ??? = l p m . has a monomial basis. Žd. Every operator from l p 1 to L my 1 Ž l p 2 = ??? = l p m . is compact. To see Žd. « Ža., note that if 1 p1
q ??? q
1 pm
G1
the operator T defined by T Ž x1 . Ž x 2 , . . . , x m . s
`
Ý
x 1n ??? x nm
ns1
is not compact. As a consequence of Corollary 1 of Section 1, 1rp1 q ??? q1rpm G 1 if and only if L m Ž l p 1 = ??? = l p m . contains a copy of l` . In particular, if p ) m, P m Ž l p . has a monomial basis and is reflexive, while if p F m, P m Ž l p . contains l` . EXAMPLE 2. Reflexive Orlicz spaces. If l M is a reflexive Orlicz space, it has an unconditional shrinking basis w19x and l Ž l M . s a M , uŽ l M . s bM , the upper and lower Boyd indexes. Thus we have Ži. If 1ra M q ??? q1ra M - 1, then L m Ž l M 1 = ??? = l M m . has 1 m monomial basis and is reflexive. Žii. If 1rbM q ??? q1rbM ) 1, then L m Ž l M 1 = ??? = l M m . contains 1 m ` l .
564
DIMANT AND ZALDUENDO
For polynomials, if a M ) m, P m Ž l M . has monomial basis and is reflexive, while if bM - m, P m Ž l M . contains l` . EXAMPLE 3. c 0 and Tsirelson space. In these cases l Ž c 0 . s uŽ c 0 . s `, l Ž l 1 . s `, uŽ l 1 . s 1 and l ŽT . s uŽT . s 1, l ŽT U . s uŽT U . s ` Žwe denote the original Tsirelson space by T *.. Since c 0 , T, and T U have shrinking bases and T is reflexive Žand its basis is unconditional., we have Ži. Žii.
L m Ž c 0 = ??? = c 0 . has monomial basis for all m, L m ŽT U = ??? = T U . has monomial basis and is reflexive for all
Žiii.
L m ŽT = ??? = T . contains a copy of l` for all m G 2,
m, and analogous results hold for the corresponding spaces of homogeneous polynomials. EXAMPLE 4. L p w0, 1x for 1 - p - `. Here l Ž L p w0, 1x. s min 2, p4 , and Ž u L p w0, 1x. s max 2, p4 . For m s 2, L m Ž L p 1 w0, 1x = ??? = L p m w0, 1x. never verifies the inequalities of Theorems 2 and 3. If m ) 2, the inequality of Theorem 3 may or may not be verified. In any case, it is easily seen that L m Ž L p 1 w0, 1x = ??? = L p m w0, 1x. always contains a copy of l` , for l 2 is isomorphic to a complemented subspace of L p w0, 1x. In the same manner, the space of homogeneous polynomials P m Ž L p w0, 1x. Ž m ) 1. contains a copy of l`. EXAMPLE 5. An application to tensor products. If E1 , . . . , Em are reflexive Banach spaces with a basis, we have already mentioned ŽCorollary 2 or Section 1. as a consequence of a result of Holub w17x, that L m Ž E1 = ??? = Em . has a basis if and only if it is reflexive. In that case we know from Theorem 1 that every linear operator from E1 to L my 1 Ž E2 = ??? = Em . is compact. By transposing we then obtain that every linear operator ˆ ??? m ˆ Em to E1U is compact. If we relate this to the from E2 m Gonzalo]Jaramillo indexes we have the following result which generalizes Pitt’s theorem w21x and a result of Pelczynski w20x. PROPOSITION. Let E1 , . . . , Em be reflexi¨ e Banach spaces with bases and such that for some j, 1 l Ž E1 .
q ??? q
ˆ1 l Ž Ej .
q ??? q
1 l Ž Em .
-
1 u Ž EUj .
.
ˆ ??? m ˆ Em to EUj is compact Ž here the Then e¨ ery linear operator from E1 m tensor products exclude Ej .. In the case of l p spaces the converse is also true. Thus we have the following result which was communicated to us by Gamelin w12x. We have since learned that it has also been proved by Alencar and Floret w3x.
BASES IN SPACES OF MULTILINEAR FORMS
565
ˆ ??? m ˆ l pm, l r . THEOREM ŽAlencar]FloretrGamelin.. E¨ ery T g L Ž l p 1 m is compact if and only if 1rp1 q ??? q1rpm - 1rr. We end with some problems which we believe are still open: 1. Is it possible for L m Ž E1 = ??? = Em . to have neither monomial basis nor a subspace isomorphic to l`? Ž E1 , . . . , Em with shrinking but non-unconditional bases.. 2. We have observed that if Bi1 , . . . , i m 4 is a Schauder basis of L m Ž E ., then Pi1 , . . . , i m 4 is a Schauder basis of P m Ž E .. Is the reciprocal true? 3. In the construction of monomial bases the ‘‘square ordering’’ seems all-important. Can a monomial basis be unconditional? 4. For m ) 1 and E infinite dimensional, can P m Ž E . and P m Ž EU . both be reflexive? Consideration of the Gonzalo]Jaramillo indexes seems to suggest a negative answer, but perhaps some stronger hypothesis is required.
REFERENCES 1. R. Alencar, R. Aron, and S. Dineen, A reflexive space of holomorphic functions in infinitely many variables, Proc. Amer. Math. Soc. 90, No. 3 Ž1984., 407]411. 2. R. Alencar, On reflexivity and basis for P Žm E ., Proc. Roy. Irish. Acad. Sect. A 85, No. 2 Ž1985., 131]138. 3. R. Alencar and K. Floret, Weak-strong continuity of multilinear mappings and the Pelczynski]Pitt theorem, in ‘‘39 Seminario Brasileiro de Analise, 1993,’’ pp. 247]265. ´ 4. R. Aron, B. Cole, and T. Gamelin, Weak-star continuous analytic functions, preprint. 5. R. Aron and S. Dineen, Q-reflexive Banach spaces, preprint. 6. R. Aron, C. Herves, ´ and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52 Ž1983., 189]204. 7. S. Dineen, A Dvoretzky theorem for polynomials, preprint. 8. J. Diestel and T. J. Morrison, The Radon]Nikodym property for the space of operators, I, Math. Nachr. 92 Ž1979., 7]12. 9. J. Farmer, Polynomial reflexivity in Banach spaces, Israel J. Math. 87 Ž1994., 257]273. 10. J. Farmer and W. Johnson, Polynomial Schur and polynomial Dunford]Pettis properties, Contemporary Math. 144 Ž1993., 95]105. 11. B. R. Gelbaum and J. Gil de Lamadrid, Bases of tensor products of Banach spaces, Pacific J. Math. 11 Ž1961., 1281]1286. 12. T. W. Gamelin, Analytic functions on Banach spaces, in ‘‘Complex Function Theory’’ ŽGauthier and Sabidussi, Eds.., Kluwer Academic, Amsterdam, to appear. 13. M. Gonzalez and J. Gutierrez, Unconditionally converging polynomials on Banach ´ ´ spaces, preprint. 14. R. Gonzalo and J. A. Jaramillo, Compact polynomials between Banach spaces, preprint. 15. A. Grothendieck, Sur certaines classes de suites dans les espaces de Banach et le theoreme ` de Dvoretzky Rogers, Bol. Soc. Math. Sao ˜ Paulo 8 Ž1956., 81]110. 16. J. R. Holub, Tensor product bases and tensor diagonals, Trans. Amer. Math. Soc. 151 Ž1970., 563]579.
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17. J. R. Holub, Hilbertian operators and reflexive tensor products, Pacific J. Math. 36, No. 1 Ž1971., 185]194. 18. J. R. Holub, Reflexivity of L Ž X, Y ., Proc. Amer. Math. Soc. 39 Ž1973., 175]177. 19. J. Lindenstrauss and L. Tzafriri, ‘‘Classical Banach Spaces I,’’ Springer-Verlag, New YorkrBerlin, 1977. 20. A. Pelczynski, A property of multilinear operations, Studia Math. 16 Ž1957]58., 173]182. 21. H. R. Pitt, A note on bilinear forms, J. London Math. Soc. 11 Ž1936., 174]180. 22. R. Ryan, ‘‘Applications of Topological Tensor Products to Infinite-Dimensional Holomorphy,’’ Ph.D. thesis, Trinity College, Dublin, 1980. 23. R. Ryan, The Dunford]Pettis properties and projective tensor products, Bull. Polish Acad. Sci. 35, No. 11]12 Ž1987., 785]792. 24. I. Zalduendo, An estimate for multilinear forms on l p spaces, Proc. Roy. Ir. Acad. A 93, No. 1 Ž1993., 137]142. 25. M. Zippin, A remark on bases and reflexivity in Banach spaces, Israel J. Math. 6 Ž1968., 74]79.
200, 548]566 Ž1996.
0224
Bases in Spaces of Multilinear Forms over Banach Spaces V. Dimant and I. Zalduendo Departamento de Economıa Uni¨ ersidad de San Andres, ´ y Matematica, ´ ´ CC 1983-Correo Central, (1000) Buenos Aires, Argentina Submitted by Richard M. Aron Received October 13, 1994
INTRODUCTION In this article we study the existence of monomial bases in spaces of multilinear forms over Banach spaces. During the past years it has become increasingly apparent that there are close relationships between the existence of monomial bases in spaces of polynomials, the reflexivity of such spaces, and compactness or weak sequential continuity Žcomplete continuity. of polynomials. Thus, for example, for reflexive Banach spaces with bases E and F, Holub w17x has proved that reflexivity of L 2 Ž E = F . is equivalent to compactness of all linear maps from E to F U . Ryan w23x shows that when E and F have shrinking bases, the same condition Žcompactness of linear operators E ª F U . is equivalent to the existence of a monomial basis in L 2 Ž E = F .. Alencar w2x proved that for reflexive Banach spaces with basis, the existence of monomial basis in the space of homogeneous polynomials is equivalent to reflexivity of this space. Also, the non-existence of monomial basis has been related in some cases w3, 7, 13x to the containment of a copy of l` . We extend a result in w14x by applying the Gonzalo]Jaramillo indexes and the estimates found in w24x concerning multilinear forms, and we use this to give conditions for the existence of monomial basis in L m Ž E1 = ??? = Em ., the space of m-linear forms over a product of Banach spaces. We begin by giving, in Section 1, a number of conditions including compactness and weak sequential continuity, which are equivalent to the existence of monomial bases. We apply these equivalences and the Gonzalo]Jaramillo indexes to the problem of existence of monomial bases in spaces of multilinear forms over spaces with upper or lower p-bounds for 548 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
549
BASES IN SPACES OF MULTILINEAR FORMS
sequences in Section 2, and in spaces of homogeneous polynomials in Section 3. Some examples, applications and open problems are given in Section 4.
1. MULTILINEAR FORMS AND MONOMIAL SCHAUDER BASES Throughout, E1 , . . . , Em will be Banach spaces over the complex field, with Schauder bases e1n4nG 1 , . . . , e nm 4nG 1 , respectively. For each i s 1, . . . , m and for each n g N, e niU will denote the coordinate functional. We will denote by L m Ž E1 = ??? = Em , F . the space of all continuous m-linear mappings from E1 = ??? = Em into F, and by L m Ž E1 = ??? = Em . we mean L m Ž E1 = ??? = Em , C.. We say that A g L m Ž E1 = ??? = Em , F . is compact if A maps the product of unit balls BE1 = ??? = BE m into a relatively compact subset of F, and that A is weakly sequentially continuous if A maps weakly convergent sequences in E1 = ??? = Em into norm convergent sequences in F. The set of all compact m-linear mappings from E1 = ??? = Em into F will be denoted by K m Ž E1 = ??? = Em , F . and the set of all weakly sequentially continuous m-linear mappings will be m Ž denoted by Lwsc E1 = ??? = Em , F .. The notation E1 = ??? = Eˆj = ??? = Em means the Ž m y 1.-product of the spaces E1 , . . . , Em , excluding the space Ej . The notation AŽ x 1 , . . . , ˆ x j , . . . , x m . for an Ž m y 1.-linear form A is interpreted analogously. Let A g L m Ž E1 = ??? = Em ., n g N, and 1 F k F m, we define 5 A 5 kn s sup < A Ž x 1 , . . . , x m . < : 5 x i 5 s 1
½
k for all i and x k g e nk , e nq1 ,...
5
and 5 A 5 n s sup5 A 5 kn : k s 1, . . . , m4 . Also, if T g L Ž Ei , F ., we write 5 T 5 n s sup 5 T Ž x . 5 : 5 x 5 s 1,
½
i x g e ni , e nq1 ,...
5.
For each Ž i1 , . . . , i m . g N m we define Bi1 , . . . , i m g L m Ž E1 = ??? = Em . by Bi1 , . . . , i m s e1i1U ??? e immU . We consider in N m the square ordering, described inductively by Ryan in w22x Žsee also w11x.: for m s 2, Ž1, 1., Ž1, 2., Ž2, 2., Ž2, 1., Ž1, 3., Ž2, 3., Ž3, 3., Ž3, 2., . . . ; and given the ordering s1 , s2 , . . . of N my 1 , the order in N m is Ž s1 , 1., Ž s1 , 2., Ž s2 , 2., Ž s2 , 1., Ž s1 , 3., . . . . Thus the notation Bi , . . . , i 4 means 1 m
550
DIMANT AND ZALDUENDO
the sequence ordered in that way, and ÝŽ i1 , . . . , i m .g N m means a series with this ordering. On the other hand, Ý`i1s1 ??? Ý`i m s1 refers to an iterated sum. In w4x E is said to have the LBP property Žfor Littlewood, Bogdanowicz, and Pelczynski. if certain analytic functions can be uniformly approximated on bounded sets by polynomials of finite type. Accordingly, we say that E1 = ??? = Em has the m-LBP property if each A g L m Ž E1 = ??? = Em . is approximable Žuniformly on the product of unit balls. by finite sums of products of linear functionals. We come now to the main theorem of this section. THEOREM 1. Let E1 , . . . , Em be Banach spaces with Schauder bases e1n4 , . . . , e nm 4 . Then the following are equi¨ alent: Ži. For all A g L m Ž E1 = ??? = Em ., 5 A 5 n ª 0. Žii. For all A g L m Ž E1 = ??? = Em ., A s Ý`i s1 ??? Ý`i s1 1 m 1 AŽ e i1, . . . , e imm . Bi1 , . . . , i m Ž uniformly on the product of unit balls.. Žiii. For each i s 1, . . . , m, e ni 4n G 1 is shrinking and E1 = ??? = Em has the m-LBP property. Živ. Bi , . . . , i 4 is a Schauder basis of L m Ž E1 = ??? = Em .. 1 m Žv. The basis e1n4nG 1 is shrinking, for all C g L my 1 Ž E2 = ??? = Em .5 C 5 n ª 0 and L Ž E1 , L my1 Ž E2 = ??? = Em .. s KŽ E1 , L my1 Ž E2 = ??? = Em ... Žvi. For each i, j s 1, . . . , m, e ni 4nG 1 is shrinking and L my 1 E1 = ??? = Eˆj = ??? = Em , EUj
ž
/
s K my 1 E1 = ??? = Eˆj = ??? = Em , EUj .
ž
/
Žvii. For each i, j s 1, . . . , m, e ni 4nG 1 is shrinking and L my 1 E1 = ??? = Eˆj = ??? = Em , EUj
ž
/
my 1 s Lwsc E1 = ??? = Eˆj = ??? = Em , EUj .
ž
/
Žviii. For each i s 1, . . . , m, e ni 4nG 1 is shrinking and there is a j g my1 Ž 1, . . . , m4 for which L my 1 Ž E1 = ??? = Eˆj = ??? = Em , EUj . s Lwsc E1 U = ??? = Eˆj = ??? = Em , Ej .. Proof. We first see the equivalence between Ži., Žii., and Žiii.. Ži. « Žii. We use induction on m. For m s 1, the coordinate functionals are a basis of EU if the basis of E is shrinking; so this is clear. For
551
BASES IN SPACES OF MULTILINEAR FORMS
m G 2, let Ž x 1, . . . , x m . g E1 = ??? = Em , 5 x i 5 s 1 for all i and let « ) 0. Note that the equality in Žii. is true pointwise: AŽ x 1 , . . . , x m . s A
`
ž
i1s1
`
s
e1i1U Ž x 1 . e 1i1 , . . . ,
Ý
i m s1
e immU Ž x m . e imm
/
`
???
Ý
`
Ý
Ý A Ž e1i , . . . , eim . Bi , . . . , i Ž x 1 , . . . , x m .
i1s1
i m s1
N1
Nm
m
1
m
1
so we have AŽ x , . . . , x . y
Ý
???
???
Ý A Ž e1i , . . . , eim . Bi , . . . , i
m
1
i1s1 N1
F
Ý
i1s1
ž
`
Ý
i 2s1
N1
Ý
Ý
i1s1
m
1
i m s1
m
1
`
Ý
???
???
Ý A Ž e1i , . . . , eim . Bi , . . . , i
ž
` i 2s1
ž
`
m
1
m
1
1
m
Nm
???
i1sN1q1
/
Ý A Ž e1i , . . . , eim . Bi , . . . , i Ž x 1 , . . . , x m .
i m s1
m
1
`
Ý
m
1
` i m s1
i 2s1
/
Ý A Ž e1i , . . . , eim . Bi , . . . , i Ž x 1 , . . . , x m .
i m s1
N2
Ý
m
Ý A Ž e1i , . . . , eim . Bi , . . . , i Ž x 1 , . . . , x m .
`
y
q A
1
Ý
Ý
m
1
Nm
???
i1sN1q1 i 2s1
s
m
1
i m s1
i 2s1
q
m
1
`
N2
y
Ý A Ž e1i , . . . , e im . Bi , . . . , i Ž x 1 , . . . , x m .
i m s1
x 1i1 e 1i1 , x 2 , . . . , x m
/
1
m
.
The second term of the last sum is at most 5 A 5 N1Ž1 q K . Žwhere K is the basis constant for e1n4. so we can choose N1 so that this term is less than «r2. For this N1 we can choose N2 , . . . , Nm so that the first term of the last sum is less than «r2, since if Ži. is valid for m-linear forms then it is valid for Ž m y 1.-linear forms and we can use the inductive hypothesis. Žii. « Žiii. Clearly Žii. implies that E1 = ??? = Em has the m-LBP property. In order to see that e nj 4n G 1 is shrinking, let w i g EiU and
552
DIMANT AND ZALDUENDO
A s w 1 ??? wm g L m Ž E1 = ??? = Em .. Suppose that for each i s 1, . . . , m, j x i g Ei , 5 x i 5 s 1, and x j g e nj , e nq1 , . . . , so we have < AŽ x 1 , . . . , x m . < s
`
Ý
i1s1
s
`
Ý
`
???
Ý
i m s1 `
???
i1s1
w 1 Ž e i1 . ??? wm Ž e i m . Bi1 , . . . , i mŽ x 1 , . . . , x m . `
???
Ý
Ý
i jsn
i m s1
w 1 Ž e i1 . ???
wm Ž e i m . Bi1 , . . . , i mŽ x 1 , . . . , x m . F
`
Ý
???
i1s1
`
`
???
Ý
Ý
i jsn
i m s1
w 1 Ž e i1 . ??? wm Ž e i m . Bi1 , . . . , i m .
Since the last of the above terms tends to 0 as n grows, it follows that 5 A 5 nj s 5 w j 5 n ª 0. Thus e nj 4n G 1 is shrinking. Žiii. « Ži. Let A g L m Ž E1 = . . . = Em .; let « ) 0. By Žiii. there exist j w 1 , . . . , w Nj g EUj such that N
Ý wi1
Ay
??? w im -
is1
« 2
.
In order to prove that 5 A 5 kn ª 0, let M s max5 w ij 5 : i s 1, . . . , N; j s 1, . . . , m4 and choose n 0 g N such that 5 w ik 5 n - «r2 NM my 1 for i s 1, . . . , N and n G n 0 . Suppose that x j g Ej , 5 x j 5 s 1 for j s 1, . . . , m and k x k g e nk , e nq1 , . . . , so we have < AŽ x 1 , . . . , x m . < F
N
ž
Ý wi1
Ay
/
??? w im Ž x 1 , . . . , x m .
is1 N
q
Ý wi1
??? w im Ž x 1 , . . . , x m .
is1
-
« 2
N
Ý < wi1 Ž x 1 . < ??? < wim Ž x m . <
q
is1
« 2 N
qÝ is1
« 2 NM my 1
M my 1 s « .
553
BASES IN SPACES OF MULTILINEAR FORMS
Thus, 5 A 5 kn - « if n G n 0 . Since k is arbitrary, we have shown that 5 A 5 n ª 0, and this completes the equivalence between the three first items. Now we prove Žii. « Živ. « Ži.. Assume that Žii. holds so Bi1 , . . . , i m 4 spans a dense subspace in L m Ž E1 = ??? = Em .. But Bi1 , . . . , i m are the coordinate functionals corresponding to the canonical basis of the projecˆ ??? m ˆ Em Žsee w22x., so they form a basic setive tensor product E1 m Ž . quence. Then iv is valid. Assume, now, that Živ. holds. Let A g L m Ž E1 = ??? = Em . and suppose k that x j g Ej , 5 x j 5 s 1 for j s 1, . . . , m and x k g e nk , e nq1 , . . . . Let m D k s Ž i1 , . . . , i m . g N : i k G n4 . By the square ordering of N m , D k ; D s Ž i1 , . . . , i m . g N m : Ž i1 , . . . , i m . G Ž1, 1, . . . , 1, n.4 . Note also that if Ž i1 , . . . , i m . f D k then Bi , . . . , i Ž x 1, . . . , x m . s 0. Hence, 1 m AŽ x 1 , . . . , x m . s
Ý Ž i 1 , . . . , i m .gN m
s
Ý Ž i 1 , . . . , i m .gD
s
A Ž e1i1 , . . . , e imm . Bi1 , . . . , i mŽ x 1 , . . . , x m .
A Ž e1i1 , . . . , e imm . Bi1 , . . . , i mŽ x 1 , . . . , x m .
Ý Ž i 1 , . . . , i m .G Ž1, 1, . . . , 1, n .
A Ž e1i1 , . . . , e imm . Bi1 , . . . , i mŽ x 1 , . . . , x m . .
Therefore, < AŽ x 1 , . . . , x m . < F
Ý Ž i 1 , . . . , i m .G Ž1, 1, . . . , 1, n .
A Ž e1i1 , . . . , e imm . Bi1 , . . . , i m ª 0.
We conclude that 5 A 5 kn ª 0 for each k so 5 A 5 n ª 0 and Ži. holds. We remark that it is clear that Ži. implies that all the bases e ni 4nG 1 are shrinking and also 5 C 5 n ª 0 if C g L my 1 Ž E2 = ??? = Em ., so we should prove only the second part of items between Žv. and Žviii.. Let us see that Žv. holds. Let T g L Ž E1 , L my 1 Ž E2 = ??? = Em .. and let AT be the associated m-linear mapping in L m Ž E1 = ??? = Em .. By Žii., AT s
`
Ý
???
i1s1
`
Ý
i m s1
AT Ž e1i1 , . . . , e imm . Bi1 , . . . , i m ,
so we have T Ž x1 . s
`
Ý
i1s1
???
`
Ý T Ž e1i .Ž ei2 , . . . , eim . e1i U Ž x 1 . ei2U
i m s1
1
2
m
1
2
??? e immU
uniformly for x g BE1. Thus T is a limit of finite rank operators and, therefore, compact. 1
554
DIMANT AND ZALDUENDO
We now prove that Žv. « Ži.. Let A g L m Ž E1 = ??? = Em . and let T be the associated linear mapping in L Ž E1 , L my 1 Ž E2 = ??? = Em ... Note that for all n g N,
a n s sup 5 A 5 kn : k s 2, . . . , m4 s sup 5 T Ž x 1 . 5 n : 5 x 1 5 s 1 4 so we will prove that a n ª 0 and that 5 A 5 1n ª 0. Let « ) 0. Since T is compact, there exist x 11, . . . , x 1N in the unit ball of E1 such that T Ž x 1 g E1 : 5 x 1 5 s 14. ; D 1 F jF N B ŽTx 1j , «r2.. By Žv., for each x 1j there is an n 0 Ž x 1j . g N such that if n G n 0 Ž x 1j ., 5 Tx 1j 5 n - «r2. Then for n G n 0 s max n 0 Ž x 1j .4 we have that 5 T Ž x 1 .5 n - « for all x 1 with unit norm. Thus, a n ª 0. It remains to prove that 5 A 5 1n ª 0. For this, note that 5 A 5 1n s 5 T 5 n . Let « ) 0 and take x 1n g E1 such that 5 x 1n 5 s 1, x 1n g w e1n , e1nq1 , . . . x, and 5 T Ž x 1n .5 ) 5 T 5 n y « for each n. Since the basis of E1 is shrinking, we have that x 1n4 converges weakly to 0, so by the compactness of T, T Ž x 1n . ª 0. Therefore, 5 T 5 n ª 0. The proof of Žii. « Žvi. « Ži. is similar to the proof of Žii. « Žv. « Ži.. Finally, we prove Žii. « Žvii. « Žviii. « Ži.. Suppose that Žii. holds. Let F g L my 1 Ž E1 = ??? = Eˆj = ??? = Em , EUj ., for arbitrary j s 1, . . . , m and take for each i s 1, . . . , m, i / j, a sequence x ni 4n G 1 converging weakly to x i g Ei . We may suppose that all these sequences and their limits have norm at most 1. By Žii.,
F Ž y1 , . . . , ˆ y j, . . . , y m . s
`
Ý
`
???
Ý
i1s1
i m s1
F e 1i1 , . . . , ˆ e ijj , . . . , e imm
ž
j ij
/že /e
1U i1
Ž y 1 . ???
e immU Ž y m . e ijjU uniformly in BE1 = ??? = BˆE j = ??? = BE m . Let « ) 0 and let N1 , . . . , Nm such that N1
FŽ y , . . . , ˆ y ,..., y j
1
m
.y Ý
???
i1s1
Nm
Ý
i ms1
F e1i1 , . . . , ˆ e ijj , . . . , e imm
ž
j ij
/že /e
1U i1
Ž y 1 . ???
e immU Ž y m . e ijjU -
« 3
555
BASES IN SPACES OF MULTILINEAR FORMS
w
for all Ž y 1 , . . . , ˆ y j, . . . , y m . g BE1 = ??? = BˆE j = ??? = BE m . Since x ni ª x i, we can choose n 0 g N such that for all n G n 0 N1
Nm
Ý
???
i1s1
Ý
i m s1
N1
y
Ý
ž
j ij
/že /e
1U i1
Ž x 1n . ??? eimU Ž x nm . eijU
Nm
???
i1s1
-
F e1i1 , . . . , ˆ e ijj , . . . , e imm
Ý
i m s1
« 3
F e 1i1 , . . . , ˆ e ijj , . . . , e imm
ž
j ij
/že /e
m
1U i1
j
Ž x 1 . ??? e immU Ž x m . e ijjU
.
So we have that 5 F Ž x 1n , . . . , x nm . y F Ž x 1, . . . , x m .5 - « for n G n 0 , and F is weakly sequentially continuous. It is clear that Žvii. « Žviii.. We now prove that Žviii. « Ži.. Let A g mŽ L E1 = ??? = Em . and let F be the corresponding map in L my 1 Ž E1 = ??? = Eˆj = ??? = Em , EUj .. Suppose that for some k, 5 A 5 nk ¢ 0, and take « ) 0 and norm-one sequences x ni 4n g Ei for i s 1, . . . , m, such that w x nk ª 0 and < AŽ x 1n , . . . , x nm .< ) « . We may assume that x ni 4n G 1 is weakly Cauchy Žfor all i / k ., because in a Banach space with a shrinking basis e n4 , every bounded sequence x n4 has a weakly Cauchy subsequence. Indeed, eUn Ž x k .4k G 1 is Cauchy for all n, so the result is a consequence of w6, Lemma 2.7x. We must consider two cases to obtain a contradiction: Ža. k / j. Since F is weakly sequentially continuous we may use w6, Lemma 2.4x Žmodified for m-linear forms from a product of different spaces. to see that 5 F Ž x 1n , . . . , ˆ x nj , . . . , x nk , . . . , x nm . 5 ª 0. Therefore < A Ž x 1n , . . . , x nm . < s < F Ž x 1n , . . . , ˆ x nj , . . . , x nk , . . . , x nm . Ž x nj . < F 5 F Ž x 1n , . . . , ˆ x nj , . . . , x nk , . . . , x nm . 5 ª 0 and this is a contradiction. Žb. k s j. By w6, Corollary 2.5x, F maps weakly Cauchy sequences into norm Cauchy sequences. Thus F Ž x 1n , . . . , ˆ x nk , . . . , x nm . is a norm Cauchy
556
DIMANT AND ZALDUENDO
sequence in EkU , so it converges to z g EkU , and 5 z 5 n ª 0 because the basis of Ek is shrinking. Hence, < A Ž x 1n , . . . , x nm . < s < F Ž x 1n , . . . , ˆ x nk , . . . , x nm . Ž x nk . < F 5 F Ž x 1n , . . . , ˆ x nk , . . . , x nm . 5 n F 5 F Ž x 1n , . . . , ˆ x nk , . . . , x nm . y z 5 n q 5 z 5 n F 5 F Ž x 1n , . . . , ˆ x nk , . . . , x nm . y z 5 q 5 z 5 n ª 0. Again, we have a contradiction, and this completes the proof. We end this section with two corollaries and a few remarks. COROLLARY 1. If the Banach spaces E1 , . . . , Em all ha¨ e unconditional shrinking bases, then either Ža. Žb.
monomials form a basis of L m Ž E1 = ??? = Em ., or L m Ž E1 = ??? = Em . contains a copy of l` .
Proof. If Ža. does not hold, by Žv. of Theorem 1 either there is a non-compact T g L Ž E1 , L my 1 Ž E2 = ??? = Em .., or a C g L my 1 Ž E2 = ??? = Em . with 5 C 5 n not tending to zero. In the first case, L Ž E1 , L my 1 Ž E2 = ??? = Em .. contains l` by a result of Diestel and Morrison w8, Theorem 2x. In the second, by Ži. of Theorem 1 monomials do not form a Schauder basis of L my 1 Ž E2 = ??? = Em .; the result follows by induction. COROLLARY 2. If E1 , . . . , Em are reflexi¨ e Banach spaces with Schauder bases, then L m Ž E1 = ??? = Em . is reflexi¨ e if and only if it has monomial basis. Proof. Holub w17x proved that if E and F are reflexive Banach spaces with bases, then L Ž E, F . is reflexive if and only if L Ž E, F . s KŽ E, F .. Holub’s result and Theorem 1 yield the proof. Remark. Give E1 , . . . , Em such that L m Ž E1 = ??? = Em . has a monomial basis, one may wonder if it has a non-monomial basis as well. Another result of Holub w17x is that if X and Y are reflexive Banach spaces with basis and their projective tensor product is not reflexive, then it has a non-tensor product basis. If any of the spaces E1 , . . . , Em is not reflexive, neither is L m Ž E1 = ??? = Em . for this contains copies of EiX . By a result of Zippin w25x not all bases of L m Ž E1 = ??? = Em . can be boundedly complete; thus there is a non-monomial basis Žall monomial bases, being dual, are boundedly complete..
BASES IN SPACES OF MULTILINEAR FORMS
557
2. m-LINEAR FORMS OVER SPACES WITH UPPER OR LOWER p-BOUNDS In this section we apply Theorem 1 to spaces with bounds for sequences. We begin by recalling the necessary definitions of lower and upper indexes of a Banach space E, introduced by Gonzalo and Jaramillo in w14x. A sequence x i 4 in E is said to have an upper p-estimate Ž1 F p F `. if there is a positive constant C such that for every n-tuple of scalars a1 , . . . , a n , n
Ý
1rp
n
ai x i F C
is1
ž
Ý < ai < p is1
/
This is equivalent w15x to the sequence being weakly q-summable where 1rp q 1rq s 1. A Banach space is said to have property S p if every weakly null semi-normalized basic sequence in E has a subsequence with an upper p-estimate. Clearly property S p implies property Sr for all r F p. The lower index of E is then defined as l Ž E . s sup p G 1 : E has property S p 4 . In an analogous manner, but using lower q-estimates one obtains the definition of property Tq Žwith Tq implying Tr for all larger r ., and u Ž E . s inf q G 1 : E has property Tq 4 . We refer to w14x for more on the Gonzalo]Jaramillo indexes, but we wish to mention that for 1 - p - `, l Ž l p . s uŽ l p . s p. It is well known that monomials form a Schauder basis in the space of k-homogeneous polynomials over l p if k - p, and that when k G p, P k Ž l p . contains a copy of l` . Theorems 2 and 3 generalize these facts. First, we require a lemma. The proof is modeled on the proof of Theorem 2.4 of w14x. LEMMA.
Let E1 , . . . , EN , and F be Banach spaces such that 1 l Ž E1 .
q ??? q
1 l Ž EN .
-
1 uŽ F .
.
Then e¨ ery A g L N Ž E1 = ??? = EN , F . is weakly sequentially continuous.
558
DIMANT AND ZALDUENDO
Proof. Choose p1 , . . . , pN , q so that Ei has property S p i , F has property Tq , and 1 p1
q ??? q
1 pN
-
1 q
.
Take x ni 4 ; Ei converging weakly but not in norm to 0 Žif convergence is also strong for some i, there is nothing to prove.. We may extract semi-normalized basic subsequences x fi i Ž k .4 with upper pi-estimate. Thus for every i s 1, . . . , N, the map l p i ª Ei mapping e k to x fi i Ž k . is continuous. Given A g L N Ž E1 = ??? = EN , F . define F g L N Ž l p 1 = ??? = l p N , F . by F Ž e i1 , . . . , e i N . s A Ž x 1f 1 Ž i1 . , . . . , x fNN Ž i N . . . We shall see that y k s AŽ x 1f 1 Ž k . , . . . , x fNN Ž k . . is a sequence with an upper Ž1rp1 q ??? q1rpN .y1 -estimate. For this, it will be sufficient w15x to verify that Ž y k . is weakly s-summable, where ss 1y
ž
1 p1
q ??? q
1 pN
y1
/
.
But given any g g F U , g ( F g L N Ž l p 1 = ??? = l p N ., and 1 p1
q ??? q
1 pN
-
1 q
F 1,
so by w24x, we obtain that g Ž y k . is s-summable. Since q - Ž1rp1 q ??? q1rpN .y1 , no subsequence of Ž y k . can have a lower q-estimate. But if some subsequence of Ž y k . were bounded from below in norm, since F has property Tq we could extract a subsequence with a lower q-estimate. Thus y k tend to 0 in norm. Since this can be done with any subsequence of x ni 4 , we obtain that A is weakly sequentially continuous at 0. Suppose now that x ni 4 tends weakly to x i. We will see by induction on N that AŽ x 1n , . . . , x nN . converges to AŽ x 1, . . . , x N . in norm. For N s 1 this is a consequence of the weak sequential continuity at 0 and the linearity of A. Note that our hypothesis Ž1rp1 q ??? q1rpN - 1rq . and the first part of the proof allows the weak sequential continuity of the k-linear functions Ž k - N . obtained by fixing some variables in A. Now AŽ x 1n y x 1, . . . , x nN y x N . may be expanded to a sum of 2 N terms Ž2 Ny 1 added and 2 Ny1
559
BASES IN SPACES OF MULTILINEAR FORMS
subtracted .. By adding the subtracting AŽ x 1, . . . , x N . 2 Ny 2 times we obtain A Ž x 1n y x 1 , . . . , x nN y x N . s A Ž x 1n , . . . , x nN . y A Ž x 1 , . . . , x N . q Ý Ž A Ž . . . , x i , . . . , x nj , . . . . y A Ž x 1 , . . . , x N . . 1
q Ý Ž A Ž x 1 , . . . , x N . y A Ž . . . , x i , . . . , x nj , . . . . . , 2
where if N is odd we have 2 Ny 2 terms in Ý1 and 2 Ny 2 in Ý2 , whereas if N is even we have 2 Ny 2 y 1 in Ý1 and 2 Ny 2 q 1 in Ý2 . In any case, 5 A Ž x 1n , . . . , x nN . y A Ž x 1 , . . . , x N . 5 F 5 A Ž x 1n y x 1 , . . . , x nN y x N . 5 q Ý 5 A Ž . . . , x i , . . . , x nj , . . . . y A Ž x 1 , . . . , x N . 5 which tends to 0 by our inductive hypothesis and the weak sequential continuity of A at 0. THEOREM 2. Let E1 , . . . , Em be Banach spaces with shrinking bases e ni 4 and such that for some j 1 l Ž E1 .
q ??? q
ˆ1 l Ž Ej .
q ??? q
1 l Ž Em .
-
1 u Ž EUj .
.
Then the monomials Bi1 , . . . , i m s e1i1U ??? e immU form a Schauder basis of L m Ž E1 = ??? = Em .. Proof. Just apply the lemma and equivalences Živ. and Žviii. of Theorem 1. COROLLARY. Let E1 , . . . , Em be reflexi¨ e Banach spaces with Schauder bases and such that for some j 1 l Ž E1 .
q ??? q
ˆ1 l Ž Ej .
q ??? q
Then L m Ž E1 = ??? = Em . is reflexi¨ e.
1 l Ž Em .
-
1 u Ž EUj .
.
560
DIMANT AND ZALDUENDO
Proof. The proof is as that of Corollary 1 of Section 1. THEOREM 3. Let E1 , . . . , Em be Banach spaces with unconditional shrinking bases, and such that for all j 1
q ??? q
u Ž E1 .
ˆ1
q ??? q
u Ž Ej .
1
)
u Ž Em .
1 l Ž EUj .
.
Then L m Ž E1 = ??? = Em . contains a copy of l` . Proof. None of the Ei ’s contain a copy of l 1 , for all have shrinking bases. Thus they either contain copies of c 0 or are reflexive. Those which contain copies of c 0 have an infinite upper Gonzalo]Jaramillo index. Therefore the inequality in our hypothesis forces at least one of the Ei ’s to be reflexive. Say Em is reflexive. We have 1
q ??? q
u Ž E1 .
1 u Ž Emy1 .
)
1 U l Ž Em .
.
U Choose p, q1 , . . . , qmy1 such that Ei has property Tq i for i F m y 1, Em has property S p , and 1rq1 q ??? q1rqmy1 s 1rp. Given the unconditional shrinking bases Ž e ij . i of Ej , by normalizing when necessary, we obtain weakly null seminormalized basic sequences from which we extract subsequences Žwith the same subindexes., such that Ž e nj i . i has a lower q j-estimate. Since Em is reflexive, Ž e imU . i is shrinking; thus in the same manner as above we extract a subsequence Ž e nmiU . i with an upper p-estimate. For each i g N define Ti g L m Ž E1 = ??? = Em . by setting
Ti Ž x 1 , . . . , x m . s x 1n i ??? x nmi . In other words, Ti s e1nUi ??? e nmiU . Given scalars a1 , . . . , a n , n
Ý aiTi is1
n
s sup j5
Ý ai x 1n
5 x F1 is1
??? x nmi
i
n
s sup j5
5 x F1
Ý ai x 1n
??? x nmy1 e nmiU i
i
is1 1rp
n
F D sup 5 x j 5F1
ž
F sup < a i < D sup i
5 x j 5F1
/ 1rp
n
ž
/
561
BASES IN SPACES OF MULTILINEAR FORMS
F 5 a 5 ` D sup
5 x j 5F1
1rq 1
n
ž
Ý < x 1n < q1 i
is1
/
1rq my 1
n
???
ž
Ý < x nmy1 < q my 1 i
is1
n
F C1 ??? Cmy1 D 5 a 5 ` sup
Ý
j5
5 x F1
/
n
x 1n i e 1n i ???
is1
Ý x nmy1 enmy1 i
i
is1
F KC1 ??? Cmy1 D 5 a 5 ` , where K is the maximum unconditional basis constant of the bases of the Ei ’s. Thus n
Ý aiTi
F C 5 a5 ` .
is1
On the other hand, for each k g N, < ak < s
n
ž
/
Ý aiTi Ž e1n , . . . , enm . is1 n
F
Ý aiTi
k
k
5 e 1n 5 ??? 5 e nm 5 , k
k
is1
so 5 a 5 ` s sup < a k < F sup 5 e1n 5 ??? 5 e nm 5 k k k
k
n
Ý ai Ti
.
is1
Therefore there are constants C, CX such that CX 5 a5 ` F
n
Ý aiTi
F C 5 a5 ` ,
is1
and the closed subspace of L m Ž E1 = ??? = Em . spanned by the Ti ’s is isomorphic to c0 . Being a dual space, L m Ž E1 = ??? = Em . must also contain a copy of l`. Note that the proof of the theorem remains valid as long as one has the inequality for a given j, with a reflexive Ej . Also, it is not necessary for the basis of this Ej to be unconditional. An example in which the inequality of Theorem 3 is verified by some but not all j’s is provided by L 2 Ž c 0 = l 2 .. This space has a basis by Theorem 2.
562
DIMANT AND ZALDUENDO
3. POLYNOMIALS OVER SPACES WITH UPPER OR LOWER p-BOUNDS Here we apply the results of the previous sections to spaces of polynomials over a Banach space E. The set D s Ž i1 , . . . , i m . g N m : i1 G i 2 G ??? G i m 4 is given the ordering induced by the square ordering of N m . For each Ž i1 , . . . , i m . g D we will note by Pi1 , . . . , i m the m-homogeneous polynomial Pi1 , . . . , i mŽ x . s Bi1 , . . . , i mŽ x, . . . , x . and by the notation Pi1 , . . . , i m 4 we mean the sequence Pi1 , . . . , i m 4Ž i1 , . . . , i m .g D ordered as above. We then have the following proposition. PROPOSITION. Let E be a Banach space with Schauder basis such that Bi , . . . , i 4 is a basis of L m Ž E .. Then Pi , . . . , i 4 is a basis of P m Ž E .. 1 m 1 m Proof. Clearly, if Bi1 , . . . , i m 4 is a basis of L m Ž E . then Pi1 , . . . , i m 4 spans a dense subspace in P m Ž E .. Since Pi1 , . . . , i m are the coordinate functionals ˆ m, s E Žthe m-fold symmetric projective tensor associated with the basis of m product., they form a basic sequence. Consequently, Pi1 , . . . , i m 4 is a basis of P m Ž E .. As simple consequences of this proposition, we have the following analogues of Theorem 2 and its corollary. THEOREM 4.
Let E be a Banach space with shrinking basis such that
Ž m y 1 . u Ž EU . - l Ž E . . Then Pi1 , . . . , i m 4 is a basis of P m Ž E .. COROLLARY.
Let E be a reflexi¨ e Banach space with a basis such that
Ž m y 1 . u Ž EU . - l Ž E . . Then P m Ž E . is reflexi¨ e. Theorem 3 can also be proved for spaces of homogeneous polynomials. As we have mentioned above, when m ) p the space of m-homogeneous polynomials P m Ž l p . contains a copy of l` Žsee, for example, w1, 7, 13x.. The following theorem generalizes that result. THEOREM 5. such that
Let E be a Banach space with unconditional shrinking basis
Ž m y 1 . l Ž EU . ) u Ž E . . Then P m Ž E . contains a copy of l` .
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563
Proof. If we put E1 s E2 s ??? s Em s E and q1 s ??? s qmy1 s q in the proof of Theorem 3 we obtain Ti s eUn i ??? eUn i g Lsm Ž E ., for the Ti ’s are symmetric. Thus c 0 ; w Ti x ; Lsm Ž E . ( P m Ž E .. Being a dual, P m Ž E . contains l`.
4. EXAMPLES AND APPLICATIONS In this section we apply our results to several concrete Banach space examples. In several cases, and particularly for the polynomial examples, these results are known, with different proofs. One should consult, for example, w5, 2, 3, 7, 12, 13x. The Gonzalo]Jaramillo indexes used can be found in w14x. EXAMPLE 1. l p spaces Ž1 - p - `.. In this case one has l Ž l p . s uŽ l p . s p. Thus the following conditions are all equivalent: Ža. 1rp1 q ??? q1rpm - 1. Žb. L m Ž l p 1 = ??? = l p m . is reflexive. Žc. L m Ž l p 1 = ??? = l p m . has a monomial basis. Žd. Every operator from l p 1 to L my 1 Ž l p 2 = ??? = l p m . is compact. To see Žd. « Ža., note that if 1 p1
q ??? q
1 pm
G1
the operator T defined by T Ž x1 . Ž x 2 , . . . , x m . s
`
Ý
x 1n ??? x nm
ns1
is not compact. As a consequence of Corollary 1 of Section 1, 1rp1 q ??? q1rpm G 1 if and only if L m Ž l p 1 = ??? = l p m . contains a copy of l` . In particular, if p ) m, P m Ž l p . has a monomial basis and is reflexive, while if p F m, P m Ž l p . contains l` . EXAMPLE 2. Reflexive Orlicz spaces. If l M is a reflexive Orlicz space, it has an unconditional shrinking basis w19x and l Ž l M . s a M , uŽ l M . s bM , the upper and lower Boyd indexes. Thus we have Ži. If 1ra M q ??? q1ra M - 1, then L m Ž l M 1 = ??? = l M m . has 1 m monomial basis and is reflexive. Žii. If 1rbM q ??? q1rbM ) 1, then L m Ž l M 1 = ??? = l M m . contains 1 m ` l .
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For polynomials, if a M ) m, P m Ž l M . has monomial basis and is reflexive, while if bM - m, P m Ž l M . contains l` . EXAMPLE 3. c 0 and Tsirelson space. In these cases l Ž c 0 . s uŽ c 0 . s `, l Ž l 1 . s `, uŽ l 1 . s 1 and l ŽT . s uŽT . s 1, l ŽT U . s uŽT U . s ` Žwe denote the original Tsirelson space by T *.. Since c 0 , T, and T U have shrinking bases and T is reflexive Žand its basis is unconditional., we have Ži. Žii.
L m Ž c 0 = ??? = c 0 . has monomial basis for all m, L m ŽT U = ??? = T U . has monomial basis and is reflexive for all
Žiii.
L m ŽT = ??? = T . contains a copy of l` for all m G 2,
m, and analogous results hold for the corresponding spaces of homogeneous polynomials. EXAMPLE 4. L p w0, 1x for 1 - p - `. Here l Ž L p w0, 1x. s min 2, p4 , and Ž u L p w0, 1x. s max 2, p4 . For m s 2, L m Ž L p 1 w0, 1x = ??? = L p m w0, 1x. never verifies the inequalities of Theorems 2 and 3. If m ) 2, the inequality of Theorem 3 may or may not be verified. In any case, it is easily seen that L m Ž L p 1 w0, 1x = ??? = L p m w0, 1x. always contains a copy of l` , for l 2 is isomorphic to a complemented subspace of L p w0, 1x. In the same manner, the space of homogeneous polynomials P m Ž L p w0, 1x. Ž m ) 1. contains a copy of l`. EXAMPLE 5. An application to tensor products. If E1 , . . . , Em are reflexive Banach spaces with a basis, we have already mentioned ŽCorollary 2 or Section 1. as a consequence of a result of Holub w17x, that L m Ž E1 = ??? = Em . has a basis if and only if it is reflexive. In that case we know from Theorem 1 that every linear operator from E1 to L my 1 Ž E2 = ??? = Em . is compact. By transposing we then obtain that every linear operator ˆ ??? m ˆ Em to E1U is compact. If we relate this to the from E2 m Gonzalo]Jaramillo indexes we have the following result which generalizes Pitt’s theorem w21x and a result of Pelczynski w20x. PROPOSITION. Let E1 , . . . , Em be reflexi¨ e Banach spaces with bases and such that for some j, 1 l Ž E1 .
q ??? q
ˆ1 l Ž Ej .
q ??? q
1 l Ž Em .
-
1 u Ž EUj .
.
ˆ ??? m ˆ Em to EUj is compact Ž here the Then e¨ ery linear operator from E1 m tensor products exclude Ej .. In the case of l p spaces the converse is also true. Thus we have the following result which was communicated to us by Gamelin w12x. We have since learned that it has also been proved by Alencar and Floret w3x.
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ˆ ??? m ˆ l pm, l r . THEOREM ŽAlencar]FloretrGamelin.. E¨ ery T g L Ž l p 1 m is compact if and only if 1rp1 q ??? q1rpm - 1rr. We end with some problems which we believe are still open: 1. Is it possible for L m Ž E1 = ??? = Em . to have neither monomial basis nor a subspace isomorphic to l`? Ž E1 , . . . , Em with shrinking but non-unconditional bases.. 2. We have observed that if Bi1 , . . . , i m 4 is a Schauder basis of L m Ž E ., then Pi1 , . . . , i m 4 is a Schauder basis of P m Ž E .. Is the reciprocal true? 3. In the construction of monomial bases the ‘‘square ordering’’ seems all-important. Can a monomial basis be unconditional? 4. For m ) 1 and E infinite dimensional, can P m Ž E . and P m Ž EU . both be reflexive? Consideration of the Gonzalo]Jaramillo indexes seems to suggest a negative answer, but perhaps some stronger hypothesis is required.
REFERENCES 1. R. Alencar, R. Aron, and S. Dineen, A reflexive space of holomorphic functions in infinitely many variables, Proc. Amer. Math. Soc. 90, No. 3 Ž1984., 407]411. 2. R. Alencar, On reflexivity and basis for P Žm E ., Proc. Roy. Irish. Acad. Sect. A 85, No. 2 Ž1985., 131]138. 3. R. Alencar and K. Floret, Weak-strong continuity of multilinear mappings and the Pelczynski]Pitt theorem, in ‘‘39 Seminario Brasileiro de Analise, 1993,’’ pp. 247]265. ´ 4. R. Aron, B. Cole, and T. Gamelin, Weak-star continuous analytic functions, preprint. 5. R. Aron and S. Dineen, Q-reflexive Banach spaces, preprint. 6. R. Aron, C. Herves, ´ and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52 Ž1983., 189]204. 7. S. Dineen, A Dvoretzky theorem for polynomials, preprint. 8. J. Diestel and T. J. Morrison, The Radon]Nikodym property for the space of operators, I, Math. Nachr. 92 Ž1979., 7]12. 9. J. Farmer, Polynomial reflexivity in Banach spaces, Israel J. Math. 87 Ž1994., 257]273. 10. J. Farmer and W. Johnson, Polynomial Schur and polynomial Dunford]Pettis properties, Contemporary Math. 144 Ž1993., 95]105. 11. B. R. Gelbaum and J. Gil de Lamadrid, Bases of tensor products of Banach spaces, Pacific J. Math. 11 Ž1961., 1281]1286. 12. T. W. Gamelin, Analytic functions on Banach spaces, in ‘‘Complex Function Theory’’ ŽGauthier and Sabidussi, Eds.., Kluwer Academic, Amsterdam, to appear. 13. M. Gonzalez and J. Gutierrez, Unconditionally converging polynomials on Banach ´ ´ spaces, preprint. 14. R. Gonzalo and J. A. Jaramillo, Compact polynomials between Banach spaces, preprint. 15. A. Grothendieck, Sur certaines classes de suites dans les espaces de Banach et le theoreme ` de Dvoretzky Rogers, Bol. Soc. Math. Sao ˜ Paulo 8 Ž1956., 81]110. 16. J. R. Holub, Tensor product bases and tensor diagonals, Trans. Amer. Math. Soc. 151 Ž1970., 563]579.
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17. J. R. Holub, Hilbertian operators and reflexive tensor products, Pacific J. Math. 36, No. 1 Ž1971., 185]194. 18. J. R. Holub, Reflexivity of L Ž X, Y ., Proc. Amer. Math. Soc. 39 Ž1973., 175]177. 19. J. Lindenstrauss and L. Tzafriri, ‘‘Classical Banach Spaces I,’’ Springer-Verlag, New YorkrBerlin, 1977. 20. A. Pelczynski, A property of multilinear operations, Studia Math. 16 Ž1957]58., 173]182. 21. H. R. Pitt, A note on bilinear forms, J. London Math. Soc. 11 Ž1936., 174]180. 22. R. Ryan, ‘‘Applications of Topological Tensor Products to Infinite-Dimensional Holomorphy,’’ Ph.D. thesis, Trinity College, Dublin, 1980. 23. R. Ryan, The Dunford]Pettis properties and projective tensor products, Bull. Polish Acad. Sci. 35, No. 11]12 Ž1987., 785]792. 24. I. Zalduendo, An estimate for multilinear forms on l p spaces, Proc. Roy. Ir. Acad. A 93, No. 1 Ž1993., 137]142. 25. M. Zippin, A remark on bases and reflexivity in Banach spaces, Israel J. Math. 6 Ž1968., 74]79.