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Bases in Spaces of Homogeneous Polynomials on Banach Spaces ✩

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Bases in Spaces of Homogeneous Polynomials on Banach Spaces ✩ View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. Math. Anal. Appl. 332 (2007) 272–278 www.elsevier.com/locate/jmaa Bases in spaces of homogeneous polynomials on Banach spaces ✩ Leonardo Pellegrini Escola de Artes, Ciências e Humanidades, Universidade de São Paulo, CEP 03828-000 São Paulo, Brazil Received 20 October 2005 Available online 7 November 2006 Submitted by R.M. Aron Abstract In this work we present some conditions of equivalence for the existence of a monomial basis in spaces of homogeneous polynomials on Banach spaces. © 2006 Elsevier Inc. All rights reserved. Keywords: Homogeneous polynomials; Basis on Banach spaces 1. Introduction The main purpose of this work is to study the existence of a monomial basis in spaces of homogeneous polynomials on Banach spaces. The study was motivated principally because of its relation with the reflexivity of the homogeneous polynomials spaces. For example, Alencar [1] proved that, for a reflexive Banach space with a Schauder basis, the existence of Schauder basis (in homogeneous polynomials spaces) is equivalent to reflexivity of this space. Let E be a Banach space with a Schauder basis (xi)i∈N. Let us remember that a basis (xi)i∈N is said to be shrinking if for all ϕ ∈ E, ϕ := ϕ| →0, as n →∞. In an equivalent n [xn+i : i∈N] way, a basis is shrinking if the sequence of coordinate functionals is a Schauder basis for E. In [5], V. Dimant and I. Zalduendo proved an analogous relation between a norm and the existence of a monomial basis for spaces of multilinear forms over Banach spaces. However, the ✩ Supported by FAPESP, Brazil, Research Grant 01/04220-8. E-mail address: [email protected]. 0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.09.058 L. Pellegrini / J. Math. Anal. Appl. 332 (2007) 272–278 273 norm-condition is just a sufficient condition for the existence of a monomial basis in spaces of homogeneous polynomials. Here we develop a relation between a norm and the existence of basis in spaces of homoge- neous polynomials. As a consequence we obtain, in Section 3, a characterization of the canonical extension of the homogeneous polynomials in those spaces. We refer to the books of Dineen [6] or Mujica [8] for background information from homogeneous polynomials on Banach spaces. 2. Main results Before stating our results, let us fix some notation and terminology. Let E be a real or complex Banach space. We will denote by P(mE) the space of all scalar valued m-homogeneous poly- m m nomials which are continuous on E. Let us recall that Pf ( E) denotes the subspace of P( E) n m generated by all polynomials of the form P = ϕ with ϕ ∈ E and Pw( E) denotes the closed subspace of P(mE) of all m-homogeneous polynomials weakly continuous on bounded sets. We will consider in Nm the square order defined by induction as follows: • If n = 1 the square order is the usual one. n−1 n • If (s1,s2,...)is the square order of N , we define the square order in N by (s1, 1), (s1, 2), (s2, 2), (s2, 1), (s1, 3), (s2, 3), (s3, 3), (s3, 2), (s3, 1), .... m For each m ∈ N,letDm be the subset of N consisting of all s = (i1,...,im) for which m i1 i2 ··· im.IfE has a Schauder basis (xi)i∈N then an element P ∈ P( E) is called a monomial of degree m if there exists (i1,...,im) ∈ Dm such that P = P ··· := x x ···x , i1i2 im i1 i2 in where (xi)i∈N represents the sequence of coordinate functional. Now, we introduce a definition and we will show that this definition is equivalent to the exis- tence of a monomial basis in homogeneous polynomials spaces. Definition 2.1. Let E be a Banach space with Schauder basis (xi)i∈N. The basis (xi)i∈N is said to be a polynomial-shrinking basis if for all m ∈ N and for all P ∈ P(mE), ∞ := ˇ m−1 −n−−−→∞→ ρn(P ) sup P xi(x)xi,x 0, x 1 i=n+1 where Pˇ denotes the symmetric m-linear map associated to P . The next proposition is an immediate consequence from this definition. Proposition 2.1. Let E be a complex Banach space. If (xi)i∈N is a polynomial-shrinking basis, then (xi)i∈N is a shrinking basis. Proof. In view of above definition, for each ϕ ∈ E,wehavethat ∞ ∞ = → ϕ n sup ϕ xi(x)xi sup ϕ xi(x)xi 0, x 1 i=n+ x 1 i=n+ = ∞ 1 1 x i=n+1 xi (x)xi and the conclusion of the proposition follows from this. 2 274 L. Pellegrini / J. Math. Anal. Appl. 332 (2007) 272–278 Now we give a criterion to recognize a polynomial-shrinking basis. Theorem 2.2. Let E be a complex Banach space with a Schauder basis (xi)i∈N. Then, the fol- lowing statements are equivalent: (i) The basis (xi)i∈N is a polynomial-shrinking basis; (ii) For all m ∈ N, the monomials of degree m are a Schauder basis of P(mE); m m (iii) The basis (xi)i∈N is a shrinking basis and Pf ( E) = P( E), ∀m ∈ N; m m (iv) The basis (xi)i∈N is a shrinking basis and Pw( E) = P( E), ∀m ∈ N. Proof. (i) ⇒ (ii): By [7], the monomials of degree m are the coordinate functionals correspond- m ing to the canonical basis of π,sE, the completion of the projective symmetric tensor product m [ ]= m π,s E. Hence they are a basic sequence. So, it is just necessary to prove that Pi1···im P( E). In order to do this, it is enough to show that, for each P ∈ P(mE), N N ˇ P = lim ··· P(xi ,...,xi )Pi ···i . N→∞ 1 m 1 m i1=1 im=1 If m = 1, by Proposition 2.1 we know that (xi)i∈N is a shrinking basis and so N N = = ϕ lim ϕ(xi)xi lim ϕ(xi)Pi, N→∞ N→∞ i=1 i=1 for each ϕ ∈ E. Let us assume that the property is true for all polynomials of degree less than m, and let P ∈ P(mE) and ε>0. First, we observe that, for each x ∈ E, we have that ∞ m ∞ ∞ P(x)= Pˇ x (x)x = ··· P(xˇ ,...,x )x (x)x (x) ···x (x) i i i1 im i1 i2 im i=1 i1=1 im=1 ∞ ∞ = ··· ˇ P(xi1 ,...,xim )Pi1···im (x). i1=1 im=1 Since (xn) is a polynomial-shrinking basis, there exists N0 ∈ N such that ε ρ (P ) < , N0 2(1 + Cm) where C is a basic constant of (xi)i∈N. For each i1 = 1, 2,...,N0, let us consider the (m − 1)- ˇ homogeneous polynomials Qi = Pxi . By the inductive hypothesis, there exists Ni such that 1 1 1 N N ˇ ε Q − ··· Q (x ,...,x )P ··· < , ∀N N . i1 i1 i2 im i2 im i1 2 xi N0 i2=1 im=1 1 Hence, if N max{N0,Ni : i1 = 1,...,N0}, for each x ∈ BE it is 1 N N ˇ P(x)− ··· P(x ,...,x )P ··· (x) i1 im i1 im = = i1 1 im 1 ∞ ∞ N N ˇ ˇ = ··· P(x ,...,x )P ··· (x) − ··· P(x ,...,x )P ··· (x) i1 im i1 im i1 im i1 im i1=1 im=1 i1=1 im=1 L. Pellegrini / J. Math. Anal. Appl. 332 (2007) 272–278 275 N0 ∞ ∞ ˇ ··· P(x ,...,x )P ··· (x) i1 im i1 im i1=1 i2=1 im=1 N0 N ˇ − ··· P(x ,...,x )P ··· (x) i1 im i1 im i =1 i =1 1 m ∞ ∞ ∞ ˇ + ··· P(x ,...,x )P ··· (x) i1 im i1 im i1=N0+1 i2=1 im=1 N N N ˇ + ··· P(x ,...,x )P ··· (x) i1 im i1 im i1=N0+1 i2=1 im=1 N0 ∞ ∞ ˇ x ··· P(x ,...,x )P ··· (x) i1 i1 im i2 im i1=1 i2=1 im=1 N N ∞ ˇ ˇ − ··· P(x ,...,x )P ··· (x) + P x (x)x ,x,...,x i1 im i2 im i i i2=1 im=1 i=N0+1 N N N + ˇ P xi(x)xi, xi(x)xi,..., xi(x)xi i=N0+1 i=1 i=1 N0 N N ˇ = x · Q (x) − ··· Q (x ,...,x )P ··· (x) i1 i1 i1 i2 im i2 im i =1 i =1 i =1 1 2 m ∞ + ˇ P xi(x)xi,x,...,x i=N0+1 N N N x (x)xi x (x)xi x (x)xi + Pˇ i , i ,..., i Cm C C C i=N0+1 i=1 i=1 N0 ε m x + ρN (P ) + ρN (P )C i1 0 0 2 xi N0 i1=1 1 ε ε < + 1 + Cm = ε. 2 2(1 + Cm) N N ˇ Therefore, P − ··· P(x ,...,x )P ··· <ε, ∀N N . i1=1 im=1 i1 im i1 im 0 ⇒ = = m = (ii) (iii): For m 1, it is Pi xi . Then, (xi)i∈N is a basis of P( E) E and so (xi)i∈N is a shrinking basis. ∈ N m [ ]= m [ ]⊂ For each m ,as(Pi1···im ) is a basis of P( E), then Pi1···im P( E). Since Pi1···im m m m Pf ( E), we conclude that Pf ( E) = P( E).
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