A CONSTRUCTION OF A BASE FOR THE M FOLD TENSOR PRODUCT OF A BRANCH SPACE

Let E be a Banach space with Schauder base (x) Let E denote the n n (1 completion of the m fold tensor product with respect to a reasonable cross norm a. We show that the set {x i x: xi. (Xn)n

A CONSTRUCTION OF A BASE FOR THE M FOLD TENSOR PRODUCT OF A BRANCH SPACE ROHAN HEMASINHA The University of West Florlda Pensacola, Florida 32514 (Received August 3,1987) ABSTRACT.
Let E be a Banach space with Schauder base (x) Let E denote the n n (1 completion of the m fold tensor product with respect to a reasonable cross norm a.We show that the set {x i x: xi. (Xn)n can be enumerated so that for each positive integer k, the first k m ter3mS are precisely all the elements of the form with i l,...,i {l,...,k} and the set so arranged is a Schauder base x i x i m for m E m KEYWORDS AND PHRASES.
m fold tensor product, projective, fnJectlve and reasonable cross norms, Schauder bases.

I. INTRODUCTION.
Given a Banach space E, we will denote by E the m fold algebraic tensor product and by E its completion with respect to reasonable norm .For m > 2 the projective (inJectlve) tensor norm on mE will be denoted by m(%).
If m 2 this norm will be denoted by T(1).
In [I], [4] it is shown that if EI,E 2 are Banach spaces with Schauder bases (Xn)n' (Yn)n then for any reasonable norm , the space E E 2 has a Schauder base (z) with the following properties.(Zn) n is obtained by enumerating the set (Xn)n (Yn)n; (2) the enumeration is such that for any positive integer k, the first k 2 terms are preclsely all elements of the form x, y, In this paper we show that for any positive integer m, the space ,oE has a m Schauder base obtained by enumerating -(Xn) n in such a way that for any positive integer k, the first k m terms are precisely all elements of the form x. (R) x with il, 'ira {l,...,k}.m Our proof is via an induction argument and in a forthcoming paper we utilize this construction to derive some properties of the symmetric tensor algebra of a Banach space.
Note that even for the case m 3, iteration to (E" E) E of the enumerating scheme described in [I] will not yield a base with the above mentioned properties.
We denote the following property (*) of the projective tensor norm.For m > 2 let u be an element of mE It can easily be shown that (*) v (u) where 7(u) is the projective norm of u when u is considered as an element of F E with F E being endowed with the norm Xm-l" We shall make use of this fact subsequent ly.
We now state our theorem.
fn Xn E fn E E' be biorthogonal system such that (Xn) n is THEOREM.Let (x n, m a Schauder base for E. Let e be a reasonable norm on oE.Then there exists for m E a biorthogonal system (z ,gn with the following properties.(2) The enumeration of (Zn,gn) is such that for each positive integer k the first k m terms are precisely all the tensors of the form x i x i m f. of.with i I i m e{l k}.
(3) The sequence (Zn)n is a Schauder base for oE.

PROOF.
It is known that [2]   for any reasonable norm .Also m m moE, c_ (m E)' "m Thus any enumeration of m-(Xn) n togetherm with the corresponding enumeration of (fn) n yields a biorthogonal system for OE.It is easily shown that m linear span of o(x .)n is dense in -E and we shall first establish the Suppose thatm a blorthogonal system (Zn,gn) with the stated properties can be constructed for E. Y m m+l Consider the sequence (Wn)n in E defined by the scheme in the accompanying figure.For each positive integer k, the terms wkm+l+l, wkm+l+2 W(k+l)m+l are described by the tensor beneath each term.{gp fqlgp (gn), fp (fn) according to the same scheme.the set In view of the inductive hypothesis the system (Wh,hn) is a biorthogonal system with the property that the first k m+l terms are all of the form Xil (R) x f f im+I, I (R) ira+ with i l,...,Im+ e {l,...,k}.
We utilized (*) in deriving ( 2) and (3).Hence, Ym+1(Wn(U)) ( (Mm+1 + 2M 2 + 2M2)ym+l(u) in all three cases.It now follows from the uniform boundedness principle that (II W n is bounded. To complete the proof let us recall that if is a reasonable norm then g a g Y Furthermore, E is the completion of mE with respect to a. Consequently for all such , the system (z ,gn is a complete btorthogonal system n whose sequence of partial sums T is pointwise bounded and hence bounded in operator n m norm.This means that (z) is a Schauder base for 8 E.
n n a CONCLUDING REMARKS.
It would be interesting to find out whether the base for m E obtained by Ym iteration of the process in [I] is equivalent to the base described in tnls paper.We hope to investigate this problem in a future paper.
n e N} (fn)n" theorem for Y m Now for m 2 the base constructed in [I] has the stated properties.

Figure
Figure ' (Zn) are bases the sequences (in),(T n) are bounded in operator norm by some constant M. To show that (w) is a base it thus suffices to show that n for some positive M', W M' for all n([4], p. 25).n Now, given a positive integer n the defining scheme for (wj)(see fig.I) shows that W can be expressed as sums of tensor products of the operators S T and f n m m Indeed, let us consider the three possible cases.T(k+1)m Tkm) S + (Tkm+r Tk m) f+l Case 3. If n k(k+l)m + r with r < (k+l) m then w =. n Sk + (T(k+1)m-Tkm) Sk + Tr "fk+l