Internat. J. Math. & Math. Sci.
VOL. 12 NO 2 (1989) 291-296
291
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)
Let
E denote the
Let E be a Banach space with Schauder base (x)
(1
n n
completion of the m fold tensor product with respect to a reasonable cross norm a. We
ABSTRACT.
show
{x
that the set
i
positive integer k, the first k
x
x
i
mE
for
i
m
with i l,...,i
KEYWORDS AND PHRASES.
xi.
x:ter3mS
(Xn)n
m
of the form
are precisely all the elements
{l,...,k}
m
can be enumerated so that for each
and the set so arranged is a Schauder base
m fold tensor product,
projective,
fnJectlve
and reasonable
cross norms, Schauder bases.
1980 AMS SUBJECT CLASSIFICATION CODE. 46B15, 46M05.
I.
INTRODUCTION.
Given a Banach space
E
and by
E, we will denote by
(xn
(R)
i
>
x
i
e
x
m
(Xn)
(Xn)n’ (Yn)n
n n
is a Schauder
m}.
J
2 the projective (inJectlve) tensor norm on
2 this norm will be denoted by
In [I],
(x)
If
will denote the set
{x
If m
.
the m fold algebraic tensor product
its completion with respect to reasonable norm
base for E then
For m
E
mE
will be denoted by
m(%).
T(1).
[4] it is shown that if
EI,E 2
then for any reasonable norm
,
are Banach spaces with Schauder bases
the space
E
E
2
has a Schauder base
(z) with the following properties.
n
(I)
(2)
k
(Zn) n
is obtained by enumerating the set
(Xn)n
(Yn)n;
the enumeration is such that for any positive integer k,
2 terms are
preclsely all elements of the form
x,
y, 2
i
l,i 2
e
{l,...,k}.
the first
292
R. HEMAS INHA
m
In this paper we show that for any positive integer m, the space ,oE
has a
m
in such a way that for any positive
Schauder base obtained by enumerating
n
m
integer k, the first k terms are precisely all elements of the form
-(Xn)
x.
x m with il,
(R)
{l,...,k}.
’ira
Our proof is via an induction argument and in a forthcoming paper we utilize this
to derive some properties of the symmetric tensor algebra of a Banach
construction
E of the
3, iteration to (E" E)
that even for the case m
Note
space.
[I] will not yield a base with the above mentioned
enumerating scheme described in
properties.
For definitions and terminology regarding Schauder bases,
products and
tensor
[4], [3], and [2].
We denote the following property (*) of the projective tensor norm.
It can easily be shown that
let u be an element of
tesnor norms we refer the reader to
For m
>
2
mE
v (u)
v(u)
(*)
where
with
7(u) is the projective norm of u when u is considered as an element of F E
E being endowed with the norm Xm-l" We shall make use of this fact
F
subsequent ly.
We now state our theorem.
Let (x n,
THEOREM.
E
fn Xn
fn
be a reasonable norm on
Let e
a Schauder base for E.
m
E a biorthogonal system (z
,gn
As sets [z n
(I)
[gn:
the
x
c_
x
i
of.
(3)
The sequence
(m
"m
enumeration of
Thus
(fn) n
Suppose
with i
such that for each positive integer k
is
all
precisely
are
any
i
I
(Zn)n
m
e{l
of
the
form
norm
.
Also
tensors
the
k}.
[2]
of
-
m
for any reasonable
m-(Xn) togetherm
n
yields a biorthogonal system for
m
.)n
oE.
is a Schauder base for
m
enumeration
dense
in
Ym
E and
OE.
we
with
the
corresponding
It is easily shown that
shall
first
establish
the
m
2 the base constructed in [I] has the stated properties.
thatm a E.blorthogonal
constructed for
(fn)n"
(Zn,gn)
is
the linear span of o(x
n
assertions of the theorem for
Now for m
oE. Then there exists for
o(x n) n
terms
It is known that
E)’
is
i
m
f.
PROOF.
moE,
k
first
(Xn) n
m
n e N}
m
m
with the following properties.
n s N}
(2) The enumeration of
E’ be biorthogonal system such that
E
Ym
system
(Zn,g n)
with the stated properties can be
CONSTRUCTION OF A BASE OF THE TENSOR PRODUCT OF A BANACH SPACE
293
m+l
figure.
(Wn)n
E defined by the scheme in the accompanying
For each positive integer k, the terms
Consider the sequence
wkm+l+l, wkm+l+2
in
W(k+l)m+l
are described by the tensor beneath each term.
Figure
k, a positive integer;
an
<
0
integer
k
wkm+l +I
wkm+ I+2
W(k+ I) re+kin(k- I)
x
x
x
zkm+l
zkm+2
(R)
Z(k+l)m
W(k+1)m+km(k-1)+1,
W(k+ I) m+km(k- I)+2,
W2(k+ I) m+km(k-2)
x
x
x
2
zkm+l
zkm+2
2
Z(k+l)m
2
w (k+l)m+km(k_)+i
w(+ I) (k+ I) m+km(k_ +
x+l zkm+l
X+l
W(k_1) (k+1)m+km+
Wk(k+l)m
x
x
k
k
zkm+l
Wk(k+ re+l,
z
Xk+l
In
(k+l)
m
exhibited
The
the
k
m
preceding
entries.
m
k((k+l)
sequence
W(k+ m+
Xk+le z2
Xk+l
tableau
k m) + (k+l)
{gp fqlgp (gn), fp (fn)
Z(k+l)m
Wk(k+ I) m+2,
each
The last row has
(h)
n
Z(k+l)m
m--
is
(k+l)
Z(k+l) m
of
m
the
k
entries.
first
k
rows
contains
(k+l)
m+l
obtained
entries, and so altogether we have
m+l
by
enume rating
according to the same scheme.
the
set
R. HEMAS INHA
294
(Wh,h n)
In view of the inductive hypothesis the system
Xil
with
l,...,Im+
i
(x
systems
n
Since
e
f
(R)
I
ira+
respectively denote the
n,gn
(Xn)’ (Zn) are
fn
(w
and
M’,
n
bases
n
th
partial sun operators of the
hn).
the sequences
(T n) are bounded in operator
(in),
(w) is a base it thus suffices to show that
To show that
norm by some constant M.
for some positive
f
im+ I,
n
M’ for all n([4], p. 25).
W
n
Now, given a positive integer n the defining scheme for
that
W
n
is a biorthogonal system
terms are all of the form
{l,...,k}.
Tn’ Wn(z
Sn,
Let
x
(R)
m+l
k
with the property that the first
(wj)
(see fig. I) shows
can be expressed as sums of tensor products of the operators
S
T
m
and f
m
Indeed, let us consider the three possible cases.
Case I.
If
km+l
n
for some positive integer k then
m+l
W
OSk,
n
Case 2. If
wn
=m+.1 Sk
Case 3.
If
wn
In u
(k+l) m +
n
(
r ((k+l)
+
(T(k+1)m
n
=. Sk
is
in
S
the m+l fold tensor product of
m
k
m
Tkm)
k(k+l)m +
+
km(k-)
<
+ r, with 0
S
+
r with
(Tkm+r Tkm)
r
<
(k+l)
m
(2)
(3)
Ym+l(m+elSk(U))
Sk + Tr "fk+l
E then property (*)
together with the
[Sk m+l Vm+1(u)
Ym+1((T(k+1)m- Tkm)(R) S (u))
< HT(k+1)mTkm
f+l
then
reasonable yields the following inequalities
(i)
k and
then
(T(k+1)m- Tkm)
m+l
k.
S Ym+l (u)
Ym+l ((Tkm+r Tkm) " f+1(u))
llTkm+r- Tkm llf+iHYm+l(U).
fact
that
Ym+l
is
CONSTRUCTION OF A BASE OF THE TENSOR PRODUCT OF A BANACH SPACE
We utilized (*) in deriving (2) and (3).
Ym+1(Wn(U))
all
in
three
(II W
+ 2M 2 +
(Mm+1
It
follows
now
To
complete
g a g
Y
m
the
let
proof
Furthermore,
Consequently for all such
,
2M2)ym+l(u)
from
the
uniform
principle
that
reasonable norm
then
boundedness
This means that
us
E is
recall that
the
the system
if
completion
(z
,gn
is
of
a
mE
with
respect
to
a.
is a complete btorthogonal system
n
is pointwise bounded and hence bounded in operator
n
m
(z) is a Schauder base for 8 a E.
n n
whose sequence of partial sums
norm.
Hence,
is bounded.
n
m
cases.
(
295
T
CONCLUDING REMARKS.
It
would be interesting to find out whether the base
iteration of the process in
[I]
for
mYm E
obtained
is equivalent to the base described in tnls paper.
by
We
hope to investigate this problem in a future paper.
REFERENCE S
I.
2.
3.
4.
GELBAUM, B.R. and GIL DE LA MADRID,
J.
Bases for Tensor Products of Banach
Spaces, Pacific J. Math. II (1961), pp. 1281-1286.
CIGLER, J. LOSERT, V. and MICHOR, P. Banach Modules and Functors on Categories
of Banach Spaces.
Lecture Notes in Pure and Applied Mathematics, Marcel
Dekker, Inc. (1979).
Annals of Mathematics Studies, 26,
A Theory of Cross Spaces.
SCHATTEN, R.
Princeton University Press, (1950).
SINGER, I. Bases for Banach Spaces, Vol. I. A Series of Comprehensive Studies
in Mathematics, 154.
Springer-Verlag, (1970).