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142

Z. ALTSHULER ET AL.

Israel J. Math .,

P~'~+1

L

Yn =

CX·X·
z. z.

i = Pn+ 1

for n = 1, 2, ···,is a block basic sequence of a symmetric basis {xn}, and for each

n, an is a permutation of {Pn + 1,pn + 2, ···,Pn+ 1 }, then {Yn) ~ {zn} where
zn = l:f~;~+ 1 j cxa(i) jxi, n = 1, 2, · · ·. Therefore, when working with block basic
sequences {Yn} of a symmetric basis {xn} we will always assume that cxPn+t
~ex
+z ~
Pn
-

··· >
cx Pr. + 1
-

~

-

0 for n = 1' 2 ' · ··.

Let {xn} be a symmetric basis in a Banach space X. Define
n

Ill x Ill =

sup
aen

sup
1Pd:::1

L

f3 ifi(x )xa(i) ,

i= 1

XEX,

H

1<n<+oo

where {.fn} is the sequence of biorthogonal functionals of {xn} in X*. Then the
symmetric norm

Ill x Ill, x EX, is an equivalent norm on X. Throughout this paper,

we shall assume that every Banach space with symmetric basis is equipped with
the symmetric norm.

1. Preliminaries
In this section we state some simple and well-known facts on symmetric basic
sequences in Banach spaces.
PROPOSITION 1. Every sym1netric basic sequence in a Banach space is either

weakly convergent to zero or is equivalent to the unit vector basis of 11 •
It is known that in the lP spaces, 1 ~ p < oo, all symmetric bases are equivalent
[12, p. 573J.As a consequence of Proposition 1, we have
CoROLLARY 1. In the spaces X= c0 or lP, 1 ~ p <

+ oo,

all symmetric basic

sequences are equivalent.
PROPOSITION 2. Let X be a Banach space with a symmetric basis {xn}· If

every bounded block basic sequence of {xn} is symmetric, then {xn} is equivalent
to the natural basis of c0 or lP for some p, 1 ~ p < + oo.
PROOF. Let {y11 } be a bounded . block basic sequence of {X 11 }. Since {y11 } is
symmetric, {Yn} ~ {y 211 }. Choose a subsequence {xni} of {xn} such that

Yzi if n = 2i,
zn =

Xni

if n

= 2i + 1,

i

= 1 2 ...
' '

i = 1 2 ...

' '

'
'

is a bounded block basic sequence of {x11}. Then, since {zn} is symmetric,