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141

SYMMETRIC BASIC SEQUENCES

Vol. 15, 1973 ·

showed that d( a, p) is a reflexive Banach space which, in general, is distinct from
the [P-spaces. See [1] for further references on other work on d(a, p).
Another class . of Banach spaces with symmetric basis is that of the Orlicz
sequence spaces. J. Lindenstrauss and · L. Tzafriri [ 6, 7] have shown that every
orlicz sequence space has a subspace isomorphic to some

[P.

They have also shown

that there are Orlicz sequence spaces which have at least two nonequivalent
symmetric bases. We show that d( a, p) has a unique symmetric basis for all a and p
and that every infinite.dimensional subspace X of d(a, p) has a subspace isomorphic
to [P which can be chosen to be complemented in X if X has a symmetric basis.
The Lorentz sequence spaces which have exactly t\vo nonequivalent symmetric
basic sequences are characterized. Finally, an example of a Lorentz sequence
space having a subspace with symmetric basis which is isomorphic neither to lP
nor to any Loret?-tz sequence space is given.
· .We introduce a new type of block basic sequence of a symmetric basis Vihich
has the property that it always has a symmetric subsequence. In the spaces d(a,p),
these are the only symmetric block basic sequences of the unit vector basis {xn}
of d(a,p) which are not equivalent to the unit vector basis of

[P.

The notations and terminology in this paper are essentially those of I. Singer

[12]. A sequence {xn} of a Banach space X is ca11ed a basis of X if every x EX
has a unique expansion of the form x

= Lf' 1 anxn- Let 1 ~ p < + oo; a basis

{xn} of X is called p-Hilbertian if 'L;' 1 tt11 Xn converges in X for every {an} E lP.
A basis {xn} is q- Besselian, 1 ~ q < + oo, if 'L;> 1 anxn converges in X implies
that {an} E lq.
If {xn} is a basis of a Banach space X, a sequence {y11 }in X is said to be a block
basic sequence of {x if there is an increasing sequence of natural numbers {Pn}
such that y n = 'Lfn ;~ + 1aixi for n = 1, 2, · · · . A block basic sequence {Yn} is said
to be bounded if 0 <; inf1 <n< +co II Yn ~ sup 1 <n< +co Yn j < + oo. We will denote
by [{Yn}J the closed linear span of the sequence {Yn}· If {x and {Yn} are bases
of X andY, respectively, we saythat {xn} ~ominates {Yn}, and write {xn} > {Y 11 }, if
11

}

11 }

:L:

1 tt11 Xn

00

converges in X implies that Ln

1

tt11 Yn converges in Y. The basis {X11 }

is equivalent to the bas'is {y11 }, and we write {xn} rv {Yn}, if {xn} > {Yn} and {Yn}
> {xn}. It is clear that a basis {xn} is equivalent to ·the unit vector basis . of [P. if
and only if {xn} is p- Hilbert ian and p- Bessel ian.
If {X 11 } and {Yn} are symmetric bases, it is easy to show that {xn}
.

.

rv

{Yn} if and

onl:y if for any sequence of scalars a 1 .~ a2 ~ • • • ~ 0, 'L~ 1 anxn converges in X if
and only · if · Ln~ 1 ttnYn converges in Y. We also note that if