ON LINEAR INDEPENDENCE OF SEQUENCES IN A BANACH SPACE
P . ERDÖS AND E . G . STRAUS
1 . A . Dvoretzky has raised the following problem :
Let x t , x 2 , • • • , x n , . • • be an infinite sequence of unit vectors in a Banach
space which are linearly independent in the algebraic sense ; that is,
k
(i = 1, . . . , k) .
ci x n . = 0 ===> ci = 0
i =1
Does there exist an infinite subsequence {
in a stronger sense?
xni 1
which is linearly independent
We may consider three types of linear independence of a sequence of unit
vectors in a normed linear space :
00
I.
F,
=->c n
c n x n =0
=0
(n= 1, 2,
• • ).
n=1
II .
If
0 (k)
> 0 is any function defined fork = 1, 2, • • • , then
1, nk) t <
(n,k=1,2, • • • )
5(k)
and
00
lim
k-00
lim
k-+ co
cn(k) xn = 0
n=1
I
(n = 1, 2,
c n(k) = 0
Received October 29, 1952 .
Pacific J. Math. 3 (1953), 689-694
689
).
690
P . ERDÖS AND E . G . STRAUS
k--k)
n =
lim
k=i
III .
0
>
lim
k-.-
n
c kl
= 0
(n = 1, 2,
..).
It is obvious that III implies both II and I ; and if
lim inf y' (k) > 0
k-then II implies I . It is easy to show that the converse implications do not hold .
In this note we give an affirmative answer to Dvoretzky's question if independence is defined in the sense I or even II for arbitrary 0 (k ) . However the
answer is in the negative if indepence is defined in the sense III .
2 . The negative part is proved by the following example due to G . Szegö
[1 ; I, p .86] :
THEOREM .
the functions
I
If An } is a sequence of positive number with An -0 cc, then
1/(x + fi n ) } are complete in every finite positive interval .
I
Obviously every infinite subsequence of
of the theorem and is therefore complete .
I
1/(x + A n ) } satisfies the condition
3 . For the affirmative part of our result we prove the followingTHEOREM . Let Ixni} be an infinite sequence of algebraically linearly
independent unit vectors in a Banach space and let ~(k) > 0 be any function
defined for k=1, 2, . . . . Then there exists an infinite subsequence Ix,,, } such
that l ci ml l < c5(i) (i, m=1, 2, - .-)and
lim
m- .~
cc ml x ni = 0
i=1
imply
lim
ci m)
= 0
(i = 1,2, . . .) .
m _M
It was pointed out to us by the referee that it suffices to prove the theorem
for a separable Hilbert space . The separability may be assumed since we may
restrict our attention to the subspace spanned by I x n } . Now every separable
Banach space can be imbedded isometrically in the space C(0 ' 1) of continuous
I
691
ON LINEAR INDEPENDENCE OF SEQUENCES IN A BANACH SPACE
functions over the interval (0, 1) ; and C (0, 1) C L 2 (0, 1), where linear independence, in any of the above defined senses, in L 2 implies the same independence in C . Let I z n } be the orthonormal sequence obtained from I x n } by
the Gram-Schmidt process ; then
n
xn -
anm zm,
M=l
with I anm
Since
th at
I
I
< 1 and ann ?~ 0.
a nm } is bounded for fixed m, we can select a subsequence
Jim an L•m
I
xni } such
= bm
.W
I--
exists for every m .
If we prove the theorem for 0(k) > 0(k), then it is proved a fortiori for
y5 (k ) . Hence we may set
(n)=max{l,~(1), . . .
~( n )},
0
I
0
so that 0 (n) > 1 and Vi (n) is nondecreasing.
If the theorem we false then for every infinite subsequence
there would exist a sequence of sequences I ckm ) } with
I
Yk
} of
I
x ni }
(k, m = 1 , 2,
1,k(-)l <0 (k)
and
W
lim
c k(M) Yk = Q
M _o0
k=i
while
I
lim sup' I cko
( )
m ~
0
for some fixed k o .
We can then select a subsequence of sequences
I
lim ckm i ) = c k
i -~
ckm i )
I
such that
692
P . ERDÖS AND E . G . STRAUS
exists for every k, and
~ 0. For convenience of notation we assume
ck 0
lim c (m)
n-- k
.o
Since
ck o ~
I
ck
= c
k
.
0, there would exist a least k t > k o such that
I< 2 k-k0
0
(k)
i
C
k0
for all k > k i
I
.
This implies
c (m)
k
(1)
I
< 2
k_ki
0
(k)
I
c (m)
kt
for all k > k t ;
I
m > mo .
Case A : b n, i . = 0 for j=1, 2,--- .
In order to simplify notation we assume b ni = 0 for all i=1,2,--- by omitting all terms with ni ~ ni l from our subsequence . V e select the subsequence
( Yk } as follows :
YI
= xnt' Yk+ i =
xn ik+i
where
I
I
We write
I
an ik+l ,n j
I
<
anj, nj
I
for
4 k+1 0(k+1)
j=1,2, • • • , ik .
Yk = xi k .
If the theorem were false then there would exist a sequence of sequences
( with the above properties such that
c ( nd)
k
c km) Y k
II
II=
(E m -* 0
as
k=t
If we take the k i defined in (1), then
(2)
Z
k=kt
but for all m > m o we have
ckm)
tz1k, 111
< Em ;
m -) oo .
ON LINEAR INDEPENDENCE OF SEQUENCES IN A BANACH SPACE
Ic~m)I < 2
693
k_kt
(k)Ic kt
+1, . .-),
(k=k t , k t
and hence
I
00
I
ckC m)
.1k, 1 k
k=k t +t
t
2
00
k_k t
yi(k)I
ck 1 (
al kt, 1 k1 I
2 -2 k t
L.
k=k t +t
We can now choose
(M) - c kt
1,k,
=
m
< 2 -akt
jalk
t
ckm)
al k' 1,
k=k t
(2)
lk
and
cki )
ckt
(alk
t
a lk
lkt
I-
I
.lkrl_2-akt Icktl
alkt 1k
cam) al k
, lk t
k=kt+t
t '
I
< 2 -akt
ilk
I .
t
we obtain
1
c kt
Em
t
la lkt,l
> lckIUelkr
> 2 -akt
'
so large that
c kt
Then for the left side of
I ck t ~( al k t . 1 k t
(k)
4k
1
-22kt
I
I ckiI Ial kt' l kt l
t j'
while for the right side of (2) we have
E m < 2 akt
(ckt
I
(al
kt
,lk
t
a contradiction .
Case B : b n1
0 except for a finite number of i .
0 for all i by omitting a
Without loss of generality we may assume b ni
finite number of elements from ( x ni } . We select the subsequence I y k I as follows :
y t = xn t '
Yk+t = xnik+t '
P . ERDÖS AND E . G . STRAUS
6 94
where
I an
ik+t' nj
- b
nj
bn ik+t I
for j =
I <
l, 2, .-
4k+t 0(k+ 1)
For simplicity we again write
y
1
_
=
x
ik .
If the theorem were false then there would exist sequences {
foregoing properties such that
ckm)YkH = Em
II
If we let
kt
1
I
bl kt
c (m)I
ki
> Ickl ) I
m-*co .
.l kt
k=k t
C (M)I
_
00
1
ck(m) al k
I
> I
with the
be defined as in (1), then on the one hand we have
00
Q=
as
---*0
Z'
ckm)
c km)
alk
. lk t
+
t
b ik t +I
2 I ckm)
00
ki
I
Y,
4kt
k=k t +t
22k-kt
.
°°
1
4 k ~J(k)
y'~(k)
1-
1
> 2
4 k ~(k)
4 t
1
Ickl ) I
>
Ic kt I>0
4
for all m > m o ; on the other hand, we have
1
(m)
ck
yk
bl kt+t
k=t
1
+
1
em
b1k2
<-Ic
4
k
i
for all sufficiently large m, a contradiction .
REFERENCES
1 . R . Courant, D . Hilbert, Mothoden der mathematischen Physik, Berlin, 1931 .
NATIONAL BUREAU OF STANDARDS, LOS ANGELES
UNIVERSITY OF CALIFORNIA, LOS ANGELES