fiK ~i~=la~d. fiK A Banach space with basis constant > 1.

A Banach space with basis constant > 1.

:PEg ENI~LO

University of Stockholm and University of California at Berkeley

~-~ n ~ e i l X

I f a B a n a c h s p a c e h a s a S c h a u d e r b a s i s fl, t h e n

fiK

SUp~,~ll~i=l i h/H /[ exists, where x =

~i~=la~d.

Inf/?K t a k e n o v e r all fi is called t h e basis c o n s t a n t of t h e B a n a c h space. I t is obvious t h a t if t h e B a n a c h space B has t h e basis c o n s t a n t p, t h e n e v e r y f i n i t e - d i m e n s i o n a l snbspace C of B can be a p p r o x i m a t e d b y subspaces D , of B - b y a p p r o x i m a t i n g a set of basis v e c t o r s of C w i t h v e c t o r s o f f i n i t e e x p a n s i o n s in some basis - such t h a t each Dn can be e m b e d d e d i n t o a finite- d i m e n s i o n a l subspace E~ of B, o n t o which t h e r e is a p r o j e c t i o n f r o m B of n o r m a r b i t r a r i l y close to p.

I n this p a p e r we c o n s t r u c t a separable i n f i n i t e - d i m e n s i o n M B a n a e h space B with a t w o - d i m e n s i o n a l subspace C 1 with t h e following p r o p e r t i e s : T h e r e is a p > 1 such t h a t , i f D is a t w o - d i m e n s i o n a l subspace o f B s u f f i c i e n t l y close to C 1 a n d E is a f i n i t e - d i m e n s i o n a l subspaee o f B c o n t a i n i n g D, t h e n t h e r e is no p r o j e c t i o n f r o m B o n t o E o f n o r m ~ p. T h u s t h e basis c o n s t a n t o f this 13anach space is > p. This seems to be b y n o w t h e s t r o n g e s t r e s u l t in n e g a t i v e d i r e c t i o n on t h e w e l l - k n o w n basis problem. T h e p r e v i o u s l y s t r o n g e s t r e s u l t seems to be G u r a r i i ' s e x a m p l e of a B a n a c h space w h e r e

fiK

> 1 for e v e r y ft. (See Singer [1]

pp. 218--42.)

W e n o w s t a r t b y giving a general a n d s o m e w h a t u npreeise d e s c r i p t i o n o f t h e ideas b e h i n d t h e c o n s t r u c t i o n a n d o f t h e p r o b l e m s we m e e t . W e consider a t w o - d i m e n s i o n a l subspace C 1 of l ~ ( F ) , w h e r e F is t h e set of pairs of p o s i t i v e integers.

W e a s s u m e t h a t t h e p r o j e c t i o n c o n s t a n t of C 1 is > 1. N o w o u r f i r s t a m b i t i o n will be t o e m b e d C1 in a larger space El, such t h a t t h e r e is no p r o j e c t i o n of n o r m close to 1 f r o m E I o n t o spaces close t o C 1 a n d such t h a t n o subspaee C 2 o f E1 c o n t a i n i n g a subspace o f E1 s u f f i c i e n t l y close to C 1 has a p r o j e c t i o n c o n s t a n t n e a r to 1. H o w e v e r , if we t r y to do this we h a v e to get c o n t r o l o f q u i t e m a n y l i n e a r spaces. I n o r d e r to describe h o w we o b t a i n t h e n e c e s s a r y simplifications, we give n o w a d e s c r i p t i o n o f t h e w a y we e s t i m a t e n o r m s o f p r o j e c t i o n s .

104 P E R E N F L O

L e t us a s s u m e t h a t C 1 is a t w o - d i m e n s i o n a l s u b s p a c e of a B a n a c h s p a c e B a n d t h a t 0, al, a2, . . . , a~ are n d- 1 p o i n t s in ^{C 1} such t h a t Ilajll = 1; j ~ 1, 2 , . . . , n,

lla~--ajll <~1; i , j = 1 , 2 , . . . , n .

W e s a y t h a t b E B is a

central point

for t h e d- 1 p o i n t s , if

Ilbll- -~

a n d

lib

- -

ajll

⁼ g, j = 1. 1 , 2 , . . . , n . I t is o b v i o u s t h a t if b is t h e o n l y c e n t r a l p o i n t in B for t h e n + 1 p o i n t s 0, a 1 , a 2 .. . . , a s a n d p o i n t s f a r a w a y f r o m b a r e f a r f r o m b e i n g c e n t r a l p o i n t s , t h e n a s u b s p a c e of B w h i c h c o n t a i n s a s u b s p a c e close t o C 1 a n d o n t o w h i c h t h e r e is a p r o j e c t i o n f r o m B of n o r m close t o 1 m u s t c o n t a i n a v e c t o r n e a r to b. F o r o t h e r w i s e b y s u c h a p r o j e c t i o n b w o u l d b e m a p p e d o n t o a p o i n t w h o s e d i s t a n c e to s o m e o f t h e p o i n t s 0, 61, a 2 . . . . , a , w o u l d be e s s e n t i a l l y l a r g e r t h a n }.

T h i s d e s c r i p t i o n s u g g e s t s h o w we c a n g e t c o n t r o l o f t h e p r o j e c t i o n c o n s t a n t s o f s u b s p a c e s o f B c o n t a i n i n g a s u b s p a c e close t o C 1. I t n a m e l y gives t h a t we c a n r e s t r i c t o u r a t t e n t i o n t o s u b s p a c e s o f B c o n t a i n i n g a s u b s p a c e close to C 1 a n d con- t a i n i n g a v e c t o r close to b. I n o u r e x a m p l e we h a v e in C 1 t w o sets 0, all , a12 . . . . ,61, , a n d 0, a21, 6 2 2 , . . . , a2, w i t h u n i q u e c e n t r a l p o i n t s bl a n d b 2 a n d so we c a n r e s t r i c t o u r a t t e n t i o n to s u b s p a c e s o f B c o n t a i n i n g v e c t o r s close to b I a n d b 2.

T h e w a y in w h i c h we will b u i l d u p o u r B a n a c h s p a c e is as follows. W e s t a r t w i t h a s u b s p a c e C 1 of l ~ ( F ) , g e n e r a t e d b y t w o v e c t o r s e 1 a n d e 2. I n C 1 we h a v e t w o p o i n t sets w i t h u n i q u e c e n t r a l p o i n t s 89 a a n d 89 4 in B. T h e c o n s t r u c t i o n is m a d e so t h a t for e v e r y t w o - d i m e n s i o n a l s u b s p a c e D o f

lop(F)

s u f f i c i e n t l y close t o C 1 t h e r e a r e p o i n t sets o f D such t h a t 89 a a n d le4 a r e t h e u n i q u e c e n t r a l p o i n t s in B for t h e s e p o i n t sets a n d p o i n t s f a r f r o m 89 ~ a n d 89 4 a r e f a r f r o m b e i n g c e n t r a l p o i n t s for t h e s e p o i n t sets. T h i s is i m p o r t a n t since it gives t h a t t h e r e will be no a c c u m u l a t i o n of a p p r o x i m a t i o n s in o u r c o n s t r u c t i o n . N o w in e v e r y s u b s p a c e o f B s u f f i c i e n t l y close to t h a t g e n e r a t e d b y e a a n d e 4 t h e r e a r e p o i n t sets w i t h t h e u n i q u e c e n t r a l p o i n t s 89 5 a n d 89 G. T h e f a c t s j u s t d e s c r i b e d are p a r t o f L e m m a 2 b e l o w a n d give a n i d e a of h o w t h e discussion goes on. W e will f i n a l l y conclude t h a t if a s u b s p a c e E of B c o n t a i n s a s u b s p a c e close t o C 1 a n d t h e r e is a p r o j e c t i o n f r o m B o n t o E of n o r m close to 1, t h e n E c a n n o t be f i n i t e - d i m e n s i o n a l .

W e n o w d e f i n e t h e v e c t o r s eY,

eJ C l~(I),

w h i c h g e n e r a t e B. W e let e~,~J d e n o t e t h e c o m p o n e n t of eJ w h i c h c o r r e s p o n d s to t h e p a i r (r, k).

W e choose e~,k = 0.611 - - (k - - 1)- 0.0ol, k = 1, 2 , . . . , 12. W e t h e n choose

2 2

e~,l = - - 0.60o a n d choose e~,k, k = 2, 3, . . . , 12 i n d u c t i v e l y b y t h e e q u a t i o n s 0.61l - - (k - - 1) 9 0.001 - - ( 0 , 9 9 9 5 ~ - 0 . 0 1 ( ] c - - 1))e~,k ⁼ 0 . 6 1 1 - - ( k - - 2 ) * 0 . O O l - - ( 0 . 9 9 9 5 -1-

0.Ol. ( / c - 1))e~,k_l. B y t h i s choice we o b t a i n t h a t e~,k k - 1, 2 , . . . , 12 lies b e t w e e n - - 0.6oo a n d - - 0.6~2 a n d t h a t t h e c o m p o n e n t (1, k) is t h e l a r g e s t of t h e 12 c o m p o n e n t s (1, 1), ( 1 , 2 ) . . . . , ( 1 , 1 2 ) o f t h e v e c t o r e 1 - (1 + 0 . o 1 . ( k - - 1))e 2, /c = 1, 2 . . . 12. W e also see t h a t i f v ~ a n d v 2 are s u f f i c i e n t l y close to e ~ a n d e 2 in

l~(F)

t h e n t h e c o m p o n e n t (1, k) will b e t h e l a r g e s t of t h e 12 c o m p o n e n t s

(1,1), ( 1 , 2 ) , . . . , ( 1 , 1 2 ) of v 1 - (1 + 0 . 0 ~ . ( k - - 1))v 2, k = 1 , 2 , . . . , 1 2 . W e p u t el, k = e~.k = 0 if k > 12. C o m p o n e n t s (r, k) w i t h r > i will be d e f i n e d later.

W e will h a v e n e e d for s u c h c o m p o n e n t s t o s a v e t h e u n i q u e n e s s o f c e n t r a l p o i n t s .

A B A N A C H S P A C E W I T t t B A S I S C O N S T A N T > 1 105

W e will n o w d e f i n e e a a n d e 4 so t h a t 89 a is t h e u n i q u e c e n t r a l p o i n t in B f o r t h e 7 p o i n t s 0, (e 1 - - (1 d- 0.o1(k - - 1))e2)/lle 1 - - (1 q- 0.ol(k - - 1))e211, k = 1, 2 . . . . , 6 a n d so t h a t 89 4 is t h e u n i q u e c e n t r a l p o i n t i n B f o r t h e 7 p o i n t s 0, (e a - - (1 q- 0.o1(k - - 1))e2)/lle 1 - - (1 q- 0.ol(k - - 1))e2][, k = 7, 8 . . . 12 a n d so t h a t t h i s h o l d s e v e n if e 1 a n d e 2 a r e r e p l a c e d b y a p p r o x i m a t i n g v e c t o r s v 1 a n d v a

a 3 . . = e 3 = 1 a n d w e c h o o s e in B. T o t h i s e n d w e p u t e 3 , 1 = e 1 , 2 = . 1,6

e 3 _{1 , 7 ,} e 3 1 , 8 , 9 ~ 9 , {31,12 a in s u c h a w a y t h a t :

(a) t h e d i s t a n c e in / ~ - n o r m b e t w e e n t h e 6 - v e c t o r 89 (%7, el,8, 9 9 -, 1,12) a n d a 3 e 3 e a c h o f t h e s e v e n 6 - v e c t o r s w h i c h c o n s i s t o f t h e c o m p o n e n t s (1, 7), (1, 8 ) , . . . , (1, 12) o f t h e v e c t o r s 0, (e 1 - - (1 q- 0.01(k - - 1))e2)/l[e 1 - - (1 q- 0.01(k - - 1))e2[I, k = l, 2 . . . . , 6, is s t r i c t l y less t h a n 1.

(b) t h e f o u r 6 - v e c t o r s ( e ~ , 7 , e l 1,8, e1,12), 1 2 2 ., 2 e3 7, e3 9 e3

s ( 1 , 1,12)

9 9 " , ( s s " ^" 1 , 8 ~ " ^{" ,}

a n d (1, 1, 1, 1, 1, 1) a r e l i n e a r l y i n d e p e n d e n t 9

W e n e e d (a) since 89 s h a l l b e a c e n t r a l p o i n t e v e n if w e a p p r o x i m a t e el a n d e2 b y v 1 a n d v 2 a n d (b) since 89 d e f i n e d b e l o w shall be a u n i q u e c e n t r a l p o i n t . 4 e4 4 in s u c h a w a y t h a t : W e p u t _{e l , 7} 4 __ e4 _{1,8 ~} 9 9 9 - - ; . 1,12 _ , e 4 a n d c h o o s e e1,i, 3,2, 9 9 9 el.6

e4 . . s

(al) t h e d i s t a n c e in / ~ - n o r m b e t w e e n t h e 6 - v e c t o r 89 9 (e~,l, 3,2, 9 , 1,6) a n d e a c h o f t h e s e v e n 6 - v e e t o r s w h i c h c o n s i s t o f t h e c o m p o n e n t s (1, 1), (1, 2), . . . , (1, 6) o f t h e v e c t o r s 0, (s _ (1 q- 0.01(k - - 1))e2)/[[e I - - (1 q- 0.01(k - - 1))e2[[, k = 7, 8 , . . . , 12, is s t r i c t l y less t h a n 89

(51) t h e f o u r 6 - v e c t o r s (e{,3, e{,2 . . . . , s j = 1, 2, 4 a n d (1, 1, 1, 1, 1, 1) are l i n e a r l y i n d e p e n d e n t .

A s a b o v e w e n e e d (al) s i n c e 1s s h a l l be a c e n t r a l p o i n t e v e n if w e a p p r o x i m a t e e 1 a n d e2 b y v I a n d v e a n d ( b l ) s i n c e 89 shall be a u n i q u e c e n t r a l p o i n t .

l~or k > 1 w e n o w c h o o s e 3 1 a n d e 4 2 W e see t h a t t h i s

-- s ~ s 1 , 1 2 ~ - k ~ s

d o e s n o t d e s t r o y t h a t ~ea a n d 89 4 a r e c e n t r a l p o i n t s f o r t h e sets d e s c r i b e d a b o v e 9 W e also o b t a i n t h a t t h e c o m p o n e n t s (1, 1), (1, 2 ) , . . . , (1, 12) o f e 3 - - (1 @ 0.01(k - - 1)) 9 e 4, k = 1, 2 . . . . , 12 a r e all < 0.25 a n d t h a t t h e c o m - p o n e n t (1, 12 @ k) is t h e l a r g e s t o f t h e c o m p o n e n t s (1, 13), (1, 1 4 ) , . . . , (1, 24) o f e 3 - - ( 1 - ~ 0 . e l ( k - - 1 ) ) . e ~, k = 1, 2 , . . . , 1 2 , a n d is > 1. T h e last s e n t e n c e is o b v i o u s l y t r u e e v e n i f e 3 a n d e4 a r e r e p l a c e d b y a p p r o x i m a t i n g v e c t o r s v 3 a n d v 4 in l ~ ( F ) .

W e n o w c h o o s e e51,k, el, k6 k = 1, 2, . . . , 12, t o be all in t h e i n t e r v M (0.0s, 0.10) a n d c h o o s e t h e m in s u c h a w a y t h a t t h e six 6 - v e c t o r s (ei,1, s . . . . , e{,6), j = 1, 2 , . . . , 6, a r e l i n e a r l y i n d e p e n d e n t a n d s u c h t h a t t h e six 6 - v e c t o r s (e{,7, eJ,8 . . . . , e{,~2) a r e l i n e a r l y i n d e p e n d e n t . T h i s is d o n e in o r d e r t o m a k e ~e a a n d 89 u n i q u e c e n t r a l p o i n t s . :For all k > ] _{- -} w e p u t s 1 , 1 2 + k ~-- s 1 , k a n d

6 4

s ~ s k"

N o w f o r all k < 1 2 j , _{- -} j > l , _{- - -} w e p u t e 6+(2j-1) _{3 , k ~} 0 a n d f o r all k > _{- -} 1 a n d

j

> 1 w e p u t e 6+(2i-1) _ 1,12j+k = el,k. F o r all k < 123", j >_ 1, w e p u t e ~ 2 j = 0 a n d 5

6 ~ 2 j 6

f o r all k > 1 a n d j > 1 w e p u t s = s k"

N o w we h a v e t o i n t r o d u c e c o m p o n e n t s (r, k) w h e r e r > 1. F o r if we d i d n o t

106 P E R E ~ F L O

do t h a t , w e w o u l d o n l y h a v e o b t a i n e d t h e f o l l o w i n g : 89 2j+1 a n d 89 2i+2 a r e b o t h c e n t r a l p o i n t s f o r sets c o n s i s t i n g o f 0 a n d l i n e a r c o m b i n a t i o n s o f e 2j-1 a n d e 2i. T h i s h o l d s e v e n if e 2j-~ a n d e 2j a r e a p p r o x i m a t e d b y v e c t o r s v 2J'-I a n d v 2j i n B.

B u t 89 2j+1 a n d 89 2i+2 will n o t b e u n i q u e c e n t r a l p o i n t s . F o r c e r t a i n l i n e a r c o m - b i n a t i o n s o f el:s, i ~ 2 j - - 2 or i > 2 j ~ 5 c o u l d b e a d d e d t o 89 2j+1 a n d 89 2i+2 w i t h o u t d e s t r o y i n g t h e i r p r o p e r t y o f b e i n g c e n t r a l p o i n t s .

2 j - 1 2j

F o r all j >_ 1 a n d k > 1, e[(j+3)/2l, k j =- l, %'+2,k = 0.1 a n d e1+2, k ~ - - 0.1.

2 j - 1 2j

T h e d e f i n i t i o n o f ej+z,k a n d ey+2,k is m a d e so t h a t 89 2j+1 a n d 89 2j+2 a r e c e n t r a l p o i n t s e v e n if e 2~-1 a n d e 2j a r e a p p r o x i m a t e d b y v e c t o r s v 2j-~ a n d v 2j in B. l % r t h o s e t r i p l e t s (j, r, k) w h e r e e~,k is n o t y e t d e f i n e d , w e p u t e]]k ~ 0.1 9 ( - - 1) [(~-1/2)1.

T h i s l a s t d e f i n i t i o n will s a v e t h e u n i q u e n e s s o f 89 2y+~ a n d 89 =j+2 w i t h r e s p e c t t o l i n e a r c o m b i n a t i o n s o f el:s, i < 2 j - - 2 or i > 2 j ~ - 5 . F o r if w e a d d s u c h a l i n e a r c o m b i n a t i o n t o 89 2j+~ o r 89 2j+2 t h e d i s t a n c e t o 0 will be > 89

W e n o w r e s u m e i n t h r e e l e m m a s t h e i m m e d i a t e c o n s e q u e n c e s o f o u r d e f i n i t i o n s o f t h e v e c t o r s e ~', j > 1.

LEMMA 1. There exists an s > 0 such that i f I]v 2j-1 -- e2j-lll < ~ and ]Iv 2j - - e2J[] <_ e holds for two vectors v 2i-1 and v 2j in B , j >_ 1, then v2i i _ (1 ~ - 0 . 0 1 ( k - 1))v ~j has its largest component on the same place as e 2j-~ - - (1 ~- 0.01(k - - 1))e 2j, k : l, 2, . . . , 12, n a m e l y some component (r, m) with r = 1, and the distance between 89 2i+1 and each of the elements 0,

(v2J

1 _ _ (1 -~- 0.01(]c - - 1))v2J)/Hv2j-1 - - (1 -~- 0.01(k - - 1))v2J[[, k = 1, 2 , . . . , 6, is 89 Obviously a s i m i l a r result holds f o r 89 2j+2 f o r k --~ 7, 8 . . . 12.

LEMMA 2. Let v 2j-I and V 2j be as in L e m m a 1. T h e n 89 2j+1 is the only vector of B such that the distance between the vector and each of the vectors 0,

_ _ 1

(v 2j-x - - (1 ~-~ O.01(k ]))v2J)/llv2j-1 - (1 -~- O.01(k - 1))v2Jl[, k : 1, 2 , . . . , 6, is ~2 and there are positive constants K and 81 such that i f g E B and []g -- e2/+1]1 z 5, then the distance between 89 and some of the

O, (V 2j 1 - - ( l ~ - 0.01(]~ - - 1))v2J)/]]V - - (1 ~ - 0.01(]~ - - ]))v2J[], ]~ = 1, 2 , . . . , 6,

is at least 89 -~ 1 K m i n (5, el). A similar result holds f o r 89 2j+2 f o r k = 7, 8 . . . . , 12.

2j--2A-t 2j--2+t 2j--2+t\

Proof. T h e l e m m a h o l d s since t h e six 6 - v e c t o r s [ e 1 , 1 2 j ~ _ 1 , e l , 1 2 j + 2 , . . . , e 1 , 1 2 j + 6 ) ,

t = 1, 2, . . . , 6, a r e l i n e a r l y i n d e p e n d e n t a n d since (j § 2, k), k > 1, a r e p a i r s w h e r e ej+z.k = 1. T h e l a s t p r o p e r t y g i v e s t h a t 2j'~l lie 2j+1 ~- hll >>

IleeJ+ll I

+ 0, 1 9

IIhL[

i <_2j 2 or i > 2j + 5.

if h is a l i n e a r c o m b i n a t i o n o f v e c t o r s e,

W e c a n a s s u m e t h a t s, s 1 a n d K in L e m m a 1 a n d 2 a r e t h e s a m e f o r e 2i+1 a n d e 2j~2.

A B A ~ A C I : [ S P A C E ~VITI-I B A S I S C O N S T A N T ~ 1 107 L]~MMA 3. L e t V 2 j - 1 a n d v 2j be as i n L e m m a 1. I f C i s a s u b s p a c e o f B , v ~j-~ E C a n d v ~j C C, a n d infgec llg - - e2J+llI = ~, then there i s n o p r o j e c t i o n f r o m the space D generated by C a n d e 2j+~ onto C o f n o r m ~ 1 q - K . m i n (0, el) w h e r e K a n d e 1 are d e f i n e d as i n L e m m a 2. A s i m i l a r r e s u l t holds f o r e 2j+:.

P r o o f . T h e l e m m a h o l d s since b y a n y p r o j e c t i o n f r o m D o n t o C t h e v e c t o r 89 2j+l is m a p p e d o n t o a v e c t o r w h o s e d i s t a n c e t o s o m e o f t h e e l e m e n t s O, (v 2j 1 _ (1 q_ 0.01(k - - 1))v"J)/Hv 2i-1 - - (1 q- 0.01(k - - 1))v2.iH, k : 1, 2 , . . . , 6, is

89 -~ 89

m i n (~,

ex).

W e n o w p r o v e that B h a s the basis c o n s t a n t ~ p ~ I. W e c h o o s e p =

1 q - 8 9

el) w h e r e e is t h a t o f L e m m a 1 a n d K a n d s 1 a r e t h e s a m e as i n L e m m a 2. L e t D b e a t w o - d i m e n s i o n a l s u b s p a c e o f B g e n e r a t e d b y v 1 a n d v ~, v 1 a n d v 2 as i n L e m m a 1. I f E i s a s u b s p a c e o f B , D G E , s u c h t h a t t h e r e e x i s t s a p r o j e c t i o n f r o m B o n t o E o f n o r m ~ p , t h e n b y L e m m a 3 E m u s t c o n t a i n v e c t o r s w h o s e d i s t a n c e s t o e a a n d e 4 a r e ~ e. T h u s b y L e m m a 1 a n d L e m m a 3 E m u s t c o n t a i n v e c t o r s w h o s e d i s t a n c e s t o e 5 a n d e 6 a r e ~ e. B y i n d u c t i o n w e see t h a t E m u s t c o n t a i n v e c t o r s w h o s e d i s t a n c e s t o e 2j-~ a n d e 2j a r e ~ s f o r e v e r y j ~ 1. T h u s B h a s t h e b a s i s c o n s t a n t ~ / 9 .

R e m a r k . I t h a s b e e n s h o w n b y J . L i n d e n s t r a u s s t h a t t h e c o n s t r u c t i o n a b o v e c a n be m o d i f i e d so t h a t w e g e t a u n i f o r m l y c o n v e x B a n a c h s p a c e i s o m o r p h i c t o H i l b e r t s p a c e w i t h basis c o n s t a n t > I.

References

1. SIZ~GER, I., Bases in Banach spaces, Springer-Verlag, Berlin 1970.

Received March 27, 1972 Per Enflo

Department of Mathematics University of California Berkeley, California U.S.A.