On the separation properties of the duals of general topological vector spaces
M. RIBE
Link6ping University, Link6ping, Sweden
Consider a non IocaIly convex topological vector space E. Suppose there is a non-zero point x in the subset E 1 c E of all points which cannot be separated from the origin b y a n y continuous linear form on E. One might ask whether x can be separated from the origin by a continuous linear form which is defined only on E 1. I t will be shown b y means of examples t h a t this m a y be the case. Since E 1 is a linear subspaee -- namely, the intersection N f-l(0) of all closed hyperplanes
f E E '
through the origin1 -- this fact gives rise to a more general question: L e t E~ be the subspace in E 1 of points which cannot be separated from the origin b y a n y continuous linear form; and then define recursively subspaces E a ~ E 4 ~ . . . I n a natural w a y we thus get a transfinite decreasing ~>sequencc~) (indexed b y the ordinals) of closed linear subspaces of E. Obviously, this sequence must become stationary at some ordinal ~(E). The observation just mentioned shows t h a t this need not happen at once, i.e., we m a y have E 1 ~= E2, so t h a t ~(E) > 2. Thus we ask: Which ordinal values can be assumed b y ~(.)? Our aim in I I below is to answer this question b y constructing examples to show t h a t even for locally bounded spaces, ~(.) m a y assume a n y ordinal value.
I n I below, we investigate some general properties of the transfinite ~>sequence)) mentioned above.
As a by-product of the constructive methods employed in II, we obtain (Section II.3) a certain isometric imbedding of metric spaces into p-normed spaces, which has universal and funcforial properties.
For brevity, we will write tvs for topological vector space(s). Further, E ' will denote the (topological) dual of a tvs E, and ~ is as usual the least transfinite ordinal. All tvs are supposed to have the same scalar field, which m a y be the real or the complex number field. A linear subspace of a tvs is always topologized by the subspace topology.
280 ARKIV FOIr lgATEMATIK. V o l . 9. N o 2
I a m m o s t g r a t e f u l t o t h e late Professor H a n s R g d s t r S m for his v i v i d i n t e r e s t a n d v e r y v a l u a b l e criticism.
I. General considerations 1. Definition. F o r a t v s E we w r i t e
A E = N f - l ( 0 ) 9 feK"
B y t r a n s f i n i t e i n d u c t i o n we define for e v e r y ordinal A~E so t h a t
A r E = A ( [-I A t E ) 9
v'<~v
v a closed linear subspace
1.1. PROPOSITION. Under ~ , the class {A~E}~ is well-ordered and has a last element. There is a least ordinal vo such that A~oE = A ~~ 1E.
1.2. Definition. F o r a t v s E , t h e ordinal Vo of 1.1 is d e n o t e d a(E).
1.3..Remark. T o see t h a t D e f i n i t i o n s 1 a n d 1.2 are worth-while, it is o f course essential t o show t h a t t h e r e are t v s E w i t h c~(E) ~ 2. As p o i n t e d o u t in t h e i n t r o d u c t i o n , it will be seen below t h a t a(.) m a y assume a n y ordinal v a l u e (see II.2.1 a n d ef. also S e c t i o n 7 o f I).
2. PnoPOSlTIO~. F o r tvs E and F and a continuous linear m a p p i n g f : E --> F , we have, for every ordinal v, that f(A~E) c A ' F . I f f is open and f-l(O) c A~~
f o r some ~o, we have f ( A ~ E ) = A ~ F for v <_v o.
Proof. W e see t h a t
f ( A E ) = f ( e N 9-1(0))
c : f ( 1 7 / - 1 V - l ( 0 ) ) = FI V-l(0) = A F ,
wEF' ~vEF'
~rhieh gives t h e f i r s t par~ in t h e case v = 1. I f f is open a n d f - l ( 0 ) ~ A E , e v e r y e l e m e n t in E ' is o f t h e f o r m ~0 o f w i t h ~ E F ' ; h e n c e we m a y replace t h e inclusion b y e q u a l i t y .
F o r a r b i t r a r y v, t h e relations are n o w o b t a i n e d b y s t r a i g h t f o r w a r d t r a n s f i n i t e i n d u c t i o n .
2.1. COnOLLAI~Y. Let E be a tvs. For every quotient space E / H , say, such that H C A~(E)E, we have o:(E/H) = o~(E).
OIq- T H E SEI:'ARATIOI~ I:'I~OPEI%TIES O F T H E D U A L S 281 2.2 COrOLLArY. A tvs E has a quotient space E / H with ~ ( E / H ) = v when- ever v ~_ a(E).
Proof. T a k e H = AVE.
3. P~OFOmTm~. A tvs E has precisely one subspace L C E such that (i) L' = { 0 } and
(ii) every subspace H c E / L , H =/= {0}, has H ' =/= {0}.
Proof. 1 ~ Uniqueness: Suppose (i) a n d (ii) to be satisfied for L = L 1 as well as for L = L0, say. I f L 1 c~ Lo, t h e subspace (L 0 + L J / L o o f E / L o is d i s t i n c t f r o m {0} a n d has a n o n - v a n i s h i n g e o n t i n u o a s linear f o r m f o n a c c o u n t o f (ii) for L ---- L o. T h e n f induces a n o n - v a n i s h i n g c o n t i n u o u s linear f o r m on L a, a n d t h u s raises c o n t r a d i c t i o n against (i) for L ~ L r H e n c e L 1 c L o, a n d similarly L o o t 1 .
2 ~ E x i s t e n c e : L e t L 0 be t h e linear hull o f all subspaces w i t h dual (0]. A g i v e n c o n t i n u o u s linear f o r m on L o vanishes on all subspaces w i t h dual {0], a n d t h u s on each o f a class o f sets s p a n n i n g L 0, a n d so on L o. H e n c e L0 = {0]. T h e s u b s p a e e L o is also t h e largest one w i t h dual {0}.
Suppose H is a n y s u b @ a c e ~: (0} o f E / L o. I t s canonical inverse image H 1 in E has a n o n - v a n i s h i n g c o n t i n u o u s linear f o r m
f
b y t h e p r e c e d i n g p a r a g r a p h (since H 1 \ L o 4= O). B u t f(Lo) = O, so f i n d u c e s a n o n - v a n i s h i n g c o n t i n u o u s linear f o r m on H = H1/L o. H e n c e (i) a n d (ii) are satisfied w h e n L ---- L 0.T h e p r o o f also gives t h e n e x t p r o p o s i t i o n (same n o t a t i o n ) .
3.1. 1)ROrOSITION. The subspace L is the largest one which has dual {0}.
3.2. THEOREM. When E and L as in 3, we have L = A~(E)E.
Proof. Since A~'(E)E clearly has dual {0], we get A~(E)E c L f r o m 3.1.
Conversely, we show b y t r a n s f i n i t e i n d u c t i o n t h a t L c A~'E for e v e r y v.
T h e assertion is t r i v i a l l y v a l i d for v = 1; t h u s a s s u m e t h a t L C A t E w h e n e v e r v' < v. B u t L c N A t E = P,, say, c e r t a i n l y implies f ( L ) = 0 w h e n e v e r f E P~.
H e n c e L c AVE b y D e f i n i t i o n 1.
3.3. PROPOSITION. When E and L as in 3, we have, for every v, that (A"E)/L = AV(E/L). Further, ~ ( E ) = o~(E/L).
Proof. Follows f r o m 3.2, 2, a n d 2.1.
3.4. Definition. A t v s will be called an A-space if e v e r y subspace d i s t i n c t f r o m {0} has dual distinct f r o m {0}.
282 nl~XlV ]~sR MATEMATIK. Vol. 9. NO 2
3.5. Remarlc. W h e n E a n d L as in 3, notice t h a t (i) E has dual {0} if a n d o n l y if E---L, a n d t h a t (ii) E is an A-space if a n d o n l y i f L----{0}.
F u r t h e r m o r e , L is t h e smallest subspace for which E / L is a n A-space. (Follows f r o m 2.)
W e f i n a l l y s t a t e y e t a n o t h e r c h a r a c t e r i z a t i o n o f L.
3.6. 1)ttOPOSITIO~. When E and L as in 3, every continuous linear mapping f : E - * - F into an A-space may be canonically decomposed according to
E ~ E / L h_~ F ,
where g and h are continuous and linear. ,Moreover, L is the largest subspace for which such a decomposition may be accomplished for all f and F.
_Proof. I f t h e image in F o f L is distinct f r o m {0}, it has a n o n - v a n i s h i n g c o n t i n u o u s linear f o r m ~, say. T h e n ~ o f does n o t v a n i s h on L, w h i c h is c o n t r a - d i c t o r y .
T h e last s t a t e m e n t n o w follows if we t a k e F ~- E / L a n d let f be t h e canonical m a p p i n g .
4. Consider an A-space E . I f E is n o t locally c o n v e x , a c o n t i n u o u s linear f o r m m a y of course n o t always be e x t e n d e d f r o m a subspace to E . H o w e v e r , if E has s e p a r a t i n g dual - - t h a t is, if a ( E ) ---- 1 - - a c o n t i n u o u s linear f o r m d e f i n e d on a finite-dimensional subspaee m a y always be e x t e n d e d t o E . B u t if a(E) ~ 2, n o t e v e n t h a t is t r u e ; j u s t let t h e g i v e n f o r m be d e f i n e d a n d n o n - v a n i s h i n g on a one- dimensional subspace o f A E . T h e b e s t we can accomplish in such a case is t o e x a m i n e w h e t h e r t h e f u n c t i o n a l m a y be e x t e n d e d t o c e r t a i n subspaces or not. T h u s we shall s a y t h a t a class ~r o f subspaces o f E is an extension-class if t h e following c o n d i t i o n is satisfied:
F o r e v e r y subspace K a n d n o n - v a n i s h i n g c o n t i n u o u s linear f o r m f d e f i n e d on it, t h e r e is a space C E<~ such t h a t
(i) f ( g [7 C) :4= {0} a n d
(ii) for e v e r y f i n i t e - d i m e n s i o n a l subspace F o f K [7 C, t h e r e s t r i c t i o n o f f to F c a n b e e x t e n d e d t o C.
Of course, t h e class o f all one-dimensional subspaces f o r m s a v e r y trivial e x a m p l e of a n extension-class. To o b t a i n s o m e t h i n g m o r e significant, we r e s t r i c t o u r a t t e n t i o n t o e x t e n s i o n classes t h a t are totally ordered u n d e r inclusion. I f a ( E ) is finite, t h e r e is such an extension-class, i.e., t h e class %~ ---- (A'E}~ O {E}. N a m e l y , if f a n d K are given, t a k e C ---- An~ w h e n n o is t h e least i n t e g e r such t h a t f ( K rl AnoE) =~
(0}.
B e f o r e t ~ f i n g t o a m o r e general situation, we i n t r o d u c e a n o t h e r n e w n o t i o n . W e s a y t h a t E is a n AP-space if e v e r y 0 - n e i g h b o u r h o o d contains all b u t f i n i t e l y m a n y o f t h e spaces o f ~:~. - - N o t i c e t h a t E m a y be an A P - s p a c e o n l y i f ~(E) _<
ON T H E S E P A R A T I O N P R O P E R T I E S OF T t I E D U A L S 283 a n d f'l A~E = {0}; a n d f u r t h e r , t h e t o p o l o g y on E m a y be c o a r s e n e d t o an A P -
V<e~
space t o p o l o g y i f a n d o n l y if t h e l a t t e r conditions are fulfilled. I f ~(E) is finite, E is t r i v i a l l y an A P - s p a c e .
F u r t h e r m o r e , let us b y an A'-spaee u n d e r s t a n d an A - s p a c e w h i c h has a 0-neigh- b o u r h o o d base o f w h i c h e a c h e l e m e n t U has t h e p r o p e r t y t h a t I"1 9 U is a linear
o > 0
subspaee. This l a t t e r c o n d i t i o n is n o t a u t o m a t i c a l l y fulfilled in a n o n locally c o n v e x tvs; h o w e v e r , it is, for instance, in t h e case o f a supremum topology of locally bounded topologies. (In v i e w o f 7.7 or II.2.2, it is clear f r o m t h e r e m a r k o f t h e p r e c e d i n g p a r a g r a p h t h a t t h e r e are, indeed, AP-spaces o f t h e l a s t - m e n t i o n e d k i n d w i t h a = co.) 4.1. I~ROPOSITm~r _For an A'-space E, the following statements are equivalent:
(i) E is an AP-space,
(ii) E has an extension-class that is well-ordered under ~ , and
(iii) E has an extension-class that is a decreasing sequence under inclusion.
Proof. (i) implies (iii): T h e a r g u m e n t a b o v e for ~(E) < ~ is applicable also in t h e case a ( E ) = ~. F o r suppose f a n d K t o b e given. T h e n f is b o u n d e d on t h e i n t e r s e c t i o n o f K w i t h some 0 - n e i g h b o u r h o o d U, say. Since E is a n A P - s p a c e , U contains An~ for some n o < w. T h e n f ( K f] A'~ ~ {0}, a n d we can p r o c e e d as before. So ~=~ is an extension-class.
(iii) implies (ii): Trivial.
(ii) implies (i): L e t (~ be a well-ordered extension-class.
A i m i n g a t a c o n t r a d i c t i o n , we a s s u m e E n o t t o b e a n A P - s p a c e . On a c c o u n t o f t h e well-ordering, t h e class ~ has t h e n , for e a c h n < oJ, a largest space Cn w h i c h does n o t c o n t a i n A~E. T h e n C, contains An+IE; otherwise, t h e defining conditions for a n extension-class w o u l d n o t be fulfilled if we let K be a o n e - d i m e n - sional subspace o f A~+IE n o t in C n. F o r if C~ ~ , t h e n C O has i n t e r s e c t i o n {0}
w i t h K - - a n d t h u s we h a v e f ( K f3 C o) = 0 for e v e r y f -- except i f C O ~ Cn # O;
a n d in t h a t case, we h a v e C O ~ A~E, so a n o n - v a n i s h i n g f on K C A~+IE can c e r t a i n l y n o t be e x t e n d e d t o C o .
Since E is a n A ' - s p a e e t h a t is n o t a n A P - s p a c e , we can n o w f i n d a 0-neigh- b o u r h o o d U such t h a t N = ['1 9 U is a linear subspace w h i c h does n o t c o n t a i n
Q > 0
A"E for a n y n < ~ . So we can t a k e e l e m e n t s z ~ E A ~ E ~ N , n < w ; t h e n we t a k e for K t h e linear hull o f N U {z,]n < ~}. W e m a y a s s u m e t h a t iV ~ ['1 C~.
(Otherwise, r e p l a c e U b y U + A C~.) F o r e a c h n < ~o, we d e f i n e a c o n t i n u o u s
r t < ~o
linear f o r m g~ on K so t h a t ga(Za) # 0 a n d so t h a t g~(K rl A=+~E) = 0; this is possible, since K f'l A"+~E has f i n i t e c o d i m e n s i o n in K . F u r t h e r , b y t h e choice o f U a n d z=, we see t h a t each o f t h e g= is b o u n d e d on U [ 3 K ; so t a k e Y ~ > 0 so t h a t y , lg=[ < 1/2 ~ on U fl K . B y t a k i n g f = Zn~'ng,t o n K , we shall h a v e o u r
284 AI~KIV FOtt MATEIVIA_TIt~. Vol. 9. No 2
c o n t r a d i c t i o n against t h e defining p r o p e r t i e s o f an extension-class, l~irst, f is con- tinuous, since [fl --< 1 on U rl K . T h e n , on a c c o u n t o f f ( N C,) = 0 (seen f r o m
n ~ c o
f(TV) = 0, which follows f r o m If] --< 1 on U Cl K), it is sufficient t o show t h a t f c a n n o t be e x t e n d e d to C O ~ C,, say, f r o m t h e one-dimensional subspace s p a n n e d
b y z , + ~ E C , f l K . This is so, for C ~ a n d z,+TEA"+TE.
T o i l l u m i n a t e t h e s i t u a t i o n s o m e w h a t , we s t a t e t h e following simple proposition.
4.2. 1)~OPOSITm~. For a totally ordered extension-class ~ of an A-space E, we have c a r d ~ > c a r d m i n {c~(E), co}.
4.3. Remar]c. T h e n o t i o n o f A P - s p a c e clearly has t h e following significance.
A n A - s p a c e E is a n A P - s p a c e if a n d o n l y if it is t h e inverse limit o f some A-spaces w i t h f i n i t e c~-values (namely, E/A"E w h e r e n < o); cf. 2 above).
5. F o r a tvs E , it is n a t u r a l t o ask w h e t h e r t h e A'E-spaces m a y be c h a r a c t e r i z e d in t e r m s n o t involving these spaces. A f i r s t step in this d i r e c t i o n was t a k e n in 3.2 a b o v e , w h e r e we c h a r a c t e r i z e d A~(~)E. Also in 4 - - 4 . 1 o u r p u r s u i t was t o enlighten this question; n a m e l y , t h e A P - s p a c e p r o p e r t y , w h i c h is d e f i n e d as a p r o p e r t y o f t h e class {AYE},, is t h e r e r e l a t e d t o t h e existence o f c e r t a i n extension-classes;
a n d t h e l a t t e r n o t i o n is d e f i n e d otherwise. A n d n o t i c e t h a t t h e case a(E) = ~o t h e r e a p p e a r s as >>criticab). - - T h e r e s t o f t h e q u e s t i o n we h a v e t o leave as an o p e n p r o b l e m .
6. This is a n a u x i l i a r y section.
6.1. Definition. L e t E be a v e c t o r space. I n a c c o r d a n c e w i t h L a n d s b e r g [3], we s a y t h a t
a) a f u n c t i o n a l x --> Ilxll is a T-norm, w h e r e 0 < p _~ 1, if it satisfies (i) I[x + yll -< [[xl[ ~- HyI[,
(ii) II~xl[ = [~IPtlzII, a n d
(iii) Hx[I = 0 implies x = 0, w h e n e v e r x , y E E a n d A scalar;
b) if a p - n o r m [I'll on E is given a n d i f t h e r e b y E is c o n s i d e r e d as a t v s w i t h 0 - n e i g h b o u r h o o d base {{xllIx H < A}I~ > 0} (this is c e r t a i n l y a 0-neighbour- h o o d base o f a v e c t o r space t o p o l o g y ; cf. L a n d s b e r g [3]), t h e n E is a p-normed 8pace.
6.2. Examples. W e r e m i n d of some c o m m o n e x a m p l e s o f p - n o r m e d spaces:
a) t h e spaces I p o f sequences x ~ (41, 4 2, . . . ) such t h a t ~ 14k[ P < oo, w i t h T - n o r m [IxIl = ~]~ [4kl p. F o r P < 1, w e h e r e h a v e simple e x a m p l e s o f n o n locally
c o n v e x spaces t h a t are d u a l - s e p a r a t e d ( T y c h o n o f f [7]; cf. also L a n d s b e r g [3]).
b) t h e spaces LP(0, 1) o f m e a s u r a b l e f u n c t i o n s x(~) on (0, 1) such t h a t
O N T H E S E P A R A T I O N P R O I ~ : E I ~ T I E S O F T I I E D I J A L S 285
f
l0lx(~ ) led~ < ~- oo (or, properly, certain equivalence classes o f such functions)s
w i t h p - n o r m Iixll-~ T h e Le(O, i) with p < I are k n o w n not to
h a v e a n y non-vanishing continuous linear forms ( D a y [I]).
e) the q u o t i e n t space E / H of a given p - n o r m e d space E w i t h p - n o r m H , say, for a closed snbspace H . T h e n t h e q u o t i e n t space t o p o l o g y is given b y t h e q u o t i e n t space p - n o r m II)ll~ -- inf{lly]]y E x + H}, if t h e canonical m a p p i n g E --> E / H is w r i t t e n x --> x. tX
Also, we introduce:
d) the p - n o r m e d direct s u m of a class {E,} of p - n o r m e d spaces w i t h p - n o r m s {l['ll,}, say, as a p-normed space S as follows. Consider t h e algebraic direct s u m (~, E~ of t h e linear spaces E,, i.e., t h e linear space of all formal finite sums ~.~ x , such t h a t x~ C E, a n d all b u t f i n i t e l y m a n y of the x, are zero. T h e n t h e p - n o r m e d space S is defined as the linear space 0 , E, e n d o w e d w i t h t h e p - n o r m II']l given b y II~, x, lI = ~ IIx,[[~-
6.3. Example. We also recall t h e space S(O, 1) of measurable functions on (0, 1) w i t h t h e topology of convergence in measure. This t o p o l o g y is also given b y t h e metric jr1 I x ( ~ ) - y('r)] d r . I t is well this has dual d(x, y) = 1 @ [x(~) -- Y(~)I k n o w n t h a t s p a c e
0
{0} (cf. [5]).
6.4. W e n o w introduce some n o t a t i o n to be used henceforth. B y co X we denote t h e convex hull of a set X in a linear space E; a n d lin X will be t h e linear hull. I f E is a t v s , co X a n d lin X will s t a n d for t h e closed convex resp. linear hull. F u r t h e r , we will m a k e extensive use of t h e following simple consequence of H a h n - B a n a c h ' s theorem:
A point in a tvs E belongs to the convex hull of every O-neighbourhood i f and only if it is not separated from 0 by E' (LaSalle [4]).
Otherwise stated: L e t 9~ he a 0-neighbourhood base in E. T h e n [7 co U---- A E .
U~B
I n this connection, notice t h a t Definition 1 above m a y t h u s be rephrased in t e r m s of convex hulls of 0-neighbourhoods in subspaces:
6.5. PROPOSITION. Let E be a tvs. For a class ooi of subsets of E, write iced for the intersection of the convex hulls of the sets of d . Further, let ~ be a O-neigh-
bourhood base of E ; and define inductively
286 AYLKIV ~51~ MAT:EMATIK. Vol. 9. :No 2 icn 1 = i c ~
icn ~ : ic{ U f3 icnr l U C ~ for all ordinals v. T h e n
i e n ' = A~E for every ordinal v.
, ~ ' < v }
Proof. B y transfinite induction.
7. We now t u r n to the problem of finding tvs E with a(E) > 2. First we prove a general assertion on sufficient conditions for a given tvs to have a sub- space with a-value > 2.
7.1. T ~ E O R ~ . Let E be a metrizable tvs and K c E a closed subspace which E ~
is separable. S u p p o s e there is a decreasing sequence { i}i=l of closed subspaees of E such that
co
(i) f~ E~ = K
i = l
(ii) AE~ D K , i > l .
T h e n E has a closed separable subspace K 1 D K such that A K 1 = K .
I n particular, i f K is dual-separated, we get K 1 with a(K1) = 2; and, more generally, i f a ( K ) is finite, we get a(K1) = a ( K ) -~ 1.
Proof. Let {Vk}k~=l be a countable 0-neighbourhood base of E and {Zk}k~=l a sequence which spans K. B y condition (ii) and 6.4 above, we can take xq E E for i _> 1 and 1 < j < k~, say, so t h a t
1~ xq C E~ fl V~
2 ~ z,,Eco(xq}~=l, 1 < n < i , k i
for each i > 1 and suitable kl.
How t a k e K 1-~ lin ({xq} U K). B y construction, every point of K is in the convex hull of every 0-neighbourhood (for {zn} spans K and {Vn} is a base);
so b y 6.4, K C AK1. On the other hand, every point x in K1 ~ K is outside AK1; for, b y (i), the point x is outside E~ for some i > 2. B y construction, Ei/3 K1 has finite codimension in K1; so the dual of K~ separates x from 0, as required.
We will make particular use of a special case:
7.2. COROLLARY. I f E is a metrizable tvs and K a closed separable subspace which has separating dual and which is intersection of a decreasing sequence of sub- spaces E 1 D E 2 D . . . . each of which has dual {0}, then there is a closed (separable) subspace K 1 such that A K 1-~ K (and thus a ( K 1 ) ~ - 2).
O1~ T H E S E P A t C A T I O I ~ P R O P E R T I E S O F T t t E D U A L S 287 7.3. COROLLARY. Let E be a metrizable ,tvs such that for every closed subspace K which is an A-space, there is an (isomorphic) imbedding i : K - - > E such that E satisfies the conditions of 7.1 with i(K) in the place of K. Then, for any n < ~o, the space E has a closed subspace K~ with A ' K , one-dimensional (and thus o~(K~) =
n + 1).
Proof. Start with a one-dimensional subspace K and find K1; then take K 1 in the place of K to find Ks; and then go on step b y step.
7.4. Example. Consider the space S(0, l) of measurable functions (cf. 6.3 above).
We can then apply 7.2 in the case when K is the one-dimensional space of all constant functions. Namely, let E k be the subspace of all periodic functions with period 2 -k (/c ~ 1). Clearly, K is the intersection of all the Ek, and each Ek has dual {0} (since E k is essentially S(0, 2-k), which space is isomorphic to S(0, 1) and thus has dual {0}). Hence S(O, 1) has a closed subspace K 1 with A K 1 one- dimensional (and thus ~(K1) = 2).
(See also 7.9 below.)
7.5. Example. Similarly, LP(0, 1) with 0 < p < 1 has a closed subspace K 1 with A K 1 one-dimensional.
7.6. Examples (continuation). We will now proceed and show t h a t also the conditions of 7.3 are fulfilled for E = S(0, 1) [resp. LP(0, 1)].
T o that end, first notice that (0, I) h a s the s a m e m e a s u r e space structure as 12 = (0, 1)X(O, 1); so S(0, 1) [resp. LP(0, 1)] m a y be identified vcith the cor- responding space S ( I 2) [resp. LP(IP)] of measurable functions on 13. Thus, we can define the required imbeddings i : K - - > E of 7.3 as imbeddings S(0, 1) K • S(1 ~) [from now on we omit the phrase ~>resp. (corresponding for) Le~> after statements like this one].
So let K be a closed subspace of S(0, 1). We will find i : S(0, 1) --> S ( I 2) and subspaces E 1 D E 2 D . . . of S ( I 2) such t h a t their intersection equals i(K) and such t h a t each of them has dual {0}. Consider the elements of S(0, 1) as functions f(O) with 0 ranging over (0, 1) and the elements of S(F) as functions f(O1, 02) with 01 and 02 ranging over (0, 1). (Thus, we denote the coordinates of 12 b y
01 and 03 resp.) Whatever K m a y be, we define the imbedding i b y m e a n s of S(0, 1) = g ~f(O) J ~ f(01) e S(I2) ;
thus, we let i range in the subspace of functions t h a t are constant in the variable 03. Clearly i is an isomorphic imbedding. I t remains to give the Ek, which is done b y
Ek = lin {g(01) 9 h(02)[g e K ; h is periodic with period 2-k}.
288 AI~KIV F61~ 2'r Vol. 9. No 2
T h e n Ek has dual {0} (cf. 7.4); we m u s t show t h a t r l k E k = i(K). F i r s t , b y definitions, t h e intersection contains i(K); conversely, a n element f0(01, 02) o f t h e intersection will be shown to belong to i(K). Since a n y f u n c t i o n of E k has period 2 -4 in 02 (for almost all 01-values), t h e f u n c t i o n fo(OD 02) is essentially c o n s t a n t w i t h respect t o 02; so write fo(01, 02) = go(01). T h e n go -- regarded as a n element o f S(0, 1) -- is to be recognized as a n element of K . I t can be, for i f f(01, 02) belongs to E 1, say, then, for almost e v e r y f i x e d 0 ~ t h e f u n c t i o n 01-->
f(01,0~) - - regarded as an element of S(0, 1) -- belongs to K ; in particular, g o E K .
Thus, 7.3 applies a n d gives:
_For every positive integer n, the space S(O, 1) [resp. LP(O, 1)] has a closed sub- space K~ with A ~ K , one-dimensional (and thus ~ ( K ~ ) = n + 1).
7.7. Examples (continuation). We call n o w easily t a k e one more step: First, notice t h a t t h e spaces Sk ~ S(2 -k, 2-~+~), where k > 1, m a y be regarded as subspaees of S(0, 1); further, there are n a t u r a l continuous a n d open projections zk : N(0, 1) --> S k (defined as t h e restrictions of functions in S(0, 1) to (2 -k, :2 -k+l) [resp. corresponding for LP]. B y 7.6, t a k e for each /~ > 1 a space K2 ~ Sk such
k o
t h a t A Kk is one-dimensional; a n d let K ~ c S(O, 1) be t h e closed linear hull of t h e K2. Now, proposition 2 -- applied to ~k a n d to t h e i m b e d d i n g S k - + S(0, 1)
says t h a t ~k(A~K ~ ~ ~
-- ~ A K k for all v a n d k. This gives on one h a n d A ~ K ~ {0} for n < co, a n d on t h e other h a n d N n < ~ A " K ~ C n k ~k-l(0) = {0}. Thus we f i n d t h a t S(0, 1) [resp. LP(O, 1)] has a closed subspace K ~ with o~(K ~ = co.
7.8. _Remark. W h y are we p a r t i c u l a r l y interested in finding closed subspaces in 7.1--7.77 -- Because these are complete w h e n we s t a r t w i t h a complete space E (such as S(0, 1) or LP(0, 1)); a n d in general, ~(.) is not invariant under completion.
(See II.2.3 below.)
7.9. Remark. Consider t h e situation in 7.4. I t m i g h t be interesting t o see explicitly w h a t t h e f o u n d subspace K 1 m a y look like in this particular case. To t h a t end, let us go b a c k a n d e x a m i n e t h e p r o o f of 7.1. W h a t we w a n t is t h e n to give elements xq which are of t h e kited m e n t i o n e d there a n d which thus, t o g e t h e r w i t h K , span
K1.
Noticing t h a t K is g e n e r a t e d b y t h e c o n s t a n t f u n c t i o n 1 on (0, 1), we t h u s need functions xii(~) on (0, 1) which satisfy
(1') x~ i e El
(1") Xq-+ 0 ( i n d e p e n d e n t l y of j) as i--> ~ (convergence in measure) (2) 1 Eco{xii}i for e v e r y i ~ l ,
where t h e E~ are as d e f i n e d in 7.4. Denoting b y
ON- T I I E S E P A R A T I O N P R O P E R T I E S OF T H E :DUALS 2 8 9
N + I sin 2 - - v
1 2
- - , N > I
F N ( T ) N + 1 T - - '
sin e -- 2
t h e Fej6r kernel on the unit circle, we claim t h a t we m a y take
, ,
X q ( T ) = f ~ \ 2 ~ \ 2 ' ~ : @ , 1 ~ j ~ i , i 2 1.
Namely, (1') and (1") follow from well-known properties of the lq'ej6r kernel;
for (2), notice t h a t we have 1 = (1/i)(xa -[- 9 9 9 -]- xu) from the well-known relations
2 N
Fs(~) = 1 - 4 - N @ 1 ~ ( N + 1 - - n ) cosnT and
j = l
We have shown t h a t the space K 1 found in 7.4 [resp. 7.5] m a y be the space lin 1, F~ 2~ 2~W ~- / ~ l, 1 _~ j _< i .
I I . C o n s t r u c t i v e m e t h o d s
I n this chapter, we shall first give a general m e t h o d to construct p-normed spaces with certain properties. Then we will use this to get the announced example which will show t h a t definitions I 1 and I 1.2 are meaningful for all ordinals. The constructive m e t h o d to be employed will be presented in a somewhat stronger form t h a n required for our chief purpose. For the sake of completeness it m a y be remarked, t h a t to fulfil this purpose, we could have got away with a simpler but less illuminating device.
1. Suppose there to be given, for a fixed number p (0 < io < 1) (i) a set ~g of p-normed spaces,
(ii) for each E E ~ g , an element
eEEE,
(iii) a metric space M (with no linear structure), and
(iv) a function
}P:~--->M•
such t h a t d(prl~(E), p r 2 T ( E ) ) = ]]e~[l, and such t h a t I m ( p r l T ) U I m ( p r e T ) = M,where
prk:MxM--->M
is the projection onto the first resp. second factor and where d(., .) is the distance on M.290 ARKIV F6R MATEYIATIK. Vol. 9. No 2
W e i n t r o d u c e t h e following n o t a t i o n . F o r (m',m") C M x M , we w r i t e -- (m', m") = (m", m'). F u r t h e r , b y a cycle we u n d e r s t a n d a f i n i t e sequence o f ele- m e n t s in M • M o f t h e f o r m
(m o, ml), (ml, m2), (m2, m3) . . . (m,_l, mn), (mn, m0).
W e shall n o w d e f i n e a space H(Cg, {eE}, M, T), w h i c h we also d e n o t e 11(~) or H w h e n t h e r e is n o risk for confusion. ( F u r t h e r , if ~g' c ~E a n d if M ' = I m (prl T ) U I m (pr2 ~ ) , we d e n o t e also H(~g ', {eE}, M', T ) b y H ( ~ ') - - in spite o f t h e change f r o m M t o M ' . )
1.1. PROPOSITION. Given the objects of (i) to (iv), there is a unique complete p-normed space 11 ~ II(C~, {eE}, M, ~J) such that
1 ~ for each E C (g, there is an isometric imbedding i E : E - - > 171, 2 ~ there is an injection j : M - - > I I related to the iE as
iE(e~) ---- j(pr2 T ( E ) ) - - j(prl T ( E ) ) , and
3 ~ if 111 is an arbitrary complete p-normed space satisfying 1 ~ and 2 ~ with i 1 E
and j l , say, in the places of iE and j resp., there is a linear mapping ~ : 11--> 111 for which the diagram
E
H ) H 1
is commutative for each E C cc:, and which is continuous with operator p-norm not larger than one (i.e., supEl~ll= 1 ll~b(x)ll _~ 1). (Universal property.)
[Concerning 3 ~ also notice: if I m T is c o n n e c t e d as g r a p h (cf. 1.4), t h e d i a g r a m M
H ~b ) H1
is c o m m u t a t i v e u p t o t r a n s l a t i o n (i.e., jx ~ ~b o j ~- c for some c o n s t a n t e l e m e n t c in HX); this follows f r o m t h e condition t h a t t h e i~ a n d j l s a t i s f y 2~
B e f o r e p r o v i n g 1.1, we s t a t e some p r o p e r t i e s o f / / w h i c h will be consequences o f t h e c o n s t r u c t i o n .
O N T H E S E P A R A T I O N P R O P E I ~ T I E S O F T H E D U A L S 291 1.2. I)ROPOSITIO~. With notation and assumptions as in 1--1.1 and with E~, E 2 C ~ , we have
~{0} i f T(E~) =~ T(E2) and =~ -- ~J(E2) i(E1) N i(E2)
[lin {eE,} i f T ( E ~ ) = T(E2) or -~ -- T(E2).
1.3. 1)~O~OSITIO~. With notation and assumptions as in 1--1.1 (notice especially the last paragraph before 1.1), let ~d' be a subset of ~, consisting of all except finitely many of the elements of ~ Then there is a canonical imbedding g (which is iso- morphic but in general not isometric) of II(C8 ') onto a closed subspace of TI(~8).
(~)Canonicab) here means t h a t the diagram E
- C C /
1I( ~, ) > I m Z C 11(~8) Z
is commutative whenever E e ~;'.)
1.4. Remark. I n the sequel, we will not distinguish between E and i(E). Thus, the spaces of c~ will be regarded as subspaces of / / .
Notice t h a t it is an immediate consequence of the universal property (3 ~ of 1.1) t h a t
lin {i(E)IE 6 c8} = / / .
Intuitively, we m a y think of I m k~ as the set of edges of a graph in M. W h a t we intend to do is to paste the spaces of c+~ at this graph by identifying the vectors eE with the (oriented) edges T(E).
Notice t h a t in general the injection j : M -+ H is not isometric; however, we know t h a t the distances d(prl T(E), pr~ T(E)) are preserved, since these equal t h e p-norm values
Ileal] (of.
(iv) of 1 and 1 ~ of 1.1).1.5. Proof of 1.1--1.3. To define TI(CS, {eE}, M, T), we start with the p-normed direct sum S (cf. I 6.2.d) of the spaces of c8. Consider the subspace H c S given b y
H = lin { ~ ekeEk]elT(E1) , . . . , snT(En) is a cycle for some s k = ~= 1, n _~ 2}. (a)
k ~ l
Then we take H as the completion of the p-normed quotient space S/H.
292 AttIs FSR 2r Vol. 9. ~No 2
To give t h e p - n o r m ]I'll~ on S / H more explicitly, we denote b y x t h e canonical image in S / H of a n element x E S. (This tilde n o t a t i o n will n o t be used except in this proof.) T h e n I['II_ is d e f i n e d b y
]]~xk[ I = i n f (~ltyk][ ] ~ x k ~- ~Yk) (b)
where t h e xk a n d yk belong to different spaces of c3, a n d where t h e i n f i m u m is t a k e n over all preimages of ~ x ~ u n d e r t h e canonical m u p p i n g S ----> S/H. (Cf.
the definitions in I 6.2.)
F r o m this construction, we see a t once h o w to define t h e iE o f 1~ for each E C ~;, let i~ be t h e composition of t h e inclusion m a p E --> S a n d t h e canonical m a p p i n g S ~ S/H. Before t h e s o m e w h a t l e n g t h y verification t h a t these iE are isometric, we prove t h e other s t a t e m e n t s .
First, 1.2 follows f r o m this definition; for, t h a t siT(E1), s2T(E~) is a cycle m e a n s precisely t h a t sl~rl(E1)= --s2~(E2). B y t h e definition of H , it m e a n s also precisely t h a t sle~, = --S2eE. A n d (a) f u r t h e r shows t h a t for xk E Ek, we can h a v e x l = x 2 only if x k = • ~e~ k for k = 1 , 2 a n d some ~.
Second, for 1.3 we consider t h e case w h e n ~ ~ ~ ' is a singleton {E0}, say;
t h e general assertion t h e n follows b y induction. L e t So be t h e snbspace of S which is s p a n n e d b y t h e spaces of ~ ' , a n d let H 0 C So be t h e subspace defined b y (a) w i t h t h e spaces El, . . . , E , ranging over ~ ' . D e n o t e b y q~ : S ~ S / H t h e cano- nical mapping. Since H 0 = S o n H , t h e i d e n t i t y m a p p i n g S----~S induces a bijective linear m a p p i n g Zo:So/Ho--~ ~(So) , which will be shown to be a n iso- morphism. I f x C S o a n d i f 7x a n d H'H= is t h e canonical image of x in, resp. t h e q u o t i e n t p - n o r m on, So/Ho, we h a v e b y (b)
l l x l l = inf( ~. ]]Ykl] ]Yk in spaces of ~ ) a n d
x ~ - ~ Y k rood H
IIx[I z = inf( ~ ]lYkll I Y~ in spaces of ~ ' ) .
x ~ X y k rood//o
Since t h e first i n f i m u m is t a k e n over a larger set, it is smaller, so ]]s < [[x][=.
Conversely, let Yk be vectors in spaces of ~ so t h a t x ~ ~. Yk m o d H . Two cases m a y occur: (1) all Yk are in spaces of ~g', a n d t h e n x ~ ~ y k m o d H o (since H 0 = S 013H); a n d (2) ~ Y k = S 0 ~ - A e a , where s 0 e S 0 a n d A 4 0 a scalar.
I n ease (2), ~v(E0) m u s t be an element of a cycle ~ of which all t h e other elements are in 4- T(~;'). (The last s t a t e m e n t s are consequences of the definition o f H.) So
So 4- ~eE0 ~ So + A ~ ekeE~ rood H ,
~k w(ED e J.
E k ~ E 0
wherein t h e second m e m b e r is an element in So which is congruent m o d u l o H o to x (since H o = S o 13 H). L e t t h e s u m in t h e second m e m b e r h a v e t h e length clIeEoll. Then, for a n y x C So, a n d ~Yk ~ x m o d H t h e second m e m b e r has length
O N T H E S E P A R A T I O N P R O P E R T I E S O F T H E D U A L S 293 at m o s t c t i m e s t h e l e n g t h of t h e first m e m b e r . S o in case (2) as in ease (I), it is seen t h a t for e v e r y x ' ~ x m o d H , there is x " E S O w i t h x " ~ - x m o d H o a n d IIx"II <_ c'[Ix'll for some c o n s t a n t c ' > 0 (which is i n d e p e n d e n t o f x); a n d so I1"1I: _< c ' i l l l 9
T h e n we e x t e n d ~/o b y c o n t i n u i t y t o a n i m b e d d i n g Z : H(cE ') --> H ( ~ ) ; clearly
H ( ), for is
I m Z is closed in cg H(Cs ') complete.
Finally, we v e r i f y l ~ ~ of 1.1.
F o r 2~ T a k e j(mo) = 0 for some m o C M. T o define j for t h e o t h e r p o i n t s in M , we use t h e following n o t a t i o n : B y a T-path f r o m m 1 t o mn we u n d e r s t a n d a sequence
(ml, m2), (m2, ma) . . . . , (mn_2, ran_l), (ran_l, ran)
where all t h e (ink, ink+i) belong to I m T U ( - - I m T ) a n d w h e r e n > l . L e t m C M be a n y e l e m e n t for which t h e r e is a T - p a t h f r o m m o t o m - - or, as we can say, t h a t lies in t h e same T-path-component o f M as m 0. L e t e , T ( E 1 ) , . . . , e~T(E~) (ek = i 1, n _> l) be a T - p a t h f r o m m0 to m. T h e n we d e f i n e j(m) = ~ = 1 ekeE k.
This d e f i n i t i o n is consistent. For, let s11T(E~), . 9 . , s , I T ( E , , ) be a n o t h e r T - p a t h 1 1 f r o m n~0 t o m; t h e n
s i T ( E 1 ) , 9 .. , s n T ( E ~ ) , - - s~T(E~) ~ 1 . . . -- s , , , T ( E , , , ) 1 (c) is a cycle, so b y c o n s t r u c t i o n (cf. (a) above) we h a v e
n~ 1
8 k % - - ~ 8neE~ k = 0 . (C')
1 1
This r e l a t i o n says t h a t t h e d e f i n i t i o n o f j(m) is i n d e p e n d e n t o f t h e choice o f p a t h f r o m mo t o m. So, if C o is t h e T - p a t h - c o m p o n e n t o f m o in M , we h a v e con- sistently d e f i n e d Jco : Co---> Tl -- t h e r e s t r i c t i o n t o C o of t h e r e q u i r e d j . A n d jco is injective; for if j(m') = j(m") a n d i f s i T ( E l ) , . . . , s~T(E~) a n d siT(E1)1 1 . . . . ,
1 1 tt
s,#T(E~) are T - p a t h s f r o m m 0 t o m' resp. m , r e l a t i o n (e') is satisfied. B y c o n s t r u c t i o n , (c) m u s t be a cycle, w h i c h m e a n s t h a t t h e t w o p a t h s go f r o m m o to t h e same p o i n t , i.e., m ' = m".
W e h a v e t o e x t e n d Jc~ to all o f M . So, for each T-path-component C we define a n i n j e c t i o n j c : C - - ~ I I in precisely t h e same m a n n e r as before 9 B u t t h e d e f i n i t i o n o f / / (cf. (a)) shows t h a t for a suitable class o f subspaces each t w o o f w h i c h m e e t o n l y in {0}, each space o f t h e class contains precisely one set I m j c as a p r o p e r subset. So b y composing each Jc w i t h a suitable t r a n s l a t i o n , we get i n j c c t i v e m a p p i n g s Jc w i t h disjoint images; these t o g e t h e r d e f i n e a n i n j e c t i o n j on all of M .
F o r 3~ T h e m a p p i n g s i~ t o g e t h e r i n d u c e a linear m a p p i n g ~ : S --> H 1 (defined b y il--(~Xk)=~i~k(Xk) for ~ x k C S , xk EEk). F i r s t n o t i e e t h a t ~ i s c o n t i n u o u s w i t h o p e r a t o r p - n o r m a t m o s t l, for
294 AttKXV 1~61~ MATEMATIK. Vol. 9. No 2
where ~x~ C S, w i t h xk E E k a n d t h e spaces Ek distinct.
Now, if sxT(Ex) . . . . , s,T(E~) (sk = ~ 1, n >_ 1) is a cycle, we h a v e for a n y m o E M occurring in a n y T(Ek) ,
0 = ix(too) -- jl(mo) = ~ ej~k(eEk),
k = l
since t h e i~ a n d j l are s u p p o s e d t o satisfy 2 ~ T h u s , a n y e l e m e n t in t h e set spanning H a n d given in t h e second m e m b e r o f (a) a b o v e is m a p p e d t o 0 b y il; h e n c e iX(H) = 0. T h e m a p p i n g ~b : II----~II 1 is t h e n d e f i n e d as t h e m a p p i n g which is i n d u c e d b y iL F r o m t h e d e f i n i t i o n o f q u o t i e n t p - n o r m we see t h a t ~b has o p e r a t o r p - n o r m a t m o s t 1, for ~-x has. T h e c o m m u t a t i v i t y o f t h e diagrams, finally, is an i m m e d i a t e c o n s e q u e n c e of t h e d e f i n i t i o n o f r
F o r 1 ~ at last, let x be an e l e m e n t in E 0, say, E 0C~2. B y definition, [lxl[_ ~ Ilxl]. T o show t h a t ]]x]l >-- ][x]l, on t h e o t h e r h a n d , let an a r b i t r a r y c~nonical pre-image o f ~ in S be w r i t t e n
x ' = y ~ - 2 r ,
w h e r e y = x - - Xe 0 (notice t h a t y e E0), 2 is a scalar, a n d r = ~ r e z ~ F e F ~ e o m o d H , for a suitable f i n i t e subset ~ C ~c~{Eo} a n d suitable n u m b e r s y~, n. W e m u s t t h e n show t h a t ]]x'[] > ][x][. N o t i c e t h a t , b y definition, ][x'[l --~ ]]Yll ~- l ~ l ~ l l r l l 9 Now, b y t h e d e f i n i t i o n o f H , we k n o w t h a t r is a linear c o m b i n a t i o n of n, say, sums
n k
/ = 1
such t h a t
-- T(Eo), ~klT(Ekl) . . . . , ~k%T(Ek%)
is a cycle ~ k for each k. Or, if T(Eo) = (mo, ml) say, we m a y express this in t e r m s of t h e n o t a t i o n o f t h e p r o o f o f 2~ for e v e r y k, ~ k l T ( E k l ) , . . . , ek%T(E~n k) is a T - p a t h f r o m m 0 to m r Also notice t h a t e 0 = - ~ t ek~ekt m o d H . To a v o i d trivial complications, we e x c l u d e t h e case w h e n some Ekl equals E o (notice t h a t E o ~ ~3).
W e f i r s t t r e a t t h e case w h e n t h e r e is o n l y one such s u m (i.e., n = 1) a n d w h e n no t w o t e r m s o f this s u m are multiples o f the same v e c t o r eE. I n o t h e r words, we suppose t h a t r --~ s 1 a n d t h a t T(Ell. ) # • T(Elt~ ) w h e n l' # l". A s s u m p t i o n
(iv) a n d t h e triangle i n e q u a l i t y in M give
n~
ILx'II = Ily]L § lXlPllr]l = ]ly/I-~ ]A[P~ II%1tl]
l = l n l
--~ ]]Yll 9 ]~l p ~ g(Prx T(Ex,), pr~ T(Elt))
l = l
O1~ T H E S E P A R A T I O N P R O P E R T I E S O F T H E D U A L S 295
I t 9
= Ilyll + I~[P(d(mo, ml) -~ d(m 1, ms) -~ . . . +
d(m'n,_l, ml)), suitable ink; t mo, m 1 as a b o v e IJY[I § I~]Pd(m0, rex) = I[YI] § ][~e01l
[]Y + ~e01l = Ilxll,
since, b y a s s u m p t i o n , ~ilT(E11) . . . . , e l n T ( E l , , ) is a U - p a t h f r o m m o r m l ; so t h e assertion follows in this case.
I n t h e general case (n > 1), we w a n t t o r e d u c e t h e p r o b l e m t o t h e case n = 1.
T h a t is, i f x ' = y ~ ~r is a n a r b i t r a r y canonical p r e - i m a g e in S o f x, we w a n t t o f i n d a n o t h e r p r e - i m e g e x" = y ~- ks', w h e r e s' is a s u m o f t y p e (d), such thaV ]]x'll ~ [Ix'][. Since we a l w a y s h a v e ][y -~ 2rl] =
IlylI § IAIP]IrI[
(by t h e d e f i n i t i o n o f H'H o n S), this m e a n s t h a t we m u s t f i n d s' w i t hlls'll
_<llrII.
This is done in five.steps.
F i r s t step: W e show t h a t if we are g i v e n r ~-- e o rood H a n d write r = ~ F e F Y~eF as a b o v e , t h e n there is a U-path from m o to ml, consisting of elements in 4- T ( ~ ) only. This is n o t quite obvious; t h o u g h we k n o w - - f r o m d e f i n i t i o n o f sk - - t h a t t h e r e arre such U - p a t h s consisting o f e l e m e n t s in ~- T({Ekz}~l). F o r , one m i g h t ask: can it h a p p e n t h a t so m a n y eEk-terms are cancelled in t h e linear c o m b i n a t i o n o f t h e s k t h a t a U - p a t h o f t h e r e q u i r e d k i n d is impossible (since we can t a k e s o small t h a t YF ~= 0 for e v e r y Y)? W e will show t h a t i t c a n n o t .
W e w r i t e r = ~=10ksk. T h e n
~
0k = 1 , (e)k = l
for r ---~ sk ~ e o m o d H . T o proceed, we n e e d a simple g r a p h - t h e o r e t i c device. L e t Z b e a subset o f M , a n d t a k e ~ = {(m',m")[m' e Z a n d m" E CZ}. ( T h u s , consists o f t h e edges c o n n e c t i n g Z a n d C Z in t h e g r a p h d e f i n e d b y I m T a n d m e n t i o n e d in 1.4 a b o v e . F u r t h e r , t h e o r i e n t a t i o n o f t h e edges o f ~ is in t h e d i r e c t i o n o u t f r o m Z.) F o r e a c h cycle ~-5~k (defined in c o n n e c t i o n t o (d)), we i n t r o d u c e a scalar v a l u e d f u n c t i o n fk d e f i n e d on Z. I f a n e l e m e n t z C ~ occurs r t i m e s in
~ k , a n d if - - z occurs s times, we d e f i n e f (z) = (r - s)O ;
n o t i c e t h a t this s i t u a t i o n m a y b e described i n t u i t i v e l y as follows. W h e n we go a r o u n d one r o u n d in ~ k (in t h e n a t u r a l direction), we pass s t i m e s into Z t h r o u g h z, a n d r t i m e s o u t o f Z t h r o u g h z. A n e l e m e n t a r y c o n s i d e r a t i o n shows t h a t
5fk( ) = 0 ze2
(for, loosely spoken, w h e n we m a k e o u r r o u n d in ~ k , we m u s t , t o t a l l y , pass into, Z precisely as m a n y t i m e s as we pass o u t t h e r e of). ~ o w d e f i n e f(z) = ~ = l f k ( z ) ; s u m m i n g t h e relations j u s t s t a t e d , we get
2 9 6 ARXIV FOR MATEMATIK. V o l . 9. N o 2
f ( z ) = 0 . (f)
z e 2
W e will n o w use this result. Assume, for a while, t h a t k~ is a n injection. T h e n , i f T ( E ) = ~z, w r i t e e~ = ees (where E E ~ , e = - } - 1 , z C Z). B y definitions, we see t h a t t h e p r o d u c t o f Ok a n d t h e coefficient o f e', in sk is precisely fk(z);
a n d t h e coefficient o f e'= in r is f ( z ) (or m o r e strictly, r : ~ y~e F • f(z)e:).
F e F eF # :s z
Also, if z0 : (too, ml) e Z, we get f(Zo) = ~ k Ok : - - 1, b y (e), since - - z 0 a p p e a r s e x a c t l y once in e a c h ~ k , a n d z 0 in n o ~ k . Now, let Z be t h e set o f p o i n t s in M w h i c h c a n b e r e a c h e d f r o m m o b y a W-path, all o f w h o s e e l e m e n t s are in =k W(~). I f t h e assertion t o b e s h o w n were n o t vaIid, m x w o u l d be outside Z.
^
Aiming a t a c o n t r a d i c t i o n , we a s s u m e it t o b e so. T h e n z 0 : (m0, zrh) e Z, a n d f(Zo) ~- - 1, as we j u s t r e m a r k e d . L e t z I e Z be d i s t i n c t f r o m %; i f z x is outside
• I m k~, we h a v e d e f i n e d f ( z l ) : O, a n d if z 1 belongs t o • I m T , t h e d e f i n i t i o n o f Z says t h a t e~, is c e r t a i n l y outside {-~ er}geZ, so t h e coefficient f ( z l ) o f e:, m u s t vanish. Thus, f ( z ) -~ 0 w h e n e v e r z ~e zo, so t h a t
f ( z ) -~ f(Zo) = - - 1 , c o n t r a d i c t i n g (f).
S e c o n d step: N o t i c e t h a t t h e a s s u m p t i o n t h a t W is i n j e c t i v e is n o t essential.
For, d e f i n e H 1 r- S as H 1 : Iin{eE, - - eeE~.]T(E1) -= ~ T ( E 2 ) , e = • 1). T h e n t h e canonical m a p p i n g S ~ S / H 1 is i m m e d i a t e l y seen t o p r e s e r v e l e n g t h of v e c t o r s in spaces o f ~ (regarded as subspaces o f S). W e n o w r e g a r d S / H as a q u o t i e n t space of S / H 1 - - i n s t e a d o f r e g a r d i n g it as a q u o t i e n t space o f S. A n d n o w t h e canonical images of t h e es in S / H 1 are in b i u n i v o c a l c o r r e s p o n d e n c e w i t h t h e v a l u e s k~(E), so t h e a r g u m e n t m a y be carried o u t as before.
T h i r d step: W e show t h a t either r ~nay be written as j u s t one s u m of t y p e (d), or the ~F w i t h F C ~ are linearly dependent. B y w h a t was s h o w n in t h e f i r s t step, t h e r e is a p a t h elT(F1) . . . e,.T(F~.), Fk E ~ , ek ~ • l, n ' ~ 1, f r o m m o t o ~n 1. T h e n we can f o r m a s u m o f t y p e (d), n a m e l y s = ~kek~Fk, all o f whose t e r m s are in {=~eF}FeF. Since n o w c o = ~ = s , we g e t ~ - - s ~ - 0 , so t h a t e i t h e r r = s or ~ F e / f l ~ F : 0, for some fl~ w h i c h do n o t all vanish.
F o u r t h step: W e show t h a t i f the eF w i t h ~' E ~ are linearly dependent, then there
! !
is a n r' ~ ~ e ~ y~e~ ~- e o rood H such that at least one coefficient y~ vanishes a n d such that [lr'[I ~
[Ir[I.
T h u s , m o r e expressively, we can change r in such a w a y t h a t t h e n u m b e r o f t e r m s in t h e e~-exl~ansion o f r is r e d u c e d b y one u n i t , a n d t h a t t h e l e n g t h does n o t increase.So let ~ e F f l ~ = 0 for some fi~ w h i c h do n o t all vanish. F o r a n y scalar ~, d e f i n e
F e F F e F
ON T H E S E P A R A T I O N P R O P E R T I E S O F T H E D U A L S 2 9 7
l~otice t h a t r(0) = r a n d t h a t r(~) ~ e 0 rood H for a n y ~. W e will show t h a t r(~) assumes minimal l e n g t h w h e n some coefficient ~F -~ vile vanishes; this will give t h e assertion. B y definition of t h e p - n o r m on S,
Since t h e f u n c t i o n z--> lz] p is concave for z r 0 (since p < 1), t h e f u n c t i o n A --> ][r(2)] l is concave in t h e d o m a i n where all coefficients are distinct f r o m zero;
so t h e m i n i m u m m u s t be a s s u m e d in t h e c o m p l e m e n t of this domain, as required.
(For t h e real case, cf. fig. 1.).
' llr( )Ll
T
Fig. 1.
( p < l )
F i f t h step: W e conclude: L e t t h e canonical pre-image x' = y + 2r, r = ~.Fe F ?Fer, o f ~ be given. I f t h e vectors eF, F E 7, are linearly dependent, we can, b y w h a t was shown in t h e f o u r t h step, f i n d a n o t h e r pre-image Xl ---- y + ~rl, for which rx =
~.Fe f~ ~E F w i t h 71 consisting of t h e elements in 7 except one a n d such t h a t ]frill _< 1 e
[[r[[, i.e., [[Xl] ] < [Ix']]. N o w we a p p l y t h e same a r g u m e n t to x 1 as we d i d to x';
a n d t h e n we proceed step b y step. Thus, we get successively n e w pre-images o f in such a w a y t h a t 7 is replaced b y smaller a n d smaller subsets; a n d a n e w pre- image is never longer t h a n a n y of t h e previous ones. Since 7 is finite, t h e procedure m u s t stop some time. So, u l t i m a t e l y , we get x" = y + Xr", where r" is a linear c o m b i n a t i o n of linearly independent vectors eE, a n d ][x"l] < Nx']]. B y w h a t was shown in t h e t h i r d step, r" can be w r i t t e n as one s u m of t y p e (d). Since t h e t e r m s o f this s u m are linearly i n d e p e n d e n t , a n y t w o of t h e m are certainly n o t proportional to each other; a n d we are b a c k to t h e case first t r e a t e d . Hence, we k n o w t h a t ][x"[] ~_
[Ixl[, a n d a fortiori [[x'[] ~_ ][x]].
W e s t a t e explicitly y e t a n o t h e r p r o p e r t y of / / to be used in t h e sequel.
1.6. P~OPOSITION. With notations and assumptions as in 1--1.1, let El, E 2 be distinct spaces of ~ such that the pairs ~P(E1) and ~(E~) of elements of 2 I have one element, m ~ pr~ ~(E1) ---- p r 1 ~(Ez), say, in common, such that it belongs
298 ARKIV FOR YIATEi~ATIK. VO1. 9. No 2
to no other element pair ~(E). Let f~ be continuous linear forms defined on subspaces K~ ~ E~ with e~ ~ E~ for Ir = 1, 2. _Further, let ~ be the set obtained by replacing E~ by K~ in ~ , k ~ - 1 , 2 . Then, i f f l ( e E , ) + f 2 ( e s , ) = O , the forms f~ have a common extension to II(5~). (Notice that by 1.3, II(~X) may be regarded as a subspaca of the tvs
II(~).)
Proof. F i r s t e x t e n d f l a n d f2 t o forms f l a n d f2 on K 1 (~ K 2 which are 0 on K z resp. K 1. B y 1.3, K a --~ I I ( ~ { K 1 , K2} ) can be considered as a closed subspace of //(~.). So t h e f o r m h = f l + f 2 on _K 1 G K2 c / / ( ~ ) m a y be ex- t e n d e d to t h e l a t t e r space b y m e a n s of an extension ~ which is 0 on K 3. This definition is consistent, since )~(eE, + es,) = fl(eE,) + f2(eE,) = 0 a n d eE, + e~, spans t h e space ( K 1 $ K2) fl K3 if this is =~ {0). F o r first, L e m m a 1.7 below will show t h a t lin {eEl} D K~ fl K a (k = 1, 2). F u r t h e r , since m is in no other pair 7t(E) t h a n ~(E~) a n d 7t(E2), t h e defining relation (a) in t h e beginning of 1.5 gives lin {eE., eE,} fl K a ~ lin {e~ + eel}, whence t h e assertion.
1.7. LEM~A. With notation and assumptions as in 1.3, we have for any E o C o~,~ ~,, that
I m (g) rl i s ( G ) = {
lin {i~o(eEo)) i f there is a cycle, one element of which is and the others belong to =J= ~ ( ~ ' )
(0) otherwise.
~(Eo)
Proof. B y 1.2 a n d 1.3, we see t h a t t h e intersection is contained in lin (i~o(eEo)).
F u r t h e r , (a) in t h e beginning o f 1.5 shows t h a t iEo(e~o) is outside I m g0 (notation as in t h e p r o o f o f 1.3 in 1.5) if a n d o n l y i f T(Eo) fails t o be in a n y cycle of w h i c h t h e other elements are in =~ ~ ( ~ ' ) . B u t b y construction we clearly h a v e i~o(eE.) a t t h e same distance f r o m I m go as f r o m lin {iE(eE)[E e ~ ' ) ( C I m go). B u t since this space has finite codimension in lin {is(eE)[E e "~}, t h e v e c t o r i~o(eE,) is seen to be outside t h e r e o f -- i.e., outside I m go -- if a n d o n l y if t h e m e n t i o n e d distance is positive, t h a t is, if a n d o n l y if iE0(eEo ) is outside I m 20 = I m g. So t h e assertion follows.
2. Example. W h e n given a n y ordinal v a n d a n y n u m b e r p, 0 < p < 1, we are n o w r e a d y for t h e actual construction of a p-normed and complete tvs E such that A~E is one-dimensional.
W e use t r a n s f i n i t e induction.
A. Assume t h a t v --- v' + 1 is a n y ordinal which is of t h e first k i n d a n d ~ 2, a n d t h a t we h a v e a space E ~ for which A~'E ~ is one-dimensional. W e are going r f o r m E b y m e a n s o f t h e m e t h o d of 1. L e t Ekl, 1 ~ / c < l , l ~ l , be copies o f E ~ a n d let, for all /c, l, t h e v e c t o r ekl be a generator of l e n g t h 1/l of A"Ekl.
F u r t h e r , let M be a metric space w i t h points m0, ml, a n d mk~ for all k, l w i t h
O ~ T H E S E P A R A T I 0 1 ~ P R O P ] ~ R T I E S O F T H E D U A L S 299 l _ _ < k < l . W r i t e m o=mo~, m ~ = m u , l > l ; t h e n ff d ( . , . ) is t h e distance, a s s u m e t h a t d ( m ~ _ ~ . t , m ~ ) = l / l for all k a n d 1. Define ~ : { E ~ } ~ - - - > M • b y ~ ( E ~ ) = (m~_~,t, m~). W e m a y describe this as follows: M is t h e set o f nodes ,of t h e infinite g r a p h in fig. 2, a n d I m ~ consists o f those pairs which f o r m end- points of some edge.
~ig. 2.
m0 m I
Notice t h a t we h a v e n o t d e f i n e d M uniquely. However, this is inessential;
~cf. 1.4.
W e n o w t a k e E ~ H({Ekt}, {ekt}, M , T). W e w a n t t o show t h a t A ' E is the one-dimensional space which is spanned by ell (the v e c t o r corresponding to t h e pair (m 0, ml) ). First, e n ~ (l/l) (lell ~- . . . -~ letl), wherein I[lekt[I = Ill p . [[e~lll-= 1P-1--~ 0 as l--> ~ . B u t we also k n o w t h a t ekl E A t E , since AtEkz r (cf. 1.2).
T h u s e n E co (U N A t E ) for e v e r y 0-neighbourhood U; so ell E A A t E = AYE.
Conversely, we h a v e to show t h a t a n y point outside t h e linear hull o f e n is also outside A~E. First, we h a v e EkaN A t E ~ AtEkt = lin{ekl}. F o r t h e f i r s t e q u a l i t y is shown b y t r a n s f i n i t e induction; suppose t h a t we k n o w t h a t
Nr (Ekt N At"E) ~ Nr A'"'Ekt a n d w a n t to show t h a t Eu N A~'E ~ Ar for some v" < v'. Now, a n y given point in nr A k l ~ A Ekt m a y be s e p a r a t e d f r o m A~"Ekt b y a continuous linear form f on Nr A'"Ekt. Since ekt E A~'Ez, we h a v e f(ekO = 0; b u t E ~ has intersection lin {ekl} w i t h t h e closed linear hull of all t h e Ek, l, w i t h (k', l') # (k, l). (This follows from L e m m a 1.7.) So we m a y
9 ytrs
,extend f linearly to f , say, on h n (rlr A Ekz, E ~) c E, b y t a k i n g f ( E ~) ---- 0. The h y p o t h e s i s n o w shows t h a t t h e d o m a i n of f includes Nr Ar Since t h u s a n y p o i n t in t h e intersection o f t h e l a t t e r space w i t h Ekz can be s e p a r a t e d f r o m A~'Ekt, as :soon as it is outside this space, b y such an f , it follows t h a t Ekt ['l A ( N , , , < , , A r Ekt fl A " E ~ A~'Ekz. The converse inclusion follows, however, f r o m 1.2 applied t o the inclusion map.
Thus, since n o w Ek~ N A t E = lin {ekl}, it remains to show t h a t a n y ek~ w i t h (k, l) # (1, 1) is s e p a r a t e d f r o m 0 b y a continuous linear f o r m which is d e f i n e d on some space containing A " E - i.e., containing all t h e e~,t, w h e n k' a n d l' r a n g e over all possible values. B u t notice t h a t t h e indices (k, l) # (1, 1) are those
