A t h e o r e m o n duality m a p p i n g s in B a n a c h s p a c e s

A R K I V F O R M A T E M A T I K B a n d 4 n r 32

R e a d 2 J u n e 1954

A t h e o r e m o n duality m a p p i n g s in B a n a c h s p a c e s

By ARNE BEURLING and A. E. LIVINGSTON

l. Introduction

The problem considered in this paper originates in the theory of Fourier series of functions belonging to a Lebesgue space L ~, where L p denotes the space of measurable functions with period 2 g a n d with n o r m

2 ~

II/11. = { f I/(~)I" d ~}".

0

F o r the Fourier coefficients of ], we shall use the notation

if

c,,

(t) = ~ e-'"" I (x)

0

d x .

A classical theorem such t h a t

asserts t h a t if { a n } ~ is a given sequence of numbers l a . [ 2 < oo,

n ~ - - O 0

then there is a unique element / E L ~ with the p r o p e r t y t h a t

c . (1) = a .

for n = 0 , __1, --+2, . . . .

The above theorem admits the following extension. Let S~, 0 < ~ < oo, denote the operator t a k i n g ' t h e complex n u m b e r z into I zl a:a z. We have

so t h a t the collection { Sa} forms a group. Furthermore, Sa performs a one- to-one m a p p i n g of the finite complex plane onto itself. Applied to a space L v, we see t h a t Sa ] will t a k e L ~ onto L p/a, 0 < a_< p. I n particular, Sv-1 ] will m a p L v onto its dual space L ~, q = p / ( p - 1 ) . We then have

T h e o r e m t . l e t the integers be partitioned into two disjoint sets A and A ' , neither o/ which is empty. Let p be a given exponent such that 1 < p < ~ . Let { a, ; n E A } and (bn ; n E A ' } be given sets o/ numSers such that, /or some h E L ~ and /or some k E L q, q = p / ( p - 1 ) , we have

A. B E U R L I N G , A. E . L I V I N G S T O N , Duality mappings in Banach s p a c e s

ca (h) = an, n fi A , cn (k) = b~, n 6 A ' . Then there is a unique element [ 6 L v such that

c,~(f)=an, n e A , c,,(Sv_l])=b,~, n E A ' .

(1)

For p = 2, the above statement reduces to the cited classical result, since, in this case, S~_1 f = f.

The proof of the uniqueness is quite easy. If there were two solutions ~1 and f3 of (1), then the difference F = [ 1 - [ 3 would be an element of L p and would have Fourier coefficients vanishing on A. Similarly, G = Sp-i ~1- Sv-1 [3 would be in L e and have Fourier coefficients zero on A'. Consequently, the Fourier coefficients of the continuous function

2n

! ~'(x+t)O(t)~t,

0

which are given by c, (E)5n (G), are zero for each n. Therefore,

2:z 21t

~! fllv I N ' ~ dr. (2)

~ f

I ("-'3), ,1

,3/

Setting fl = ]fi I e'~ s = Is e'~ we find t h a t the real part of the integrand in (2) is

Ihl ~ + 1/31"-(Ihl ]f3l n - l + 1/11 "-~ 1/31) cos (01-03)>_ (1111-1/31)(Ihl " - ~ - 1131"-1) >-0.

The vanishing of (2) therefore requires t h a t almost everywhere ]h [ = I f3l and cos ( 0 1 - 0~)= 1, and the uniqueness follows.

The above proof of the uniqueness is satisfactory, b u t it gives no indication of the nature of the problem. In particular, the role played b y the operator Sp_l is not at all clear. We shall see in the next section t h a t the operator Sp-1 is only one of a large class of operators on L v to L v/(v-1) which arise quite naturally from a consideration of what we call duality maps of a Banach space onto its first conjugate space.

2. Duality mappings

We will be dealing with a complex (or real) Banach space B and its conju- g a t e space B*. The null element of B will be denoted b y 0, of B* b y 0". The norm of an element x E B is IIxH--IIxIl., of

u B*

Let (x, u) be the bilinear functional defined for x E B and u E B* and having the properties :

A]gKIV FOR MATEMATIK~ B d 4 nr 32 (a) I f 2 is a complex (or real) number, then

(~

(b) (x 1 + x~, u) = (x 1, u) + (x,, u),

(x, u 1 + u2) = (x, u l ) + (x, us) ;

(c) II ll.. = sup os I(x,

Two elements x E S and x*E S* are said to be conjugate if (x, x * ) = 1. The sets (;t x ; 0 _ ~ < o~ } and {/~ x* ; 0 _</z < oo } win then be called conjugate rays.

Under the assumption t h a t each element on S (or S*) has a unique con- jugate on S* (or S), we will consider duality m a p s of B onto B* characterized b y the following properties: T is one-to-one and takes each r a y in B onto the conjugate r a y in B* and each sphere [[x[[=r in B onto a sphere [[u[[=Q in B*

in such a w a y t h a t r 1 < r 2 implies Q1 < 0z. F r o m this definition, it follows t h a t

T(~x)=~ (2)x*, 2>0,

x and x* being conjugate elements on the respective unit spheres and ~ (~) a continuous function t h a t is strictly increasing from 0 to oo with 2. (A special t y p e of duality m a p has previously been considered b y E. R. LORC~ [4].)

With regard to such mappings, we have

L e m n a a i . Let there exist a duality map o/ B onto B*. I / x, y E B, then the relation

( z - y , T x - T y ) = O (3)

implies that x = y.

To prove this, we observe t h a t z E B and u EB* will belong to conjugate rays if and only if (z, u)=[[zili[ul]. Consequently, the assumption x r

> 0, will imply t h a t

R e (x, T y ) < Itxll tiTyll, R e ( y , Tx)<ilYII IITxi], so t h a t upon taking the real p a r t of (3), we will have

0 > ( H x l l - H y I I ) (lITxii-iiTy[I)>-o.

Therefore x = ~y, and )t = 1.

We w a n t next to find conditions on the space B in order t h a t a duality m a p shall exist. F o r this purpose, we need to recall certain definitions.

The space B is said to be uniformly c o n v e x [ l ] if there corresponds to each e, 0 < e < l , a positive n u m b e r O(e) tending to zero with e and such t h a t

A. BEURLING, A. E. LIVINGSTON, Duality mappings i n Banach spaces imply t h a t

I n uniformly convex Banaeh spaces, each closed convex subset possesses a unique element of minimal norm.

We shall say t h a t B is differentiable if for a n y x E B, x r 0, a n d a n y y E B there is a finite complex (or real) number D = D (y, x) such t h a t

+ tytl - II ll= { t D } + o (Itl)

(4)

as the complex (or real) n u m b e r t tends to zero. I t is known [3] t h a t for differentiable spaces D (y, x) is a linear functional in y of norm unity. We observe t h a t (4) implies

j D ( y ,

)l<llyll; D(x, x)-II ll;

D ( y , X x ) = D ( y , x), 0 < 4 < ~o.

An important consequence of uniform convexity and differentiability in a Banach space is the, in principle, known

L e m n a a 2. I f B is a uni/ormly convex and di//erentiable Banach space, then each element on S (or S*) has a unique conjugate element on S* (or S).

Assume first t h a t x E S is given. We are to show the existence and the uniqueness of a linear functional L ( y ) = (y, x*) of norm unity and for which L ( x ) = 1. The existence of L (y) is clear, for we m a y t a k e L ( y ) = D (y, x). As for the uniqueness, we observe t h a t for a n y y E B and for a n y complex (or real) scalar t, we have

O<-]]x+ tyli - R e { L ( x + t y ) } = i i x + t y l [ - i ] x ] [ - R e {t L (y)}.

Combining this relation with (4) yields

O <_Re {t [i) (y, x) - L (y)]} + o (I

t])

from which it follows t h a t L (y)= D (y, x).

On the other hand, if x*E S* is given, its conjugate x E S can be characterized as the element of minimal norm in the closed and convex set {y; (y, x * ) = 1} c B . Therefore, x exists and is unique.

(We point out t h a t x*E S* can be shown to have a unique conjugate x E S if we assume only t h a t B is strictly convex and t h a t the set {y; y E B , Ily]l_<l} is weakly compact. These two conditions are known to be weaker t h a n the requirement t h a t B be uniformly convex.)

I n the sequel, we will use the following definitions and notations: I f A is a closed linear subset of B, then its orthogonal complement in B* is the set A x of u E B * for which (x,u)---0 for every x E A . I f y is an element and C a set of elements, then C + y will denote the set {x; x = y + z for z E C I.

We are now r e a d y to prove

ARKIV F6R MATEMATIK. Bd 4 nr 32 T h e o r e m 2. Let B be a uni/ormly convex and di/[erentiable Banach space and T a duality m a p o/ B onto its conjugate B*. Let C be a closed, linear, and proper subset o/ B and C • its orthogonal complement in B*. 1] H E B and K C B* are given elements, then the sets C • + K and T (C + H) have one and only one element i n common.

(Note t h a t , b y v i r t u e of L e m m a 2, d u a l i t y m a p s of B o n t o B* do i n d e e d e x i s t .)

R e c a l l t h a t t h e d u a l i t y m a p T defines a c o n t i n u o u s f u n c t i o n r (~), 0_< 2 < c~, which is s t r i c t l y increasing from 0 t o c~ w i t h 2. Set

(~) = f ~ (u) d u , 0

a n d consider t h e f u n c t i o n a l

(5) = ~ (11 x II) - R e {(x, K ) }

for x E C + H . F o r Ilxll=r, we h a v e

F (x) >__ ~b (r) - r II K liB..

The r i g h t - h a n d m e m b e r of t h i s i n e q u a l i t y is a s t r i e t l y convex function of r, t e n d s t o i n f i n i t y as r - § ~ , a n d is b o u n d e d from below for r > 0. Hence,

M = inf { F (x) ; x 6 C + H }

is a finite n u m b e r . F u r t h e r m o r e , F ( x ) _ m + 1 i m p l i e s t h a t It x II < r0 < ~ - L e t {x,} c C + H be a m i n i m i z i n g sequence of F (x). W i t h o u t loss of g e n e r a l i t y , w e m a y a s s u m e t h a t II~nll t e n d s t o s o m e f i n i t e l i m i t ar (Actually, we need n o t m a k e t h i s a s s u m p t i o n , for t h e s t r i c t c o n v e x i t y of ~b (2), 0 _ 2 < c~, g u a r a n - tees t h a t II~nll t e n d s to a finite limit.) Since C + H is a c o n v e x set, we h a v e

Since r (r), r>_ O, is convex, we o b t a i n

~X m -}- Xn~

= 1 { F (~m) + F (~n)} - F ~ - - V - / = ~m n,

where emn-~0 as m, n-->c~. Therefore

lim O(ll ll)=O, ,

rn ~-~oo n--~oo

409

A . B E U R L I N G , A. E . L I V I N G S T O N ,

Duality mappings

m--~ cr

B u t in a u n i f o r m l y c o n v e x space, t h i s i m p l i e s t h a t ( x , } is a C a u c h y sequence.

Since C + H is a closed s u b s e t of B, t h e r e is a n x o E C + H such t h a t I l x n - x o l l ~ O a n d F (xo) = M. C o n s e q u e n t l y ,

F (x o + t x) - F (Xo) > 0 for x EC a n d t a c o m p l e x (or real) scalar.

Since ~b is a d i f f e r e n t i a b l e function,

r (ll xo + t x il) - r (ll xo ID = r (ll xo lD R e { t D (x, xo) } + o (I t l)

= ~ (11 Xo II) R e {t D (x, ~o)} + o (I t I).

T h u s

R e (t [~ (H xo ]]) n (x, xo) - (x, K)]) + o (I t i) __ 0, f r o m w h i c h we conclude t h a t

([[ x o [[) D (x, Xo) - (x, K) = 0 (5)

for x E C . Since 6(HXoi})D(x, Xo) is a l i n e a r f u n c t i o n a l in x of n o r m ~b(Hxo] D, i t m a y be w r i t t e n in t h e f o r m (x, no), where u o is a n e l e m e n t on t h e sphere Ilulln.=~b (llXoH). S e t t i n g x = x o gives

(Xo, ~o) = + (11 Xo II) D (xo, Xo) = [I Xo II + (11 ~o II),

from w h i c h i t follows t h a t n o = T x o. Consequently, (5) m a y b e w r i t t e n as

(x, T x o - K ) = 0 (O)

for x E C, which i m p l i e s t h a t T x 0 E Cx + K .

W e h a v e t h e r e f o r e shown t h a t T ( C + H ) a n d C ~ + K h a v e a t l e a s t one ele- m e n t in c o m m o n .

I f x E C + H a n d T x E C j . + K , t h e n (6) i m p l i e s t h a t (x o - x, T x 0 - T x) = 0.

A n a p p l i c a t i o n of L e m m a 1 shows t h a t x = x o a n d , hence, t h a t T ( C + H ) a n d C ~ + K h a v e a t m o s t one e l e m e n t in c o m m o n .

The p r o o f of t h e t h e o r e m i s now complete.

W e wish t o p o i n t o u t t h a t t h e conclusion of T h e o r e m 2 r e m a i n s v a l i d if we a g a i n a s s u m e o n l y t h a t B is s t r i c t l y c o n v e x a n d t h a t ( y ; y E B , HyH_<I} is w e a k l y c o m p a c t .

I n o r d e r t o d e d u c e T h e o r e m 1 from T h e o r e m 2, we set B = L ~', B * = L ~, a n d let C b e t h e set of f u n c t i o n s in L p w i t h F o u r i e r coefficients v a n i s h i n g for n E A . I f

410

ARKIV FOR MATEMATIK. B d 4 n r 3 2

2 ~

(x, u)= f x(t) u(t)dt

0

for xEL p, uEL e, t h e o r t h o g o n a l c o m p l e m e n t C ~ of C consists of t h o s e uEL '~

for which ~ h a s F o u r i e r coefficients zero on A ' .

I t is k n o w n t h a t L ~, 1 < p < ~ , is u n i f o r m l y c o n v e x [1] a n d d i f f e r e n t i a b l e [2], a n d it is clear t h a t Tx=Sp_l~=ixi~/x is a d u a l i t y m a p of L ~ o n t o L ~. The d e s i r e d r e s u l t n o w follows f r o m T h e o r e m 2 if we set H = h a n d K = k.

R E F E R E N C E S

1. J. A. CLARKSON, U n i f o r m l y convex spaces. T r a n s a c t i o n s of t h e Amer. Math. Soc. 40 (1936), 396-414.

2. R. C. JAMES, O r t h o g o n a l i t y a n d linear functionals in n o r m e d linear spaces. T r a n s a c t i o n s of t h e Amer. Math. Soe. 61 (1947), 265-292.

3. - - - - L i n e a r functionals a s differentials of a n o r m . M a t h e m a t i c s Magazine 24 (1950-51), 237-244.

4. E. R. LORCH, A c u r v a t u r e s t u d y of c o n v e x bodies in B a n a c h spaces. Annali di M a t e m a t i c a P u r a ed A p p l i e a t a 34 (1953), 105-112.

Tryckt den 24 januari 1962

Uppsala 1962. Almqvist & Wiksells Boktryckeri AB

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