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General vector spaces
17
CHAPTER I1
General vector spaces 81. D e f i n i t i o n and elementary p r o p e r t i e s of v e c t o r spaces. Suppose t h a t a non-empty set E is g i v e n , and t h a t t o each ordered p a i r (x,y) of elements of E t h e r e corresponds an element x + y of E ( c a l l e d t h e sum of x and y ) and t h a t f o r each number t and X E E a n element tx of E ( c a l l e d t h e product of t h e number t w i t h t h e element x ) is d e f i n e d i n such a way t h a t t h e s e o p e r a t i o n s , namely addition and ~ c a l a rmultiplication s a t i s f y t h e following c o n d i t i o n s (where x , y and 2 denote a r b i t r a r y elements of E and a,b are numbers): 1) x + y = y + x , 2 ) x + (y + 8 ) = (x + y ) + 2 , 3 ) x + y = x + 8 impties y = 2 , 4)
a(x
+ y ) = ax + a y ,
( a + b)x = ax + bx, 6 ) a(bx) = (ab)x, 7) 1.x = x. Under t h e s e hypotheses, we say t h a t t h e set E c o n s t i t u t e s a vector o r tinear space. I t is easy t o see t h a t t h e r e then e x i s t s e x a c t l y one element, which we denote by (3, such t h a t x + ( 3 = x f o r a l l X E E and t h a t t h e e q u a l i t y a x = bx where x z (3 y i e l d s a = b ; furthermore, t h a t t h e e q u a l i t y ax = ay where a * 0 i m p l i e s x = y. Put, f u r t h e r , by d e f i n i t i o n : -x = (-112 and x - y = x + (-y). Examples 1-10 of metric spaces, d e s c r i b e d on pp. 5 and 6 , a l s o s e r v e a s examples of v e c t o r spaces, when t h e u s u a l d e f i n i t i o n s of a d d i t i o n and s c a l a r m u l t i p l i c a t i o n a r e adopted. When x * y , w e understand by t h e line segment j o i n i n g x and y t h e set of a l l elements of t h e f o r m tx + ( 1 t ) y where t is any number i n the interval [0,1]. A set C S E is said t o be convex, when it c o n t a i n s every l i n e segment j o i n i n g a r b i t r a r y p a i r s of elements of C. If x1,x2, x n are elements of a v e c t o r space, t h e expression 5)
-
...,
alxl + a2x2 +
... + anxn
=
n Caixi,
i=l
where al,a2,...,an are a r b i t r a r y real numbers, is c a l l e d a tinear aombination of t h e elements x1,s2,...,xn. 52. Extension of a d d i t i v e homogeneous f u n c t i o n a l s . L e t E and E l b e two v e c t o r spaces and f a mapping i n E whose codomain lies i n El.
18
S.
BANACH
The mapping f is s a i d t o be a d d i t i v e when f o r each pair of elements x , y we have: f ( x + y) = f ( x ) + f ( y ) ; it is c a l l e d homogeneous when f o r each element x and each number t : f(tx) = tf(x). Suppose t h a t t h e r e a r e g i v e n a f u n c t i o n a l p d e f i n e d i n E s u c h t h a t f o r a l t x , y ~E
THEOREM 1. 1'
p ( x + t l ) 5 p ( x ) + P(Y) and p ( t x ) = t p ( x ) f o r t L 0 , and 2'
an a d d i t i v e homogeneous f u n c t i o n a l f d e f i n e d i n a v e c t o r subspace G s E (1.e. a s u b s e t of E t h a t is i t s e l f a v e c t o r space with t h e s a m e d e f i n i t i o n s of t h e b a s i c o p e r a t i o n s ) s u c h t h a t f o r each X E G
f(x) 4 P b ) , t h e n t h e r e atways e x i s t s an a d d i t i v e homogeneous f u n c t i o n a l F d e f i n e d i n E such t h a t F ( x ) 6 p ( x ) for each x E E and F ( x ) = f ( x ) f o r e a c h x E C . ~ By 2', for P r o o f . W e can assume t h a t G ; t E ; l e t x , E\G. x ' , x " E C w e have: f(x") f ( x ' ) = f ( x " - x ' ) 5 p ( x " - x ' ) = p"x"+x,) + (-x'-x,)l
-
p(x"+x,) + p(-2'-
whence
- ftx')
-p(-x'-z,)
The numbers m sup
ip(x"+x,)
S O ) '
- f(x").
xEG
are t h e r e f o r e f i n i t e and m 6 M . I f r o is any number such t h a t m s r , 5 M I w e have f o r each X E C (11
-
-
f ( x ) 6 r, 5 p ( l : + x o ) f ( x ) . of a l l elements y of t h e form (2) y = x + t x , where x i C and t i s a number. C l e a r l y C, i s a v e c t o r space. Put (3) g(y) f ( + ) + tr,, where t h e element y i s given by ( 2 ) ; a s x o E E x C , each y~ 0, has a unique r e p r e s e n t a t i o n i n t h e form of ( 2 ) so t h a t t h e f u n c t i o n a l g is We a l s o see t h a t g i s a d d i t i v e and homogeneous well-defined on G,. on c, and c o i n c i d e s with f on G. W e now show t h a t (4 1 g ( y ) 5 p ( y ) f o r each y E G o I n f a c t , i f one writes y i n t h e form ( 2 ) it can be assumed t h a t t t 0. P u t t i n g x / t i n p l a c e of x i n t h e i n e q u a l i t y ( 1 ) and multiplyi n g i t s r i g h t - o r l e f t - h a n d s i d e , according as t > 0 o r t <0 , by t , one o b t a i n s t r , s p ( x + t x , ) f ( x ) which by ( 3 ) i m p l i e s t h e i n e q u a l i t y -p(-x-x,)
Consider t h e s e t
0,
-
.
(4).
-
T h i s e s t a b l i s h e d , it now s u f f i c e s t o well-order t h e set E\G, o b t a i n i n g , by s u c c e s s i v e e x t e n s i o n s of f , following t h e procedure d e s c r i b e d above, a f u n c t i o n a l F s a t i s f y i n g t h e conclusion of t h e theorem.
19
General vector spaces
Given a f u n c t i o n a l p d e f i n e d i n E s u c h t h a t f o r x , y E E p ( x + y ) 5 p ( x ) + p ( y ) and p ( t x ) = t p ( x ) f u r t.? 0 , t h e r e e x i s t s an a d d i t i v e homogeneous f u n c t i o n a Z F d e f i n e d i n E such t h a t , f o r each X E E COROLLARY.
F(x) 6 p b ) . In f a c t , c o n s i d e r an x E E and denote by G t h e set of a l l elements of t h e form t x , where t 1s a n a r b i t r a r y number. G i s then a v e c t o r space. P u t t i n g f ( t x 0 )= t p ( x o ) i n G I w e w i l l have f ( t x , ) 5 p ( t x , ) f o r any t , s i n c e t 2 0 i m p l i e s t p ( x , ) = p ( t x , ) and t <0 i m p l i e s O = p ( O ) s p ( x o )+ p ( - x o ) , whence p ( x , ) L - p ( - x o ) and f i n a l l y t p ( x o )6 - t p ( - x , ) = p ( t x , ) ; t h e r e s u l t now f o l l o w s on applying theorem 1.
53.
Applications:
measure
of t h e n o t i o n s of i n t e g r a l , of
generalisation
and
limit.
of
W e a r e now going t o d i s c u s s s e v e r a l i n t e r e s t i n g a p p l i c a t i o n s of theorem 1 and i t s c o r o l l a r y . 1. L e t E be t h e set of bounded real-valued f u n c t i o n s x ( s ) d e f i n e d on a c i r c l e of u n i t circumference where s denotes a r c - l e n g t h measured from some f i x e d p o i n t , always i n t h e same sense. With t h e u s u a l a l g e b r a i c o p e r a t i o n s , E is a v e c t o r space. NOW, f o r each element x = x ( s ) of E l l e t us d e f i n e p ( x ) t o be t h e i n f i m u m of a l l t h e numbers M ( x ; a l , a 2 , . . . , a n ) of t h e form
M(x;alra2,.*.ran)
where a,,a,,...,an f u n c t i o n a l p then f a c t , i t is p l a i n t S 0. Secondly, given E > 0, there e x i s t p1,f12,...,& such
=
sup
-m
(4 k=I x ( s + a k ) ) ,
is an a r b i t r a r y f i n i t e sequence of numbers. The s a t i s f i e s a l l t h e hypotheses of t h e c o r o l l a r y . In t h a t , f i r s t l y , one always has p ( t x ) = t p ( x ) f o r two elements x = ~ ( 8 and ) y = y(s) of E and a number f i n i t e sequences of numbers a,,a2,...,au and that
(5) ~ ( x ; u ~ , a ~ ~ . . . 6, ap ~( x) ) + E and M ( y r f 3 1 , B 2 1 . . . 1 8 v ) 5 p ( y ) + E. Arranging a l l t h e numbers a ; + S j where i = l I 2 , . . . , u and j = l r 2 , . . . , v as a s i n g l e sequence y 1 , y 2 , . . . , y U v . i n some way, one h a s (6) p ( x + 21) 5 M ( x + y t y l r Y 2 r . . . r Y ~ ~ ) and it is e a s i l y v e r i f i e d t h a t
.
..
..
( 7 ) ~(x+gty,,y,,. ,yU,) 5 M ( x ; a l l a z l . . ,aU) + M ( Z / ; S ~ ~ B ~ ,,&,I. . The r e l a t i o n s (5) ( 7 ) imply p ( x + y ) 2 p ( x ) + p ( y ) + 2 ~ which , proves t h e number E > 0 being a r b i t r a r y , t h a t p ( x + y ) 6 p ( x ) + p ( y ) . T h i s e s t a b l i s h e d , c o n s i d e r t h e r e f o r e t h e f u n c t i o n a l F which e x i s t s by t h e c o r o l l a r y . Now, i f X ( S ) = 1 , w e have p ( x ) = 1 and p ( - m ) = -1 and as P ( x ) d p t z ) and F ( X ) = - F ( - x ) L - P ( - z ) , one o b t a i n s F ( x ) = 1 . I f x ( 8 ) L 0 , w e have p ( - x ) 5 0 and moreover F ( x ) = - F ( - x ) L - p ( - x ) , so t h a t F ( x ) t 0 also. Furthermore, t h e f u n c t i o n a l F h a s t h e property of s a t i s f y i n g , f o r each number s o , t h e e q u a l i t y F [ x ( s+ s o ) ] = F [ x ( s ) l . I n f a c t , if one one h a s p u t s y ( e ) = x ( s +s o ) - x ( 8 ) and a k = ( k - l ) s o f o r k=1,2,.. for each n:
-
.,
p ( y ) 5 ~ ( ~ ' ; a ~ , a , , . . . , a n=)
1
SUP
'--a <=
[=(s+nso)-=(8)l,
20
S. BANACH
so t h a t p ( y ) 5 0; one s i m i l a r l y o b t a i n s p ( - y ) 5 0 . F ( y ) = -F(-y) b - p ( - y ) , whence F ( y ) = 0.
But F ( y ) 5 p ( y ) and
Thus, using t h e symbol j x ( s ) d s t o denote t h e f u n c t i o n a l f { F [ x ( s ) +] F [ z ( l - s ) ] } , one h a s t h e theorem: T o e v e r y f u n c t i o n x ( 8 ) of t h e class E one can a s s o c i a t e a number j x ( s ) d s i n such a way t h a t t h e foEZawing c o n d i t i o n s (where z ( s ) and y ( s ) a r e a r b i t r a r y f u n c t i o n s of t h e c l a s s E and a , b , s o a r e numbers) are s a t i s f i e d : 1) j[az(s) + by(s)]ds = alx(s)de + b/y(s)ds, 2) x ( s ) d s 2 0 when z ( s ) 2 0 , 3) 1 x ( s + sO)ds = jz(s)ds, 4) ~ ( 8l ) d s 1z(8)d8, 5) Ids = 1 . I t is easy t o check t h a t t h e f u n c t i o n a l j x ( s ) d s , s a t i s f y i n g con5 1 , always l i e s between t h e lower and upper Riemann ditions 1 ) i n t e g r a l s of t h e f u n c t i o n ~ ( 8 ) .Consequently, f o r every Ri n t e g r a b l e f u n c t i o n , t h i s f u n c t i o n a l c o i n c i d e s with t h e i n t e g r a l of t h e function. For 6-summable f u n c t i o n s , t h e f u n c t i o n a l i n q u e s t i o n does n o t always c o i n c i d e with t h e i r L-integral. Nevertheless, s t a r t i n g w i t h t h e v e c t o r space G of such (L-summable) f u n c t i o n s and d e f i n i n g t h e f u n c t i o n a l f ( x ) i n G t o be t h e L - i n t e g r a l of t h e f u n c t i o n X ( S ) E G , theorem 1 f u r n i s h e s a f u n c t i o n a l F d e f i n e d i n t h e space E such t h a t t h e f u n c t i o n a l j z ( s ) d s = ~ { F [ z [ s +) ]F [ z ( l - s ) ] } c l e a r l y s a t i s f i e s conditions 1 ) 5 ) and, furthermore, c o i n c i d e s , f o r every L-summable f u n c t i o n , with t h e i n t e g r a l of t h a t f u n c t i o n . Consider now t h e c l a s s x o f a l l s u b s e t s of t h e circumference 2. of t h e c i r c l e i n q u e s t i o n and denote by A, t h e circumference i t s e l f . P u t t i n g , f o r each s e t A of t h i s c l a s s , p ( A ) = I z ( s ) d s , where x ( s ) is t h e c h a r a c t e r i s t i c f u n c t i o n of t h e set A and t h e r e f o r e a f u n c t i o n of t h e space E d i s c u s s e d i n 1 , w e o b t a i n t h e theorem: Po each s e t A of t h e class %one can a s s i g n a number w ( A ) i n such a way t h a t t h e f o l owing c o n d i t i o n s (where A and B are a r b i t r a r u are s a t i s f i e d : s e t s o f t h e cZass&j 1) b(A U B ) = b ( A ) + b(B) whenever A f l B = @ ,
I
I I
-
-
-
2)
P(A)
2 0,
3)
B(A)
=
4)
w(Ao) = 1.
V(B) i f A
B,
-
The f u n c t i o n a l b ( A ) , which s a t i s f i e s c o n d i t i o n s 1 ) 4) lies between t h e i n n e r and o u t e r Jordan measures of t h e s e t A . Cons e q u e n t l y , f o r every J-measurable set, t h i s f u n c t i o n a l c o i n c i d e s with the measure of t h e set. For a r b i t r a r y L-measurable sets, t h e f u n c t i o n a l i n q u e s t i o n does n o t always c o i n c i d e with t h e i r L-measure, b u t , j u s t a s b e f o r e , one can a r r a n g e t h i n g s i n such a way t h a t t h i s p r o p e r t y a l s o holds. 3. L e t E be t h e set of a l l bounded r e a l - v a l u e d f u n c t i o n s x ( s ) d e f i n e d i n [ 0 , + - ] ; with t h e u s u a l d e f i n i t i o n s of a l g e b r a i c operat i o n s , this is a v e c t o r space. t h e infimum of a l l For each element z = x ( s ) of E , denote by p )I(
n
..
t h e numbers 8C x ( s + ak) , where al,ao,. ,an is a n a r b i t r a r y n k=i f i n i t e sequence of p o s i t i v e numbers. One e a s i l y v e r i f i e s t h a t t h e f u n c t i o n a l p t h u s defined i n t h e space E s a t i s f i e s t h e hypotheses of
General vector spaces
21
52
~ ( 8 )the functional P I the corollary. Denoting by the symbol which exists by the corollary, one therefore has the theorem: T o e v e r y f u n c t i o n x ( e ) E E one can a s s o c i a t e a number big x ( s ) i n such a way t h a t t h e f o l l o w i n g c o n d i t i o n s (where x ( s ) and y ( e ) a r e a r b i t r a r y f u n c t i o n s o f E and a , b and s o L 0 a r e numbers) a r e satisfied: x ( e ) + b 5% y ( s ) , 1) $& [ a x ( s ) + by(s)l = a X& 2) I&g x ( s ) k 0 wfieneuer x f s ) 2 0 , x(s+ so) = & r X(8)I 3 ) Lim 84 ) L&g l = l. The functional %@ x ( s ) satisfying conditions 1 ) 4) always lies between lim ~ ( 8 )and ~ ( 8 ) . It consequently coincides with. r s Note that X ( 8 ) whenever this limit exists in the usual sense. Lim here denotes a certain generalised while lim is reserved exclusively for limit in the usual sense. 4. Let ( E n ) be any bounded sequence. Define the function x ( s ) in ( O f + - ) by: x ( s ) = En for n-1 < s S n , and n = 1 , 2 , . . . The function = ( a ) thus belongs to the set E discussed in 3. Putting f;& E n = % % X ( 8 ) in the sense of 3, one has the theorem: To each bounded sequence ( c n ) one can a s s o c i a t e a number kiet En i n such a way t h a t t h e f o l l o w i n g c o n d i t i o n s (where ( E n ) and ( n n ) a r e a r b i t r a r y bounded sequences and a and b a r e numbers) a r e aa tisf i sd: 1) (aSn + b n n ) a f;&En + b f;&n n t 2) 5,L 0 , if E n 2 0 f o r n=lr2,...r 3) f;& Em1 = +&k! 5 8 , 4) 1 = 1. thus defined Conditions 1 ) 4) imply that the functional I&y En. Consequently, for every always lies between lim En and iF= convergent sequence this functional coincides with the limit of the sequence in the usual sense.
-
BQ
b& k&
-
CHAPTER I1
General vector spaces 81. D e f i n i t i o n and elementary p r o p e r t i e s of v e c t o r spaces. Suppose t h a t a non-empty set E is g i v e n , and t h a t t o each ordered p a i r (x,y) of elements of E t h e r e corresponds an element x + y of E ( c a l l e d t h e sum of x and y ) and t h a t f o r each number t and X E E a n element tx of E ( c a l l e d t h e product of t h e number t w i t h t h e element x ) is d e f i n e d i n such a way t h a t t h e s e o p e r a t i o n s , namely addition and ~ c a l a rmultiplication s a t i s f y t h e following c o n d i t i o n s (where x , y and 2 denote a r b i t r a r y elements of E and a,b are numbers): 1) x + y = y + x , 2 ) x + (y + 8 ) = (x + y ) + 2 , 3 ) x + y = x + 8 impties y = 2 , 4)
a(x
+ y ) = ax + a y ,
( a + b)x = ax + bx, 6 ) a(bx) = (ab)x, 7) 1.x = x. Under t h e s e hypotheses, we say t h a t t h e set E c o n s t i t u t e s a vector o r tinear space. I t is easy t o see t h a t t h e r e then e x i s t s e x a c t l y one element, which we denote by (3, such t h a t x + ( 3 = x f o r a l l X E E and t h a t t h e e q u a l i t y a x = bx where x z (3 y i e l d s a = b ; furthermore, t h a t t h e e q u a l i t y ax = ay where a * 0 i m p l i e s x = y. Put, f u r t h e r , by d e f i n i t i o n : -x = (-112 and x - y = x + (-y). Examples 1-10 of metric spaces, d e s c r i b e d on pp. 5 and 6 , a l s o s e r v e a s examples of v e c t o r spaces, when t h e u s u a l d e f i n i t i o n s of a d d i t i o n and s c a l a r m u l t i p l i c a t i o n a r e adopted. When x * y , w e understand by t h e line segment j o i n i n g x and y t h e set of a l l elements of t h e f o r m tx + ( 1 t ) y where t is any number i n the interval [0,1]. A set C S E is said t o be convex, when it c o n t a i n s every l i n e segment j o i n i n g a r b i t r a r y p a i r s of elements of C. If x1,x2, x n are elements of a v e c t o r space, t h e expression 5)
-
...,
alxl + a2x2 +
... + anxn
=
n Caixi,
i=l
where al,a2,...,an are a r b i t r a r y real numbers, is c a l l e d a tinear aombination of t h e elements x1,s2,...,xn. 52. Extension of a d d i t i v e homogeneous f u n c t i o n a l s . L e t E and E l b e two v e c t o r spaces and f a mapping i n E whose codomain lies i n El.
18
S.
BANACH
The mapping f is s a i d t o be a d d i t i v e when f o r each pair of elements x , y we have: f ( x + y) = f ( x ) + f ( y ) ; it is c a l l e d homogeneous when f o r each element x and each number t : f(tx) = tf(x). Suppose t h a t t h e r e a r e g i v e n a f u n c t i o n a l p d e f i n e d i n E s u c h t h a t f o r a l t x , y ~E
THEOREM 1. 1'
p ( x + t l ) 5 p ( x ) + P(Y) and p ( t x ) = t p ( x ) f o r t L 0 , and 2'
an a d d i t i v e homogeneous f u n c t i o n a l f d e f i n e d i n a v e c t o r subspace G s E (1.e. a s u b s e t of E t h a t is i t s e l f a v e c t o r space with t h e s a m e d e f i n i t i o n s of t h e b a s i c o p e r a t i o n s ) s u c h t h a t f o r each X E G
f(x) 4 P b ) , t h e n t h e r e atways e x i s t s an a d d i t i v e homogeneous f u n c t i o n a l F d e f i n e d i n E such t h a t F ( x ) 6 p ( x ) for each x E E and F ( x ) = f ( x ) f o r e a c h x E C . ~ By 2', for P r o o f . W e can assume t h a t G ; t E ; l e t x , E\G. x ' , x " E C w e have: f(x") f ( x ' ) = f ( x " - x ' ) 5 p ( x " - x ' ) = p"x"+x,) + (-x'-x,)l
-
p(x"+x,) + p(-2'-
whence
- ftx')
-p(-x'-z,)
The numbers m sup
ip(x"+x,)
S O ) '
- f(x").
xEG
are t h e r e f o r e f i n i t e and m 6 M . I f r o is any number such t h a t m s r , 5 M I w e have f o r each X E C (11
-
-
f ( x ) 6 r, 5 p ( l : + x o ) f ( x ) . of a l l elements y of t h e form (2) y = x + t x , where x i C and t i s a number. C l e a r l y C, i s a v e c t o r space. Put (3) g(y) f ( + ) + tr,, where t h e element y i s given by ( 2 ) ; a s x o E E x C , each y~ 0, has a unique r e p r e s e n t a t i o n i n t h e form of ( 2 ) so t h a t t h e f u n c t i o n a l g is We a l s o see t h a t g i s a d d i t i v e and homogeneous well-defined on G,. on c, and c o i n c i d e s with f on G. W e now show t h a t (4 1 g ( y ) 5 p ( y ) f o r each y E G o I n f a c t , i f one writes y i n t h e form ( 2 ) it can be assumed t h a t t t 0. P u t t i n g x / t i n p l a c e of x i n t h e i n e q u a l i t y ( 1 ) and multiplyi n g i t s r i g h t - o r l e f t - h a n d s i d e , according as t > 0 o r t <0 , by t , one o b t a i n s t r , s p ( x + t x , ) f ( x ) which by ( 3 ) i m p l i e s t h e i n e q u a l i t y -p(-x-x,)
Consider t h e s e t
0,
-
.
(4).
-
T h i s e s t a b l i s h e d , it now s u f f i c e s t o well-order t h e set E\G, o b t a i n i n g , by s u c c e s s i v e e x t e n s i o n s of f , following t h e procedure d e s c r i b e d above, a f u n c t i o n a l F s a t i s f y i n g t h e conclusion of t h e theorem.
19
General vector spaces
Given a f u n c t i o n a l p d e f i n e d i n E s u c h t h a t f o r x , y E E p ( x + y ) 5 p ( x ) + p ( y ) and p ( t x ) = t p ( x ) f u r t.? 0 , t h e r e e x i s t s an a d d i t i v e homogeneous f u n c t i o n a Z F d e f i n e d i n E such t h a t , f o r each X E E COROLLARY.
F(x) 6 p b ) . In f a c t , c o n s i d e r an x E E and denote by G t h e set of a l l elements of t h e form t x , where t 1s a n a r b i t r a r y number. G i s then a v e c t o r space. P u t t i n g f ( t x 0 )= t p ( x o ) i n G I w e w i l l have f ( t x , ) 5 p ( t x , ) f o r any t , s i n c e t 2 0 i m p l i e s t p ( x , ) = p ( t x , ) and t <0 i m p l i e s O = p ( O ) s p ( x o )+ p ( - x o ) , whence p ( x , ) L - p ( - x o ) and f i n a l l y t p ( x o )6 - t p ( - x , ) = p ( t x , ) ; t h e r e s u l t now f o l l o w s on applying theorem 1.
53.
Applications:
measure
of t h e n o t i o n s of i n t e g r a l , of
generalisation
and
limit.
of
W e a r e now going t o d i s c u s s s e v e r a l i n t e r e s t i n g a p p l i c a t i o n s of theorem 1 and i t s c o r o l l a r y . 1. L e t E be t h e set of bounded real-valued f u n c t i o n s x ( s ) d e f i n e d on a c i r c l e of u n i t circumference where s denotes a r c - l e n g t h measured from some f i x e d p o i n t , always i n t h e same sense. With t h e u s u a l a l g e b r a i c o p e r a t i o n s , E is a v e c t o r space. NOW, f o r each element x = x ( s ) of E l l e t us d e f i n e p ( x ) t o be t h e i n f i m u m of a l l t h e numbers M ( x ; a l , a 2 , . . . , a n ) of t h e form
M(x;alra2,.*.ran)
where a,,a,,...,an f u n c t i o n a l p then f a c t , i t is p l a i n t S 0. Secondly, given E > 0, there e x i s t p1,f12,...,& such
=
sup
-m
(4 k=I x ( s + a k ) ) ,
is an a r b i t r a r y f i n i t e sequence of numbers. The s a t i s f i e s a l l t h e hypotheses of t h e c o r o l l a r y . In t h a t , f i r s t l y , one always has p ( t x ) = t p ( x ) f o r two elements x = ~ ( 8 and ) y = y(s) of E and a number f i n i t e sequences of numbers a,,a2,...,au and that
(5) ~ ( x ; u ~ , a ~ ~ . . . 6, ap ~( x) ) + E and M ( y r f 3 1 , B 2 1 . . . 1 8 v ) 5 p ( y ) + E. Arranging a l l t h e numbers a ; + S j where i = l I 2 , . . . , u and j = l r 2 , . . . , v as a s i n g l e sequence y 1 , y 2 , . . . , y U v . i n some way, one h a s (6) p ( x + 21) 5 M ( x + y t y l r Y 2 r . . . r Y ~ ~ ) and it is e a s i l y v e r i f i e d t h a t
.
..
..
( 7 ) ~(x+gty,,y,,. ,yU,) 5 M ( x ; a l l a z l . . ,aU) + M ( Z / ; S ~ ~ B ~ ,,&,I. . The r e l a t i o n s (5) ( 7 ) imply p ( x + y ) 2 p ( x ) + p ( y ) + 2 ~ which , proves t h e number E > 0 being a r b i t r a r y , t h a t p ( x + y ) 6 p ( x ) + p ( y ) . T h i s e s t a b l i s h e d , c o n s i d e r t h e r e f o r e t h e f u n c t i o n a l F which e x i s t s by t h e c o r o l l a r y . Now, i f X ( S ) = 1 , w e have p ( x ) = 1 and p ( - m ) = -1 and as P ( x ) d p t z ) and F ( X ) = - F ( - x ) L - P ( - z ) , one o b t a i n s F ( x ) = 1 . I f x ( 8 ) L 0 , w e have p ( - x ) 5 0 and moreover F ( x ) = - F ( - x ) L - p ( - x ) , so t h a t F ( x ) t 0 also. Furthermore, t h e f u n c t i o n a l F h a s t h e property of s a t i s f y i n g , f o r each number s o , t h e e q u a l i t y F [ x ( s+ s o ) ] = F [ x ( s ) l . I n f a c t , if one one h a s p u t s y ( e ) = x ( s +s o ) - x ( 8 ) and a k = ( k - l ) s o f o r k=1,2,.. for each n:
-
.,
p ( y ) 5 ~ ( ~ ' ; a ~ , a , , . . . , a n=)
1
SUP
'--a <=
[=(s+nso)-=(8)l,
20
S. BANACH
so t h a t p ( y ) 5 0; one s i m i l a r l y o b t a i n s p ( - y ) 5 0 . F ( y ) = -F(-y) b - p ( - y ) , whence F ( y ) = 0.
But F ( y ) 5 p ( y ) and
Thus, using t h e symbol j x ( s ) d s t o denote t h e f u n c t i o n a l f { F [ x ( s ) +] F [ z ( l - s ) ] } , one h a s t h e theorem: T o e v e r y f u n c t i o n x ( 8 ) of t h e class E one can a s s o c i a t e a number j x ( s ) d s i n such a way t h a t t h e foEZawing c o n d i t i o n s (where z ( s ) and y ( s ) a r e a r b i t r a r y f u n c t i o n s of t h e c l a s s E and a , b , s o a r e numbers) are s a t i s f i e d : 1) j[az(s) + by(s)]ds = alx(s)de + b/y(s)ds, 2) x ( s ) d s 2 0 when z ( s ) 2 0 , 3) 1 x ( s + sO)ds = jz(s)ds, 4) ~ ( 8l ) d s 1z(8)d8, 5) Ids = 1 . I t is easy t o check t h a t t h e f u n c t i o n a l j x ( s ) d s , s a t i s f y i n g con5 1 , always l i e s between t h e lower and upper Riemann ditions 1 ) i n t e g r a l s of t h e f u n c t i o n ~ ( 8 ) .Consequently, f o r every Ri n t e g r a b l e f u n c t i o n , t h i s f u n c t i o n a l c o i n c i d e s with t h e i n t e g r a l of t h e function. For 6-summable f u n c t i o n s , t h e f u n c t i o n a l i n q u e s t i o n does n o t always c o i n c i d e with t h e i r L-integral. Nevertheless, s t a r t i n g w i t h t h e v e c t o r space G of such (L-summable) f u n c t i o n s and d e f i n i n g t h e f u n c t i o n a l f ( x ) i n G t o be t h e L - i n t e g r a l of t h e f u n c t i o n X ( S ) E G , theorem 1 f u r n i s h e s a f u n c t i o n a l F d e f i n e d i n t h e space E such t h a t t h e f u n c t i o n a l j z ( s ) d s = ~ { F [ z [ s +) ]F [ z ( l - s ) ] } c l e a r l y s a t i s f i e s conditions 1 ) 5 ) and, furthermore, c o i n c i d e s , f o r every L-summable f u n c t i o n , with t h e i n t e g r a l of t h a t f u n c t i o n . Consider now t h e c l a s s x o f a l l s u b s e t s of t h e circumference 2. of t h e c i r c l e i n q u e s t i o n and denote by A, t h e circumference i t s e l f . P u t t i n g , f o r each s e t A of t h i s c l a s s , p ( A ) = I z ( s ) d s , where x ( s ) is t h e c h a r a c t e r i s t i c f u n c t i o n of t h e set A and t h e r e f o r e a f u n c t i o n of t h e space E d i s c u s s e d i n 1 , w e o b t a i n t h e theorem: Po each s e t A of t h e class %one can a s s i g n a number w ( A ) i n such a way t h a t t h e f o l owing c o n d i t i o n s (where A and B are a r b i t r a r u are s a t i s f i e d : s e t s o f t h e cZass&j 1) b(A U B ) = b ( A ) + b(B) whenever A f l B = @ ,
I
I I
-
-
-
2)
P(A)
2 0,
3)
B(A)
=
4)
w(Ao) = 1.
V(B) i f A
B,
-
The f u n c t i o n a l b ( A ) , which s a t i s f i e s c o n d i t i o n s 1 ) 4) lies between t h e i n n e r and o u t e r Jordan measures of t h e s e t A . Cons e q u e n t l y , f o r every J-measurable set, t h i s f u n c t i o n a l c o i n c i d e s with the measure of t h e set. For a r b i t r a r y L-measurable sets, t h e f u n c t i o n a l i n q u e s t i o n does n o t always c o i n c i d e with t h e i r L-measure, b u t , j u s t a s b e f o r e , one can a r r a n g e t h i n g s i n such a way t h a t t h i s p r o p e r t y a l s o holds. 3. L e t E be t h e set of a l l bounded r e a l - v a l u e d f u n c t i o n s x ( s ) d e f i n e d i n [ 0 , + - ] ; with t h e u s u a l d e f i n i t i o n s of a l g e b r a i c operat i o n s , this is a v e c t o r space. t h e infimum of a l l For each element z = x ( s ) of E , denote by p )I(
n
..
t h e numbers 8C x ( s + ak) , where al,ao,. ,an is a n a r b i t r a r y n k=i f i n i t e sequence of p o s i t i v e numbers. One e a s i l y v e r i f i e s t h a t t h e f u n c t i o n a l p t h u s defined i n t h e space E s a t i s f i e s t h e hypotheses of
General vector spaces
21
52
~ ( 8 )the functional P I the corollary. Denoting by the symbol which exists by the corollary, one therefore has the theorem: T o e v e r y f u n c t i o n x ( e ) E E one can a s s o c i a t e a number big x ( s ) i n such a way t h a t t h e f o l l o w i n g c o n d i t i o n s (where x ( s ) and y ( e ) a r e a r b i t r a r y f u n c t i o n s o f E and a , b and s o L 0 a r e numbers) a r e satisfied: x ( e ) + b 5% y ( s ) , 1) $& [ a x ( s ) + by(s)l = a X& 2) I&g x ( s ) k 0 wfieneuer x f s ) 2 0 , x(s+ so) = & r X(8)I 3 ) Lim 84 ) L&g l = l. The functional %@ x ( s ) satisfying conditions 1 ) 4) always lies between lim ~ ( 8 )and ~ ( 8 ) . It consequently coincides with. r s Note that X ( 8 ) whenever this limit exists in the usual sense. Lim here denotes a certain generalised while lim is reserved exclusively for limit in the usual sense. 4. Let ( E n ) be any bounded sequence. Define the function x ( s ) in ( O f + - ) by: x ( s ) = En for n-1 < s S n , and n = 1 , 2 , . . . The function = ( a ) thus belongs to the set E discussed in 3. Putting f;& E n = % % X ( 8 ) in the sense of 3, one has the theorem: To each bounded sequence ( c n ) one can a s s o c i a t e a number kiet En i n such a way t h a t t h e f o l l o w i n g c o n d i t i o n s (where ( E n ) and ( n n ) a r e a r b i t r a r y bounded sequences and a and b a r e numbers) a r e aa tisf i sd: 1) (aSn + b n n ) a f;&En + b f;&n n t 2) 5,L 0 , if E n 2 0 f o r n=lr2,...r 3) f;& Em1 = +&k! 5 8 , 4) 1 = 1. thus defined Conditions 1 ) 4) imply that the functional I&y En. Consequently, for every always lies between lim En and iF= convergent sequence this functional coincides with the limit of the sequence in the usual sense.
-
BQ
b& k&
-