Calculus in limit spaces
by Kent Franklin Carlson
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY in Mathematics
Montana State University
© Copyright by Kent Franklin Carlson (1967)
Abstract:
The purpose of this paper is to examine the axiomatic characterization of a differential calculus as
proposed by Wehrli (8) for the case of linear limit spaces over an arbitrary field in view of the usual
notion of a derivative. Whether or not this characterization is satisfactory is partially dependent upon an
open question in the theory of limit spaces which I have answered in the form of the following
theorem: Let E and F be linear Limit spaces over a field K with limitations Λ and Λ' . respectively, and
let Λ0and Λ'0 be the associated principle ideal limitations. Then if Λ x Λ' represents the product
limitation of E x F, (Λ x Λ')0 = Λ0 x Λ'0. With this theorem the axiomatic characterization proposed
by Wehrli is shown to be quite natural in regard to the usual notion of differentiation on linear limit
spaces. It is then shown that the remainders introduced by Binz (1), together with the requirement that
the set of derivatives consist of all linear continuous mappings, satisfy these axioms. Finally a
generalization of these remainders which I have called the Strong Binz remainders is formulated: Let K
be a separated field and suppose that there is some calculus given on K. Define χ= {(E, Λ ), (F, Λ'), (G,
Λ"),...} to be the class of all linear limit spaces over K, and let α(EΛ ,FΛ' ) be the set of all linear
continuous mappings from EΛ to FΛ' , for all EΛ , FΛ' in χ. Then define the mapping r:EΛ→FΛ' to be
a Strong Binz remainder if r(0)=0 for 0 ⊆ E, if r is continuous at 0 ⊆ E, and if for every filter &theta
converging to 0 ⊆ E with respect to Λ there exists a filter ψ converging to 0 ⊆ F such that for every N
⊆ ψ there is an M ⊆ θ and a ς such that r(λM) ⊆ ς(λ)N, for all λ in a U-set about 0 ⊆ K. Here ς
represents a remainder from the calculus on the base field. Two methods for constructing a calculus
containing the Strong Binz remainder are then introduced, which lead to the same result. CALCULUS IN LIMIT SPACES
by
KENT FRANKLIN CARLSON
A t h e s i s s u b m itte d to th e G ra d u a te F a c u lty in p a r tia l
fu lfillm e n t o f th e re q u ire m e n ts fo r th e d e g re e
of
■ ■
DOCTOR OF PHILOSOPHY
in
M a th e m a tic s
A pproved:
C h a irm a n , E x a m in in g ' C o m m itte e
D ea:
e a h o f G ra d u a te D iv is io n
MONTANA STATE UNIVERSITY
B o z e m a n , M o n ta n a
J u n e , 1967
iii
A ck n o w led g m en t
I
w is h to e x p r e s s my s in c e r e th a n k s to
P r o fe s s o r M a rtin W e h rli fo r h is g u id a n c e in th is
r e s e a r c h p r o je c t. H is s u g g e s tio n s an d c r i t i c a l
o b s e r v a tio n s h a v e b e e n o f g r e a t im p o rta n c e to m e .
T a b le of C o n te n ts
C h a p te r
V ita
A c k n o w led g m e n t
A b s tra c t
In tro d u c tio n
I-'
B ack g ro u n d
II ■
D if f e r e n tia tio n in L im it S p a c e s
III
IV
The B inz R em ain d er
x
The S trong Binz R em ain d er
L ite ra tu re c ite d
V
A b s tra c t
The p u rp o s e o f th is p a p e r is to e x a m in e th e a x io m a tic c h a r a c te r iz ­
a tio n of a d if f e r e n tia l c a lc u lu s a s ' p ro p o s e d b y W e h rli (8) fo r th e c a s e of
lin e a r lim it s p a c e s o v e r a n a rb itra ry f ie ld in v ie w of th e u s u a l n o tio n of
a d e r iv a tiv e . W h e th e r or n o t th is c h a r a c te r iz a ti o n is-, s a ti s f a c to r y is
p a r tia lly d e p e n d e n t u p o n a n o p e n q u e s tio n in th e th e o ry o f lim it s p a c e s
w h ic h I h a v e a n s w e re d in th e form of th e fo llo w in g th eo rem : L et E
an d F b e lin e a r L im it s p a c e s o v e r a f ie ld K w ith lim ita tio n s A a n d A' ..
r e s p e c t i v e l y , a n d l e t Vleia n d A0 b e th e a s s o c i a t e d p rin c ip le id e a l
l im it a ti o n s . T hen i f A x A r e p r e s e n ts th e p ro d u c t lim ita tio n o f
E x F , ( A x A ' )q = A x tT q . W ith th is th e o re m th e a x io m a tic
c h a r a c te r iz a tio n p ro p o s e d b y W e h rli i s sh o w n to b e q u ite n a tu r a l in re g a rd
to th e u s u a l n o tio n of d if f e r e n tia tio n on lin e a r lim it s p a c e s . I t i s th e n
sh o w n th a t th e re m a in d e rs in tro d u c e d b y B inz (I), to g e th e r w ith th e
re q u ire m e n t th a t th e s e t o f d e r iv a tiv e s c o n s i s t of a ll lin e a r c o n tin u o u s
m a p p in g s , s a t i s f y t h e s e a x io m s . F in a lly a g e n e r a liz a tio n of th e s e
re m a in d e rs w h ic h I h a v e c a lle d th e S tro n g Binz r e m a in d e r s - is fo rm u lated :
L et K b e a s e p a r a te d f ie ld a n d s u p p o s e t h a t th e r e is so m e c a lc u lu s
g iv e n on K. D e fin e /^= ((E, vl ) , (F, A '), (G , A"),. . . . } to b e th e c l a s s of
a ll lin e a r lim it s p a c e s o v e r K, an d l e t /((Eyl , F a ' ) b e th e s e t of a ll
lin e a r c o n tin u o u s m a p p in g s from Ea to Fa ' , fo r a ll E a , F yl- in 9( . T hen
d e fin e th e m ap p in g ' r : Ea -^Fa - to b e a S trong Binz re m a in d e r if r(0)= 0 for
0 6 E, if r is c o n tin u o u s a t 0 <=E, a n d if fo r e v e ry f il te r $ c o n v e rg in g
to 0 a E w ith r e s p e c t to A th e r e e x is t s a f i l t e r yA c o n v e rg in g to 0 & F
s u c h t h a t for e v e ry N 6 Z1th e r e i s a n M ^ / a n d a o- su c h th a t
r ( /LM) £ cr (JL )N ,
.
fo r a ll /I in a U - s e t a b o u t 0 a K. H e re cr r e p r e s e n ts a re m a in d e r from
■the c a lc u lu s on th e b a s e f i e l d . Two m e th o d s fo r c o n s tr u c tin g a c a lc u lu s
c o n ta in in g th e S trong B inz re m a in d e r a re th e n in tr o d u c e d , w h ic h le a d
to th e sa m e r e s u l t .
''
A
vi
INTRODUCTION
In r e c e n t y e a r s s e v e r a l a tte m p ts h a v e b e e n m ad e to d e f in e in g e n e r a l .
a n o tio n o f d if f e r e n tia b ility fo r to p o lo g ic a l s p a c e s .
In (3) fo r e x a m p le ,
D ie u d o n n e 1 h a s i l l u s t r a t e d a d e r iv a tiv e s tr u c tu r e fo r B an ach S p a c e s w h ic h
i s v e ry e le g a n t.
As a n e x te n s io n o f t h e s e c o n s id e r a tio n s i t s e e m s q u ite
n a tu r a l to try to g e n e r a liz e th e n o tio n of d if f e r e n tia b ility to m a p p in g s
b e tw e e n m ore a b s t r a c t s p a c e s w h ic h do n o t n e c e s s a r i l y p o s s e s s n o rm s .
L im it s p a c e s or s o - c a l l e d p s e u d o to p o lo g ic a l s p a c e s w e re fo u n d to b e
v e ry u s e f u l fo r th is d e v e lo p m e n t
( c f . • (I)
,
(6)).
In p a r tic u la r in ( I ) ,
E. Binz p ro p o s e d a n o tio n ,o f d if f e r e n tia b ilit y fo r lin e a r lim it s p a c e s
o v e r th e r e a l f i e l d .
. In (8) W e h rli h a s .o f f e r e d a c o lle c tio n o f s ix te e n ax io m s
to
c h a r a c te r iz e a^d if f e r e n tia l c a lc u lu s a n d h e h a s sh o w n t h a t c o n c e p ts of
d if f e r e n tia tio n in to p o lo g ic a l s p a c e s s a t i s f y th e m .
The q u e s tio n of w h e th e r
or n o t th e re m a in d e rs in tr o d u c e d by B inz s a t i s f y t h e s e a x io m s is in p a rt
e q u iv a le n t to a n o p e n q u e s tio n in th e th e o ry o f lirriit s p a c e s .
a n s w e re d th is q u e s tio n in th e a ffirm a tiv e ( c f . T heorem 2 . 2 ) .
I have
In th is p a p e r
I w ill a d o p t th e a x io m a tic c h a r a c te r iz a ti o n o f a d if f e r e n tia l c a lc u lu s
p ro p o s e d b y W e h rli a n d sh o w th a t it.d o e s in f a c t c h a r a c te r iz e th e u s u a l
j
n o tio n of d if f e r e n tia tio n v e ry w e ll in th e c a s e o f .lin e a r lim it s p a c e s .
In c h a p te r I a few b a s i c id e a s o f f il te r th e o ry a n d lim ita tio n s a re
c o n s id e re d ,- a n d th e . s ix te e n a x io m s in tr o d u c e d b y W e h rli a r e n o te d .
■vii
In c h a p te r II th e a b o v e m e n tio n e d r e s u l t i s sh o w n in T heorem 2 . 2 ,
an d th e a x io m s p ro p o s e d by W e h rli a re t e s t e d in v ie w of th e d e s ir e d
g e n e r a l r e q u ire m e n ts fo r a d if f e r e n tia tio n s tr u c tu r e .
In c h a p te r II I, th e re m a in d e r in tr o d u c e d b y Binz i s c o n s id e r e d an d
sh o w n to b e a n e x a m p le o f a re m a in d e r w h ic h , to g e th e r w ith th e r e q u ir e ­
m e n t th a t, th e s e t o f d e r iv a tiv e s b e m ad e up of a ll lin e a r c o n tin u o u s
f u n c t i o n s , s a t i s f i e s th e s ix te e n a x io m s w h ic h c h a r a c te r iz e a c a l c u l u s .
F in a lly in c h a p te r .IV, a n e x te n s io n o f th e re m a in d e r in tro d u c e d by
B in z , w h ic h I.h a v e c a ll e d th e S trong B inz re m a in d e r is p ro p o s e d , an d
th is n e w d e fin itio n is e x a m in e d in v ie w o f th e a x io m s w h ic h c h a r a c te r iz e
a c a lc u lu s .
C h a p te r I
B ackgro u n d
In th is c h a p te r so m e fu n d a m e n ta l n o tio n s o f f i l t e r th e o ry an d lim it
s p a c e s a re c o m p ile d w h ic h a re n e e d e d in th e d e v e lo p m e n t to fo llo w ,
( c f . (2), (4)).
L im ita tio n s
D e fin itio n
L et E b e a n o n e m p ty s e t an d s e t 'Z5(E) b e th e p o w e r s e t o f E.
A f il te r ^ on E is a n e le m e n t of P(E) w ith th e fo llo w in g p ro p e rtie s :
(1) th e em p ty s e t d o e s n o t b e lo n g to <p .
(2) for a n y tw o s u b s e ts X an d Y o f E, if X 6 0 , and
X GY, th en YC 0 . (3) for a n y tw o e le m e n ts X an d Y of 0 , X H Y £ 0 .
A s u b c o lle c tio n ^ o f a f ilte r 0 on E is c a lle d a f li t e r b a s i s o f 0 if
0 = [ X G E | f o r som e Y£2T, YCX].
It is c le a r th e n th a t a n o n e m p ty c o lle c tio n o f s u b s e ts o f E is a f ilte r
b a s i s t for som e f il te r 0 on E if an d o n ly if
(1) th e em p ty s e t d o e s n o t b e lo n g to t ,
(2) an d if Be
an d B1& if , th e n t h e r e - e x is ts a B" e /s u c h ,
th a t B" £. B f iB 1. '
N ow l e t E b e a n o n e m p ty s e t , a n d c o n s id e r th e fo llo w in g n a tu ra l
o rd e r A of a ll f ilte r s Jf(E) on E.
If j a n d 9 a re f il te r s in Ji (E ), th e n
$ 4 $ m e a n s th a t for e v e ry M £ 0 th e re e x is t s a n N £ .0 s u c h th a t N S M . W e
s a y in th is c a s e th a t (9i s . fin e r th a n f or ft i s c o a r s e r th a n & , an d
9 is c a lle d a n o v e r f ilte r ot ft
,
W e d e fin e th e u ltr a f i lte r of ^ ( E ) .t o b e th e m ax im al e le m e n t o f Jy (E ).
D e fin itio n
a
A
An e le m e n t I of th e pow er s e t f { -J/ (E)) o f ' J (E) is c a lle d
I d e a l in
Ji (E)
if a n d o n ly if
(1) w h e n e v e r 0 & I , a l l o v e r f ilte r s in J / (E) o f $ l ie in I , and
w h e re (J)^d , c a ll e d th e jo in of
(2) w h e n e v e r ^
■(j) a n d 9 , c o n s i s t s o f a l l s e t s in E c o n ta in in g s e t s from
(A UB I A6 ^ ,
B69j .
W e c a l l I a A P rin c ip le id e a l
J
(a A - H a u p tid e a l in G erm an ) in ,
(E) if I c o n s i s t s of a l l o v e r f ilte r s of a f il te r 0, <s^.(E).
I is th e n
s a id to b e g e n e r a te d b y 0 .
A gain le t E b e a n o n e m p ty s e t .
D e fin itio n
A m apping
A : E —>
/Z?(4(E)) is c a lle d a lim ita tio n o r sim p ly a
c o n v e rg e n c e s tr u c tu r e on E (L im iterung in G erm an) if an d o n ly if
(1) for e v e ry x z E, A (x) is a Tl id e a l in J^ (E ),
(2) an d for e v e ry x <? E, x 6 A & ) , w h e re & is th e f il te r
c o n s is tin g of a l l s u b s e ts of E c o n ta in in g x .
The f il te r s in A W a re d e fin e d to b e c o n v e rg e n t to x £ E r e la tiv e to
A
o T hen th e s e t E to g e th e r w ith th e s tr u c tu r e
A
i s c a lle d a lim it
3
sp ace
( L im esrau m
in G erm an ) .
In th is p a p e r (E, A
) a n d Ea
w ill b e u s e d in te r c h a n g e a b ly to d e n o te th is lim it s p a c e .
W e n o te th a t e v e ry to p o lo g y is a lim ita tio n s in c e th e c o n v e rg e n t
f il te r s to a n y p o in t c le a r ly form a A i d e a l .
c o n v e rg e n c e of f ilte r s ' to a p o in t x
O f c o u r s e th e d e f in itio n of
in a to p o lo g ic a l s p a c e
X re q u ire s
t h a t e v e ry n e ig h b o rh o o d o f th e p o in t c o n ta in a s e t of th e f i l t e r .
c le a r ly c o n s i s t e n t w ith th e re q u ire m e n t t h a t x
D e fin itio n
L et A b e so m e lim ita tio n on s e t E.
T h is is
c o n v e rg e to x .
If fo r e v e ry x £ E, th e
f il te r s c o n v e rg e n t to x r e la tiv e to A form a .p rin c ip le i d e a l , A is
■
c a lle d a p r in c ip le id e a l lim ita tio n on E. (H a u p tid e a l L im itie ru n g in
G erm an)
. .
A p r in c ip le id e a l lim ita tio n is a to p o lo g y on E if a n d o n ly if for e v e ry
x £ -E A
, th e ' c o a r s e s t f i l t e r
0 (x) £ A (x) s a t i s f i e s th e re q u ire m e n t, th a t
to e v e ry U £ ^ ( x ) , th e r e e x i s t s a V £ ^ (x) s u c h t h a t fo r a l l y £ V,
U<= f ( y ) .
:
. L et x £ Ea
a n d .c o n s id e r A (x ).
to b e th e f il te r
g e n e r a te d by
a re th e f il te r s in A (x ).
A
(x) .
D e fin e th e lo w e r b o u n d o f A (x)
U Ny^ ,
£/>\(x), w h e re th e
^ (x)
T his f il te r of c o u r s e d o e s n o t a lw a y s b e lo n g to
If o n e c o n s tr u c ts fo r e v e ry p o in t x £ E , th e f il te r
(j) (x ), and
d e f in e s i t to b e lo n g -to A (x ), th e n a ,p r in c ip le id e a l lim ita tio n i s form ed
o n Ea
w h ic h is u n iq u e ly d e te rm in e d b y . A
. W e d e n o te th is b y ' A 0 .
4
D e fin itio n
L et
b e /L o p e n
(E, ' A ) b e a lim it s p a c e .
A s u b s e t A ^ E i s s a id to
if a n d o n ly if fo r e v e ry x £ A,. A^ <£ , fo r e v e ry f il te r (W l(x ).
F is h e r h a s sh o w n th e fo llo w in g in (4):
' T heorem 1 . 1 .
The s e t o f a l l A o p e n s e t s Q in (E, A q ) form s a to p o lo g y
on E, in w h ic h th e s e t s o f Q a re th e o p e n s e t s .
A
W e c a l l th is th e to p o lo g y a s s o c i a t e d w ith
N ow l e t A b e n o n e m p ty s u b s e t of s e t E.
.
’
' j;
1
A ll f i l t e r s o f a A id e a l in
A (E) w h ic h h a v e n o n e m p ty i n te r s e c tio n w ith A, g e n e r a te in A (A )
a A .id e a l.
set A
W e n o te .h e r e t h a t n o n e m p ty i n te r s e c tio n of a f i l t e r ft w ith
m e a n s th a t a l l s e t s of ft h a v e n o n e m p ty i n te r s e c tio n w ith A.
D e fin itio n
L et E^
b e a lim it s p a c e a n d A a n o n e m p ty s u b s e t of E .
v
A s s o c ia te w ith e v e ry x 6 A th e A i d e a l A
(x) in A (A) w h ic h is
s
g e n e r a te d in A by A (x) th ro u g h in t e r s e c t i o n o f f il te r s in
■
'
A
.
(x) w ith .
■■
;
■:
A, w h e re of c o u r s e o n ly f il te r s in A (x) w ith n o n e m p ty i n te r s e c tio n w ith
A a re c o n s id e r e d .
T hen A ^ : A
c a lle d th e lim ita tio n in d u c e d
( J? (A)) i s a, lim ita tio n .o n A a n d is
on A 'b y A V
S u p p o s e th a t A , a '- a re b o th lim ita tio n s d e fin e d on E.
D e fin itio n
,
/
■ A i s s a id to b e fin e r th a n A if a n d o n ly if fo r e v e ry x £ E,
/
■
,
A (x )-v l (x:). W e d e n o te th is s y m b o lic a lly b y A A A
th a t A
'
•
.' ' W e a ls o s a y
is c o a r s e r th a n A . ■
N ow l e t E a n d F b e tw o n o n em p ty s e t s , a n d l e t f b e a m a p p in g ffom E
.
I1
I
5
to F .
If sZi i s a f il te r in jf(E ), i t i s m a p p ed b y f in to th e s e t
, in F 7 w h ic h form s a f i l t e r b a s is on F .
D e fin itio n
A m a p p in g f:E ^—' Fy
fo r e v e ry f il te r f a A ( x ) ,
is c o n tin u o u s a t x 6 E
if a n d o n ly if
f( <f) ) is a f i l t e r b a s i s o f a f i l t e r in
A z (f(x.)).
O f c o u r s e .if th e m a p p in g f is c o n tin u o u s a t a ll x £ E 7 i t i s s a id to b e
c o n tin u o u s on E .
T his d e f in itio n o f c o n tin u ity i s e n tir e ly c o n s i s t e n t w ith th e d e fin itio n
g iv e n in th e c a s e o f to p o lo g ic a l s p a c e s .
p ro v e n in p a rt b y F is c h e r in
W e n o te th e fo llo w in g th e o re m
(4) :
'
x
The d e f in itio n of c o n tin u ity fo r a fu n c tio n in .th e c a s e of
T heorem 1 .2
\
lim ita tio n s i s e q u iv a le n t to th e d e f in itio n o f a c o n tin u o u s fu n c tio n fo r '
to p o l o g i e s .
Proof
S u p p o s e f i r s t o f a l l th a t w e h a v e a c o n tin u o u s fu n c tio n f-E-yF 7
f
w h e re E a n d F a r e to p o lo g ic a l s p a c e s . .L e t x —» y , w h e re x a n y p o in t in
E 7 a n d l e t (f) be' a n y f i l t e r 'o n E c o n v e rg in g to x ln ^ th e to p o lo g ic a l s e n s e .
C o n s id e r fh e n f ( ^ ) .
If U is a n y o p e n n e ig h b o rh o o d of y 6 F th e n by
d e fin itio n f. ^ (U) i s a n o p e n n e ig h b o rh o o d o f x $ E . . T hen th e r e e x is t s
a V 6 ^ s u c h t h a t V £ f ^ (U ).' T h e re fo re f
■*
f(V) S U 7 a n d U .6 f ( 0 ) .
(U) 6
It fo llo w s th e n t h a t
W e h a v e sh o w n th e n th a t fo r a n y o p e n n e ig h b o r-
6
h o o d U o f y £ F , a n im a g e s e t f (V) o f a s e t V in th e f i l t e r cp c o n v e rg in g
to x i s c o n ta in e d , in U .
T hus th e im a g e of 0 u n d e r f form s a f ilte r
c o n v e rg e n t to y .
■ Now s u p p o s e a g a in th a t E a n d F a re to p o lo g ic a l s p a c e s , a n d f:E-*F
i s c o n tin u o u s m ap p in g
b u t now in th e s e n s e of l im it a ti o n s .
b e a n y o p e n s e t in F , a n d l e t M = I- ^ (N ).
x £ M an d l e t f(x) = y £ N .
a n y f il te r c o n v e rg in g to x .
L et N
C h o o s e a n a r b itr a r y p o in t
W e m u st sh o w th a t f""1"(N) £ O , w h e re 0 is
By th e c o n tin u ity o f f ,
f il te r in F , an d th e r e f o r e , N£ f(© ) .
s e t M 1 C O su c h .th a t f(M ') G N..
f ( ©) g e n e r a te s a
S in c e N £ .f ( 0 ) , th e r e e x is t s a
It fo llo w s th e n th a t M = f
c o n ta in s f - "*- (f(M ')) a n d th e r e f o r e M -'.
-I
(N)
But th is m e a n s th a t M£ 0 „
T h e re fo re M is o p e n in E.
I
The n o tio n of th e lim it "sp a c e c a n n o w b e e x te n d e d to p ro d u c t s p a c e s .
L et Eyl
,F 4 ' b e lim it s p a c e s , an d on th e s p a c e E x F c o n s tr u c t th e
c o a r s e s t lim ita tio n fo r w h ic h th e p ro je c tio n m a p p in g s p r^:E x F-*E
an d prp:E x F ->F a r e c o n tin u o u s .
T h is is c a ll e d th e p ro d u c t lim ita tio n
an d is d e n o te d -Zl x v d / .
The fo llo w in g th e o re m sh o w n b y B inz
(i)
i s q u ite u s e f u l w h en
w o rk in g w ith p ro d u c t lim ita tio n s :
T heorem 1 .3
L e t Eyj
a n d F4 ' b e tw o lim it s p a c e s .
A f il te r
(E x F)
c o n v e rg e s to (x ,y ) 6 E x F r e l a t i v e to y l x y l / i f a n d o n ly if p r^ ( 0 )
(x)
7
($) e S l ' f y ) .
a n d pr
X
I t c a n b e re m a rk e d h e re th a t c le a r ly if0 M (x ) an d
g e n e r a te s a f i l t e r in ( E x F t A x A ' ) c o n v e rg e n t to (x , y ) .
, th e n p-x. 9 ■
On th e o th e r
h a n d if Z' i s a f i l t e r in (E x F l A x A ' ) c o n v e rg in g to ( x , y ) , th e n from
I
■
T heorem 1 .3 , p^. piAfc) a n d p^ Pt AW) . T h is m e a n s o f c o u r s e t h a t pr
g e n e r a te s a filter< ^7i(x), an d jx; /" g e n e r a te s a f i l t e r ^^/!.(y),a n d m o re o v e r, '
f t . i s fin e r th a n f) x 6
.
To sh o w th is l e t X5€f >. T hen pr U = V1 €. <p
an d i t fo llo w s th a t U ^ V^ x V ^. T hus ^ l ^ x 6
a n d p r.U
.
T h e s e r e s u l t s on p ro d u c t lim ita tio n s w ill b e u s e d fre q u e n tly ,in th e
p re se n t p a p er.
‘ ^
L im it S p a c e s w ith A lg e b ra ic O p e ra tio n s
F is c h e r in (4). d e f in e s a lim it g ro u p (G , A ) to b e a s e t G on w h ic h
i s d e fin e d a lim ita tio n A s u c h t h a t
(a)
G i s a g ro u p .
(b)
a n d (G ,A ) i s a lim it s p a c e s u c h t h a t th e g ro u p o p e ra tio n s
■ , -I ■ ■
.
(x ,y ) - jS-Xy,. a n d x - » x • a r e c o n tin u o u s m ap s from G x G in to G ,
a n d G in to G r e s p e c t i v e l y .
The d e f in itio n s fo r lim it f i e l d s , lim it r i n g s , an d lin e a r lim it s p a c e s a re
s im ila r .
For i n s ta n c e th e d e f in itio n o f a lim it f ie ld i s s im ila r to th e
d e f i n i t i o n fo r a lim it g ro u p in th a t c o n tin u ity is re q u ire d fo r b o th o p e ra tio n s
a n d b o th in v e r s io n m a p p in g s . '
8
N ow l e t E b e a l in e a r s p a c e o v e r a lim it f ie ld
D e fin itio n
K.
.A lim ita tio n yl o n E is '.s a id to b e a d m is s a b le fo r E if an d
I
o n ly if th e m a p p in g s ( A ,x ) —
a n d ( x , y ) - > x + y , from K x E to E
a n d E x E to E r e s p e c t i v e l y , a re c o n tin u o u s .
'
D e fin itio n ■ If A is a n a d m is s a b le lim ita tio n o n th e lin e a r s p a c e E o v e r
f ie ld
K, th e n (E, A ) i s c a ll e d a lin e a r lim it s p a c e ( lim itie r te r V ektorraum
I_
in G erm an ) .
A xiom s of a D iff e re n tia l C a l c u l u s .
-In (8) W e h rli p ro p o s e d th e fo llo w in g s tr u c tu r e to d e s c r ib e a d if f e r e n t­
i a l c a lc u lu s o v e r ^a f ie ld
K.
M u ch u s e of th is s tr u c tu r e w ill b e m ad e in
th e p r e s e n t p a p e r .
■
'
W e -s a y th a t w e h a v e a d if f e r e n tia l c a lc u lu s o v e r a f ie ld
.
\
fo llo w in g a re g iv e n :
.I.
. b e c a ll e d o b je c ts
II.
.
'
A c l a s s % o f l in e a r s p a c e s [ E ,F , „. . j
K w h e n th e .
o v e r th e f ie ld
K w h ic h w ill
o f th e d if f e r e n tia l c a lc u lu s -.
For a n y tw o o b je c ts E ,F
of ^
th e r e a r e tw o s e t s (Z-(Ef F ), an d
(E,F) o f m a p p in g s from E to F .
T he m a p p in g s .of (2.(E,F) w ill b e
c a lle d -d e r iv a tiv e s from E to F , th e m a p p in g s of "^(E., F) w ill b e c a ll e d
re m a in d e rs from E to F, an d th e y w ill s a t i s f y th e fo llo w in g a x io m s:
A x io m -O
- Axiom I
The' fie ld , -K
is a n o b je c t of th e d if f e r e n tia l c a l c u l u s .
"/f (Ef F) is a lin e a r s p a c e o v e r K ' fo r a n y E ,F £
9
Axiom 2
-fi (E,F) i s a lin e a r s p a c e o v e r
fo r a n y E , F 6 y^f.
Axiom 3
L e t r e "^(E , F ) . If rj o n e 'd im e n s io n a l s u b s p a c e L of E is lin e a r ,
K.
th e n r j l = 0
Axiom 4
If A 6 ^ ( E ,F ) a n d B e ^ ( F f G ), th e n BA€ ^ ( E v G) ■
Axiom 5
r ( A +p ) ^ ^ ( E 1G) W h e n r e f ( F zG ),
.
S
A e ^ (E / F ) ,a n d ^ e ^ ( E / F ) .'
Axiom 6 I f A c-^(F ,G ), a n d re"^(E, F ), I h e n A r ^ lf ( E zG ).
Axiom 7
L e t r £ i[(E, F ) , a n d
£ K.
T h en th e m ap p in g f (x) = r( Ax)
i s a re m a in d e r from E to F , iev
Axiom 8
fe T^(EzF) „
L e t E 1F , ' a n d G 'b e o b je c ts in X. , a n d l e t G ^ E .
I f r e ^ ( E zF )z
th e n r |G i s c o n ta in e d in"jf(G ,F) „
.
Axiom 9
L et E zF zG b e o b je c ts- in'
a n d l e t E b e a o n e d im e n s io n a l
- ■ <
'
1
■
.
s u b s p a c e of E z a n d F a o n e d im e n s io n a l su b s p a c e o f F . If
-
.
r e /f(E ,G ) a n d L is a lin e a r is o m o rp h is m of F^ in to E^ , . t h e n th e r e
i s a ^ p e - ^ ( F zG) s u c h t h a t r*L =Zj I f 1 . ■
■Axiom 10 Le t E ,F ,' an d G b e o b je c ts in ^
'
-
'
.•
•
z I e t G - ^ E z a n d r e ^ ( G zF ).
T h ery 3^ ( E zF ). .
Axiom 11 Le t r , , r '£ ^ ( E zF ), a n d l e t r:E -» F s u c h th a t w h e n x & E z an d
1
2
r ( x ) r^ (x ), th e n r(x) = r^ (x ). T h e n r e j^ ( E zF ) 0
‘
Axiom 12 Le t E z F z G b e o b je c ts in ^ • and l e t M. b e a s u b s e t o f E.
L et
;
1I
-
10
r e ^ (E , F) s u c h t h a t r ' | M = c , a n o n z e ro c o n s t a n t .
T hen no
m a tte r w h a t f:M -*G i s , th e r e e x is t s , a/>^(E,G ) w ith y (M= L
Axiom 1 3 L et E , F £ ^
and le t
O ^-zeE, y e F .
T hen th e re , e x i s t s a n
(E, F) s u c h t h a t r (z) = y .
. Axiom 14 L et E, F ,G £ ^ a n d l e t M b e a s u b s e t o f E x E s u c h t h a t th e re
e x i s t s a n r<=^(E,F) w ith th e p ro p e rty t h a t r(x + y) = c , a n o n z e ro
c o n s ta n t for (x ,y ) £ M . T h en th e r e e x i s t s a G i n ^ a n d p ^ ( E / G)
s u c h t h a t y 5 (x) ta k e s o n ly tw o v a lu e s a n d s u c h t h a t fo r (x ,y ) 6
.
M
or J? (y) i s n o t z e r o .
Axiom 1 5 L e t.E , F , G , G b e o b je c ts i n l a n d l e t M b e a s u b s e t of K x E.,
s u c h th a t th e r e e x is t s a n r£ ^ (E ,F ) w ith r(Xx) = d , a n o n z e ro c o n '
s t a r i t , for (A,x) € M . T hen th e r e is ayoey[(E, G) w h ic h a s s u m e s o n ly
x
-.
_
tw o v a l u e s , an d th e re e x is t s ay o e / f (K, G ) , s u c h t h a t f o r . ■
(
M ,yO(x) a n d jo( ^~) a re n o t z e ro s im u lta n e o u s ly .
C h a p te r. I I
D iff e re n tia tio n in L im it S p a c e s
i
P re lim in a ry M a te r ia l
to W e h rli (9 ).
C o n s id e r f i r s t of a l l th e fo llo w in g th e o re m d u e
L e t (X, / ] ) a n d ( Y , A ) b e lim it s p a c e s .
Theorem 2 . 1 " If x c X, a n d y £ Y, a n d f:
f
x -»y is c o n tin u o u s a t x , th e n f: (X1Yl0)
(x, A
)
.(Y1 A' ) w h e re
t
(Y1YIq) i s c o n tin u o u s a t x .
(H ere o f c o u r s e A 0, a n d A oa r e th e a s s o c i a t e d p rin c ip le id e a l l im it a ti o n s .)
I t i s im p o rta n t to re m a rk a t th is p o in t t h a t if Eyl a n d Fyl' a te lim it
s p a c e s a n d f i s a c o n tin u o u s fu n c tio n from Eyl to Fyl- , th e n
f i s of
c o u r s e , b y T heorem 2 . 1 , c o n tin u o u s r e la tiv e to th e a s s o c i a t e d ' lim ita tio n s
■A
f:E/i
o a n ^ a 'o . .In v ie w o f T heorem 1 .2 w e s e e t h a t th is m e a n s th a t
F y i s th e re fo re a l s o a ' c o n tin u o u s m ap p in g in th e to p o lo g ic a l
yiO-
o
sense.
W ith th e a id o f T heorem 2 .1 th e fo llo w in g im p o rta n t r e s u l t c a n b e
o b ta in e d .
' '
T heorem 2 .2
L e t (X1A ) an d (Y1A ) b e lim it s p a c e s , a n d c o n s id e r
( X x Y 1A x / f ) / w h e re A x A', i s th e p ro d u c t lim ita tio n .
T h en (X x Y1 A q x / y =
( X x Y 1 (A xA ') L
o
Proof
f i l t e r in
L et (x ,y ) b e a n y p o in t in X x Y 1 a n d l e t
(A x AO q c o n v e rg in g to ( x ,y ) .
i s a p rin c ip le id e a l .lim ita tio n .
in A
^
x A
n
b e th e c o a r s e s t
C le a r ly ^ e x i s t s s in c e
( A xA ) q
In a d d itio n l e t o ^ x b e th e c o a r s e s t f il te r
c o n v e rg in g to ( x , y ) .
12
Lemma 2 „ I
:
cy
o
Proof o f Lemma 2 .1
x a
o
i s n o t fin e r th a n
•o
L e t V b e a n y s e t in of x ^5o .
s e t of th e form V1 x
T hen V c o n ta in s a
w h e re V €of^and
From th e d e fin itio n o f
of a n d /5 w e know th a t V ■= UV1
an d V = UV^ w h e re V, a n d V a re
o / o
I
IA
2 2 /^
l/(
2/^
s e t s from e a c h o f th e f ilte r s y x a n d ^ in A (x)and A '(y )re s p e c tiv e ly .
Then
vizx V=AtVx V i-
V1 X V 2
Now w h a t is th e n a tu re o f f0 ?
The s e t s o f f0 c o n ta in s e t s o f th e form.
U (W1 x W g) w h e re W^ a n d W ^ a re s e t s from e a c h of th e f il te r s w h ic h
c o n v e rg e in
x a r e
A
and
A
to x
and y re s p e c tiv e ly .
The s e t s th e n of
a l s o s e t s of f Q , a n d o ^ x ^ i s n o t fin e r th a n
Lemma 2 .2
-----------------
Vo is n o t fin e r th a n
Proof of lem m a 2 .2
f0 .
(XO x/ /5
Q ..
B a s ic a lly th e a p p ro a c h in th is p ro o f i s to sh o w th a t
th e r e i s a lim it s p a c e
(S , f ) , an d a m ap p in g f from (S, P ) in to
-
(X x Y, vl x a ' ) w h ic h is c o n tin u o u s a t a p o in t u e S w ith f(u) = ( x , y ) , an d
w h ic h m ap s a f il te r c o n v e rg in g to u in (S, P ) in to th e f i l t e r of
x
Z9 '
(X x Y,
(A x A ) ) \
O
T hen of c o u r s e
V
O
c o u ld n o t b e fin e r th a n
xA
Ol O
in
.
To c o n s tr u c t th is s p a c e S f i r s t of a ll form th e d ir e c t p ro d u c t S1= P X ^
fo r v w h e r e % i s a s e t w h ic h in d e x e s th e f il te r s
fo r a ll . % . T hen s im ila r ly form Sg = "P Y^for
f il te r s d^A(y) , an d l e t Y^ =
"IT X ,
i
*
x
TTY.
X
T hen S =
Y.
(x ), a n d l e t X^= X
th e in d e x s e t of a ll
N ow form th e d ir e c t p ro d u c t S = S1 x S^ =
X x X x . . . x Y x Y x . . . , in th e a b o v e d im e n s io n
13
N ow d e fin e th e fo llo w in g lim ita tio n P o n ' S .
L et u = ( x , x , . . . , y , y , ...)
.rn S b e th e p o in t w ith x in e a c h o f th e X c o m p o n e n ts , a n d y in e a c h o f th e
Y c o m p o n e n ts . .
I.
For f i l t e r s ^ a n d ^ o n S , d e fin e
PCu) an d ^C (u) w h e n ^ i s g e n e r a te d
by
, ^ X i(x ),a n d w h en ^ i s g e n e r a te d b y
(XzX, . .
■(x,x,...,y,y,...,
• • • •),,
w h e re j u s t o n e x c o m p o n e n t
is o c c u p ie d b y th e p o in ts of a f i l t e r ^ c o n v e rg in g to x in (X1A )
a n d a l l o th e r c o m p o n e n ts a re id e n tic a l w ith th o s e o f u , or w h e re
j u s t o n e Y c o m p o n e n t is o c c u p ie d .’b y -th e p o in ts o f a f ilte r
c o n v e rg in g to y in (Y, A' ) an d a l l o th e r c o m p o n e n ts a re id e n tic a l
w ith th o s e ' o f u .
II.
If
A
n,
,° L ' a re a f in ite s e t o f f il te r s c o n v e rg in g to
v
u in th e s e n s e of I , d e fin e th e ir jo in to c o n v e rg e to u .
• II I . . F in a lly d e fin e a f il te r & to c o n v e rg e to u if ^ i s fin e r th a n a
f il te r c o n v e r g e n t to u in th e s e n s e o f II.
For a n y o th e r p o in t z ^ u in S d e fin e z to b e th e o n ly f il te r in S
c o n v e rg in g to z .
.
The q u e s tio n th a t m u s t b e a n s w e re d now is w h e th e r or n o t ILis a
lim ita tio n on S .
C o n s id e r th e n th e d e f in itio n o f a lim ita tio n .
The f ir s t
c o n d itio n of th is d e f in itio n r e q u ir e s th a t P ( z ) , fo r a ll z € S 1 m u s t b e a
A
id e a l in J/( S ) . If z ^ u - ', th e n P(z) c o n s i s t s of j u s t 2 .
C e r ta in ly
•14 '
£ A z = z , a n d a n y f il te r c o n ta in in g z i s j u s t z b e c a u s e ± i s th e u l t r a f i l t e r ,
i . e . th e f i n e s t f i l t e r c o n v e rg in g to z .
T hus r ( z ) is a Tl id e a l fo r z ^ u .
I f z = u , th e n th e f a c t th a t f (u) is a /I id e a l fo llo w s d ir e c tly from th e
d e f in itio n o f P . T he' s e c o n d c o n d itio n of th e d e f in itio n o f a lim ita tio n
r e q u ir e s t h a t fo r a l l z <£ S , z & P ( z ) .
P ' for a l l z Tt u .
c o n v e rg in g to u
T his is tru e b y th e d e f in itio n of
I t i s a l s o tru e for u s in c e u i s fin e r th a t a n y f il te r
in th e s e n s e o f I I . ■T h e re fo re P i s a lim ita tio n o f S .
Now d e fin e f: (S , P ) -^ (X x Y, vl x A z ) so th a t f m ap s p o in ts in th e
fo rm ,■
'
(x, x , . . . , s , . . . , x , = . . y , y , . . . ) in to (s , y ) , w h e re th e p o in t i s
■
c o n s tr u c te d b y r e p la c in g o n e of th e x 's in u b y s £ S;
.
'
p o in ts in th e form (x , x , . . . , y , y , . . . , t , . . . ) in to ( x / t ) , w h e re
\
th is p o in t is form ed b y r e p la c in g o n e of th e y ' £ in u b y t £ S ,
an d '
■joints in th e form (x , x , . . . , s , x , . . . y , y , . . . , t . , , , ) in to
-/•
. ■
(s ,t) / w h e re t h i s p o in t i s form ed b y r e p la c in g o n e of th e x 's a n d o n e o f th e
y ’s in u b y s a n d t r e s p e c tiv e ly .-
If o n e d e f in e s f to m ap a ll o th e r p o in ts
o f S in to (x ,y ) th e n f is a c o n tin u o u s m ap from (S ,P ) in to (X x Y ,/ l x / 1 ) =
To sh o w th is l e t ^ c o n v e r g e to u a c c o rd in g to form I , i . e . ^ i s g e n e r a te d
b y (x, x , . . . , x , 0 X , yi, . . . ,
y, y
W c o n ta in s a s e t of th e form (x, x ,
U £ g) .
,
wh e r e ^ 4 ( x ) .
If W 6 Px, th e n
, U-, x , . . . x , y , y , . . . ) ,
w h e re ';
T his s e t is m a p p ed by f in to (U , y ) , s o f(W) c o n ta in s (U , y ) .
15
All Ue ^ a p p e a r in im a g e s u n d e r f in th e form (U 7 y ) , i . e . e v e ry Ue (4.
g e n e r a te s a s e t We. A 7 a n d th e im a g e o f a n y s u c h W .u n d e r
c o n ta in s (U 7 y ) . I t i s c le a r th a t
f ' th e n
■
[(U 7 y) | U e ^ g e n e r a te s a f il te r J in
(X x Y fS l x A ' ) . At th is p o in t i t i s c o n v e n ie n t to u s e T heorem 1 .3 s ta te d
in C h a p te r I .
The p ro je c tio n m ap s o f X x Y in to X. an d Y 7 s y m b o lic a lly
p r^ X X-Y-t X a n d p r^ X x Y - t Y 7 a re c o n tin u o u s , a n d s in c e pr% If g iv e s
a l l U €r 0 x e A (x ), w h ic h form a b a s i s of ^ i n A (x ), a n d pr
X g iv e s a ll
s e t s in (Y7ZL ) w h ic h c o n ta in y , i . e . th e u ltr a f i lte r y e A ( y ) , b y Theorem 1 .3
i t fo llo w s th a t If c o n v e rg e s to ( x 7y) in ( X x Y 7A x V l'). A s im ila r arg u m en t
w ill s u f f ic e if a f i l t e r , c o n v e rg e n t in form I I 7 is c h o s e n s in c e o n ly f in ite
jo in s a r e in v o lv e d . A f ilte r
c o n v e rg e n t to u in form III i s o f c o u rs e fin e r
th a n so m e f il te r x-f c o n v e rg e n t to u in form I I .
But th e im a g e o f X^ u n d er
I g e n e r a te s a f il te r c o n v e rg e n t in (X x Y 7Xl xxT) to (x 7y ) .
u n d e r f c le a r ly g e n e r a te s a f il te r in (X
The im a g e o f
x Y , A x A z) w h ic h is fin e r th a n
a n d i s th e re fo re c o n v e r g e n t to (x , y ) .
f th u s is c o n tin u o u s a t u .
In tro d u c e now th e a s s o c i a t e d p r in c ip le id e a l lim ita tio n s
(A x A ') Q on S a n d X x Y r e s p e c t i v e l y .
u 6 S r e la tiv e to t h e s e n ew l i m i t a t i o n s .
■
and .
By T heorem 2 .1 f i s c o n tin u o u s a t
The t a s k now i s to c o n s tr u c t a
f il te r in S w h ic h c o n v e rg e s to u in Pq an d i s m ap p ed b y f in to a f il te r b a s is
o f ^ o X j2>Q .
C o n s tr u c t th e n a l l f il te r s ^ a n d O^in (S7P q) .
c o n v e rg e s to u&S b y form I .
C le a rly th e s e ts
/
E ach o f t h e s e
16
(x , x , . . . , x A
,
X,
. .. , y, . . . )
(x , x , . . ; , y , Y i . . . , 7 AA , y , . . . )
fo r a l l 0 X a n d
in th e
/t a n d ^
c o m p o n e n ts r e s p e c t i v e l y , e a c h
g e n e r a te a f i l t e r c o n v e r g e n t to u b y form I I 0 D e n o te t h e s e f i l t e r s by
and
re s p e c tiv e ly .
N o te now t h a t th e fo llo w in g s e t s g e n e r a te t h e s e
filte rs ,
for a l l Xe
(XzX , . . . , x U Ux , x ,'. . . , y , . . . ) , a l l
e ^
(x, x , . . . y ,
£ 6 ^ fo r a l l
o. , y ^
, y , y , . . . ), a ll
. N ow c o n s tr u c t th e f il te r ¥ g e n e r a te d b y
If f i s a p p lie d to. th e s e t w h ic h i s fo rm ed W hen o n e f ix e s x U
,
w h e re U e ^ f o r.so m e IKtyJfl, in .a n X co m p o n en t; r e q u ir e s t h a t a l l o th e r X
\
c o m p o n e n ts b e o c c u p ie d b y sim p ly x t X; a n d a llo w s e a c h Y c o m p o n en t to
b e o c c u p ie d by a d i s t i n c t V ^ e
, fo r a
l l th e n th is s e t i s m ap p ed
in to
.
■ u V X.
N ow if a d i s t i n c t x U l^,
• '
.U ^ (^ fo r
i s a llo w e d to o c c u p y e a c h X
c o m p o n e n t of th is s e t , i t is m ap p ed b y f ■in to
th e o rig in a l s e t b e lo n g e d to
0 a n d th u s w h e n f
a l l s e t s in th e fo rm •
.
x UV^.
O f c o u rs e
is a p p lie d to th is f ilte r
■1
fUU.
\
A
x
VVJ
vI vI
for al l
Uf(Z) and V d A ,
K
a
.
fL
I
17
a re o b ta in e d '.
All o th e r s e t s in
Y m ap in to o v e r s e ts o f th e s e .'
/
C le a rly [U U a
g e n e r a te s th e f i l t e r
x
0
p ro je c tio n m a p p in g s prx a n d .p r^ o f th is f i l t e r g e n e r a te
re s p e c tiv e ly .
T h e re fo re by the. c o n tin u ity o f f ,
(x ,y ) w ith r e s p e c t to
(A
x Jl ) Q, an d
in X x Y, an d th e
( A x A ) = A qx A q .
a n d [3 Q
cr q
<x Q x yd o c o n v e rg e s to
c a n .not b e fin e r th a n
Lem m as 2 .1 a n d 2 .2 th e n to g e th e r im p ly th a t ^ q=
th e re fo re th a t
'
ex' 0 x yS
an d
x
T h is c o n c lu d e s th e p ro o f of
T heorem 2 . 2 .
The fo llo w in g th e o re m s on lim it g ro u p s a n d lin e a r lim it s p a c e s a s
in tro d u c e d in c h a p te r I a re a lm o s t im m e d ia te c o n s e q u e n c e s o f Theorem 2 .2 .
Theorem 2 .3
g ro u p .
Proof
If (G , A ) is a lim it g ro p p , th e n (G , A
. '
'
•■
F ir s t o'f a l l , s in c e x —» x
i t i s c o n tin u o u s r e la tiv e to A
r e la tiv e to ( ' A x A ' , a n d
r e la tiv e
) i s . a to p o lo g ic a l
A
to ( A x Vl )0 a n d A Q.
.
is c o n tin u o u s r e la tiv e to Yl , b y T h e o re m .2.1,
S e c o n d ly , if (x ,y ).-» x y is c o n tin u o u s
, th e n b y T h e o re m ; 2 . 1 i t i s c o n tin u o u s
But b y T heorem 2 . 2 ,
s o (x , y h x y i s c o n tin u o u s r e la tiv e to A Q x A 0 an d A
(A
.
x A ) Q = A qX A q ,T h e re fo re (G , A q )
is a lim it g ro u p , a n d of c o u r s e th e n a to p o lo g ic a l gro u p w ith s tr u c tu r e A q
s in c e b y T heorem 1 .1 A q is a to p o lo g y vbn G/. ■
• \
I
A s im ila r th e o re m for lin e a r lim it’s p a c e s is a s f o llo w s .
T heorem 2 .4
If (E, A ) i s a lin e a r lim it s p a c e o v e r th e lim it f ie ld
18
(K, A
Proof
) , th e n (E; Vl
i s a to p o lo g ic a l s p a c e o v e r (K, A Q) .
From th e d e f in itio n o f a lin e a r lim it s p a c e th e o p e r a tio n s of
v e c to r a d d itio n a n d s c a l a r m u ltip lic a tio n a re c o n tin u o u s from E x E to E
a n d E x K to E r e s p e c t i v e l y .
S in c e x
= - x , th e f a c t t h a t x
x _1 is
c o n tin u o u s fo llo w s from th e c o n tin u ity o f s c a l a r m u ltip lic a tio n , an d E
i s a to p o lo g ic a l g ro u p .
T h e re fo re (E7ZIq ) i s a to p o lo g ic a l s p a c e o v er
(K, A Q) w ith th e .to p o lo g y o f th e g ro u p .
R e q u ire m e n ts fo r a. G e n e ra l D iff e re n tia tio n S tru c tu re
S u p p o se w e a r e g iv e n a n o tio n o f d if f e r e n tia tio n in th e c l a s s
l in e a r lim it s p a c e s o v e r a lim it f ie ld
K.
o f a ll
It is q u ite n a tu ra l to a s k w h a t
c o n d itio n s m u s t h o ld , or w h a t re q u ire m e n ts r e a lly c h a r a c te r iz e th e
s tr u c tu r e , in v ie w of th e a x io m s p ro p o s e d fo r a d if f e r e n tia l c a lc u lu s by
\
■W e h r li.
By a 'c o n c e p t of d if f e r e n tia tio n w e m e an f i r s t of a l l t h a t b e tw e e n
a n y tw o lin e a r lim it s p a c e s E yl a n d Fa ' o v e r K, th e r e e x i s t s a s e t of
lin e a r c o n tin u o u s fu n c tio n s
Cl (E a .,F a ' ) a n d a s e t of fu n c tio n s 7f(EA ,F ^ ')
w ith th e p ro p e rty t h a t if ' r6/^, th e n r(0 ). = 0 , a n d r is c o n tin u o u s a t z e r o .
T hen f : E- ^ F i s s a id to b e d if f e r e n tia b le a t th e p o in t z e E i f a n d o n ly if
th e r e e x i s t f u n c tio n s
A £ ^ ( E y 1 , Fa ' ) a n d r e "^(Eyl ,F yl/ ) s u c h th a t
f (z + h). - f(z) = Ah + r ( h ) , ^or a l l h £ E.
■A th e n is s a id to b e th e d e r iv a tiv e of, f(x) a t x = z , a n d is u s u a lly
19
d e s ig n a te d f ' ( z ) . r(h) is s a id to b e a re m a in d e r .
It a p p e a rs q u ite n a tu ra l
now to re q u ire t h a t th e z e ro m ap p in g from Eyl to F y
b e d if f e r e n tia b le
fo r a ll z £ E, an d h a v e d e r iv a tiv e e q u a l to z e ro on a ll s u c h p o in ts .
A lso
i t s e e m s n a tu ra l to s tip u la te th a t d if f e r e n tia tio n b e a lo c a l p r o c e s s .
To
e x p a n d on th is n o tio n s o m e w h a t, w e m ean th a t w h en a fu n c tio n h a s a
d e r iv a tiv e a t a c e r ta in p o in t th is d e r iv a tiv e i s in d e p e n d e n t of th e n a tu re
o f th e fu n c tio n o u ts id e a A o p e n n e ig h b o rh o o d o f th e p o in t.
It fo llo w s
th e n from th is a n d th e f a c t th a t th e z e ro m ap p in g is d if f e r e n tia b le th a t
a n y fu n c tio n f from E a to
o f z e ro is a re m a in d e r .
Fy w h ic h v a n is h e s on a / A o p e n n e ig h b o rh o o d
T h is c a n b e sh o w n b y c o n s id e rin g th e fo llo w in g
e x p r e s s io n w h ic h re s u lt's from th e f a c t t h a t f .is d if f e r e n tia b le a t zero:
f(0 + h)
- f (0) = Ah +r(h) ,h € .E .
•
' ,If f v a n is h e s on th e- A o p e n n e ig h b o rh o o d of z e ro U in E, th e n w e h a v e
in U th e z e ro m a p p in g an d s in c e th e z e ro m ap p in g h a s d e r iv a tiv e z e ro ,
th e a b o v e e q u a tio n r e d u c e s to f(h) = r( h ) , h e
re m a in d e r .
U £ E.
-
T h e re fo re f is a
.
N ow a s s u m e th e fo llo w in g s e p a r a tio n a x io m p ro p o s e d b y F is c h e r (4)
fo r our s p a c e s E ^ , Fyl- , e t c . :
If x , y 4 E e
an d x # y , th e n y^vi(x). .W ith -
th is we' c a n p ro v e th e fo llo w in g lem m a:
Lemma 2 .3 if x , y £ E, s u c h th a t x ^ y , th e n C y > w h e re C y i s th e c o m p le m e n t
o f y in E, is a A o p e n n e ig h b o rh o o d of x .
20
L et z e C y , a n d sh o w .th at C y € ( p ^ A { z ) , w h e re ^
P roof of lem m a 2 .3
is a r b itr a r y in A ( z ) .
Cy ^ ^ .
y £ A (z ).
S u p p o se th e n th a t th e r e e x is t s a<f)iA(z) s u c h th a t '
But th e n y i s fin e r th a n (f) an d
T hen U €j^ im p lie s t h a t y £ 'U.
T h is o f c o u r s e c o n tr a d ic ts th e s e p a r a tio n a x io m s in c e y ^ z .
T h e re fo re C y £ (Z)W( z ) , a n d s in c e z w a s a r b itr a r y in Cy an d . 0 .w a s a rb itra ry
in A ( z j , C y is a A o p e n n e ig h b o rh o o d o f x .
T hus fo r a n y tw o p o in ts x a n d y in E , w h e re x ^ y , a,A . o p e n n e ig h b o r­
h o o d o f x c a n b e fo u n d w h ic h d o e s n ’t c o n ta in y .
■ ■,
Axiom 13 p ro p o s e d by W e h rli fo r a d if f e r e n tia l c a lc u lu s c a n b e e a s ily
sh o w n by f i r s t a p p ly in g Lemma 2 .3 a n d fin d in g a A o p e n n e ig h b o rh o o d U
o f z e ro w h ic h d o e s n o t c o n ta in z .
N ow d e fin e
N
'
,
f 0,
. r(x)
=
x £U
) y ,x = z
a rb itra ry ,. x ^ U a n d x ^ z
C e r ta in ly r £
‘
) s in c e i t v a n is h e s o n a A o p e n n e ig h b o rh o o d
of z e ro .
In Axiom 12
fo r a d if f e r e n tia l c a lc u lu s i t i s g iv e n t h a t
a re o b je c ts in
^(E a , F a '' ) ,
,F ^ , an d
a n d M i s a s u b s e t o f E s u c h t h a t fo r so m e
r
r)M = c , a n o n z e ro c o n s t a n t . : T hen fo r a n y f :M - » G , w e
m u s t sh o w t h a t th e r e e x i s t s Byse-C(E7G) w ith th e p ro p e rty t h a t p | M = f .
in
-
.
21
D ra w in g u p o n Lemma 2 .3 a A o p e n n e ig h b o rh o o d U of. z e ro in (E1A ' ),
o
th e a s s o c i a t e d to p o lo g ic a l s p a c e / is o b ta in e d w h ic h d o e s n 't c o n ta in
c € F .' S in c e r i s c o n tin u o u s a t 0 € F b y T heorem 1 . 2 , th e in v e r s e im a g e
o f U u n d e r r i s a A o p e n n e ig h b o rh o o d W o f z e r o 'in (E, A
W H M i s e m p ty .
) , a n d c le a r ly
O
N ow d e fin e
0, x 6W
f
(x) =
vf(x ),
x^W
C e r ta in ly yP^"^(Eyl , Gyj*) s i n c e . i t v a n is h e s o n W , andy?jM = f .
Axiom 14
p ro p o s e d by W e h rli c a n a ls o b e s h o w n :
E. , F . -
, G A-' a re
to b e o b je c ts in ^ an d M i s a s u b s e t of E x E s u c h th a t th e r e e x is t s an
\
r £ 'KiEjy , E JY1 ) w ith th e p ro p e rty th a t r(x + y) = c , a n o n z e ro c o n s ta n t,
fo r (x ,y ) € M . '.If th is is th e c a s e Axiom 14 r e q u ir e s th e - e x is t e n c e o f a
G yl- in
an d a
E-^ E y v G
) s u c h t h a t y>(x)or j> (y) i s n o t z e r o .
r
L et th e n cT b e th e m ap from E x E.into- E w h e re ( x , y ) - ^ x + y .
o f c o u r s e is c o n tin u o u s .
•
N ow c o n s tr u c t on (E, A ) ,
(E , A ) , an d
'
/
(E x. E , /Ix A )'.the a s s o c i a t e d p rin c ip le id e a l lim ita tio n s
( Ax-A ) .
T his
Aq,
A q , and
By T heorem 2 . 1 , r an d S a re s t i l l c o n tin u o u s a t z e r o , an d
b y T heorem 1 . 1 , (E7 A q )/
( F 7A L q ) , a n d (E
x
E, ( A x A )
) a re
to p o lo g ic a l s p a c e s .■ T heorem 2 .2 im p lie s t h a t ( A x A I 0 - A
w e s e e t h a t (E x E ,
x A q an d
(A x A ) ) i s j u s t a to p o lo g ic a l s p a c e w ith th e
'p ro d u c t to p o lo g y .
By lem m a 2 .3 we c a n fin d a A o p e n n e ig h b o rh o o d W
o f z e ro in F s u c h th a t c ^ W .
L e t' U b e th e in v e r s e im a g e o f W u n d e r r .
S in c e r i s c o n tin u o u s , b y T heorem 1 .2 U i s a /I o p e n n e ig h b o rh o o d of
T hen if th e im a g e o f M u n d e r S i s N £ E, c le a r ly U H N i s
z e ro in E.
e m p ty .
B e c a u s e <f i s c o n tin u o u s a t (0, 0) w ith r e s p e c t to th e to p o l o g i e s ,
th e in v e r s e im a g e of U u n d e r 'r i s a A o p e n n e ig h b o rh o o d U ' o f (0 ,0 ) in
E x E.
C e r ta in ly U 1ft M i s e m p ty .
S in c e w e a r e d e a lin g w ith th e p ro d u c t
to p o lo g y on E x E, a n d U ' i s o p e n in th is to p o lo g y , U 1 c o n ta in s a s e t in
x Vg , w h e re
th e form
(E, A Q) .
T hen
an d
a re A o p e n n e ig h b o rh o o d s o f z e ro in
ft V ^'= V i s a A o p e n n e ig h b o rh o o d of z e ro in (E, A q ) .
N ow d e fin e /3 (x) to b e s u c h th a t ■
'
\
\
f0 6 G,
/M x W
%6 V
'
^ ^ 0 6 G, ' x
C le a r ly J d (x ) 6 ^ (E yl , G ^ - ) s in c e i t v a n is h e s o n a A o p e n n e ig h b o rh o o d of
z e r o , an d if
■Axiom 15
(x) a n d 'y " (y) b o th a r e z e r o , th e n
c a n b e v e r if ie d in a s im ila r f a s h io n .
(x , y) ^ M . •
D u p lic a te th e p re v io u s
a rg u m e n t to th e p o in t a t w h ic h a A o p e n n e ig h b o rh o o d o f
( 0 , 0 ) , c o n ta in in g
V1 x V9 in K x E i s f o u n d , w h e re V1 a n d V a re A o p e n n e ig h b o rh o o d s of
■1
^
1 V
2
z e ro in K a n d E r e s p e c t i v e l y .
N ow d e fin e
23
"0€G ,
xeVg
^ O £ G,
x
rO 6 G,
Vg
V1
and
C le a r ly _/> (x)c ^(E yl ,G ^ „ ) a n d
(^-)
=
0,
( ?-)£ ^ ( K vGyl- ) , a n d if
(x)
( . ^ 1/ x) 4 M .
Thus A xiom s 1 2 , 1 3 , 1 4 , 15 fo llo w a s a c o n s e q u e n c e o f th e c o n tin u n ity '
o f re m a in d e rs a t z e r o , a n d o f th e f a c t t h a t a fu n c tio n v a n is h in g o n a
A
o p e n n e ig h b o rh o o d o f z e ro i s a re m a in d e r .
U n d er (th e ‘g iv e n c o n d itio n s Axiom 11 c a n a t l e a s t b e v e r i f i e d ’fo r "the
\
s u b s e t of our s e t o f re m a in d e rs m ade up o f th o s e w h ic h v a n is h o n a A o p e n
n e ig h b o rh o o d o f z e r o .
o'-t-E ,F
). ,w h ere r i I
D e n o te th is s e t b y ^ ( E ^ ,F yl/
).
If r ^ , r^
a re in
= 0 a n d r ^ l U^ = 0 , U j , Ug A o p e n n e ig h b o rh o o d s
o f z e ro in E, th e n c le a r ly r^ | U 1 /I Ug = 0 a n d rglU-^/l Ug = 0 .
U^/T'Tj'g i s a -/!open n e ig h b o rh o o d o f z e ro in E.
T h e re fo re r e g a r d le s s of
h o w r is d e f in e d , a s lo n g a s .r ( x ) = r, (x) or r(x) = r
J-
■
O f c o u rs e
I
(x ), c e r ta in ly
r | U j A ' Ug = 0, an d r is a re m a in d e r .
Axiom 10
c a n b e q u ic k ly v e r if ie d fo r re m a in d e rs in our s e t
(Eyl , F a ' )
s in c e th e lim ita tio n A on G i s in d u c e d b y th e lim ita tio n on E , i . e . t h e
'
24
to p o lo g y /I q on G is of c o u rs e th e n j u s t th e r e la tiv e to p o lo g y w ith
r e s p e c t to (Et A 0).
S in c e r ^ 0 C G . , E ^
),
r v a n is h e s o n a A o p e n .
But- U = U 'A G fo r so m e A o p e n
rr (x ), x e G
T hen if P(x) = j
LO, x 4 E , b u t n o t in G ,
n e ig h b o rh o o d U c o n ta in in g z e ro in G .
n e ig h b o rh o o d of z e ro -U1 in E .
c le a r ly
(E ^ ,F a ' ) s in c e
v a n is h e s o n U 1.
The s ta te m e n t of Axiom 9 d e p e n d s o n w h e th e r or n o t o n e a s s u m e s th a t
\
a l l lin e a r s u b s p a c e s o f th e s p a c e s in jf a r e is o m o rp h ic .
To see. t h a t
th is i s in g e n e r a l n o t th e c a s e c o n s id e r th e fo llo w in g e x am p le:
L et IR. b e th e r e a l n u m b e rs a n d d e fin e a f il te r
to c o n v e r g e 'to a p o in t
w h e n i t is g e n e r a te d by a c o u n ta b le s e q u e n c e o f n u m b ers t h a t c o n v e rg e to
th a t p o in t in th e n a tu r a l to p o lo g y of IR .
\
The c o lle c tio n of a l l f ilte r s
c o n v e r g e n t to p o in ts in IR in th is s e n s e e s t a b l i s h e s a Iim ita tio n A on IR.
\
C le a r ly th e n a tu r a l to p o lo g y J o n IR a ls o e s t a b l i s h e s a lim ita tio n on
C o n s id e r th e n th e id e n tity m ap
id: ■ ( IR. , A ) "^ ( IR-,'A ) .
If th is is a n
iso m o rp h ism of c o u r s e c o n tin u ity m u s t.h o ld in b o th d ir e c t i o n s .
id :
( IR , A ) - ^ ( IR, j ) is c o n tin u o u s , b u t id :
IR. .
C e rta in ly
( IR , 5 ) - ” ( IR , A ) is n o t
s in c e th e n e ig h b o rh o o d f i l t e r c o n v e r g e n t to z e ro in IR w ith r e s p e c t to d
u n d e r th e id e n tity m a p p in g , d o e s n 't c o n v e rg e to z e ro in IR w ith r e s p e c t
to A .
If w e a s s u m e fo r our s y s te m th a t a ll o n e d im e n s io n a l s u b s p a c e s a re
i
is o m o rp h ic w e c a n sh o w Axiom 9 for re m a in d e rs
,F ^ J a s fo llo w s :
6
>
W e h av e th a t
and
a re o n e d im e n s io n a l s u b s p a c e s o f E an d F
r e s p e c t i v e l y , th a t L i s a lin e a r is o m o rp h ism o f F1 in to E , a n d th a t r i s ,in
1
I
/^0 (Eyji. /Gy1" ) .
in E .
T hen r | U = O , w h e re U is a A o p e n n e ig h b o rh o o d of z e ro '
But U d E j i s a A o p e n n e ig h b o rh o o d o f z e ro in E1 , a n d b y c o n tin u -
ity , th e in v e r s e im a g e u n d e r L o f U A E i s a A o p e n n e ig h b o rh o o d N 1 of
I
I
z e ro in F .
T hen th e r e is a A o p e n s e t N in F s u c h th a t N fl F1 = N 1 .■
N ow d e fin e
fr
• L (x ), Xf=F1
y ( x ) = |
VO, X f f F 1
C le a r ly / ( x ) 6 ^ 0(F1G) s i n c e / | N = 0 , w h e re N is a yl o p e n n e ig h b o r^
h o o d o f z e ro in F .
If w e do n o t a s s u m e th a t lin e a r o n e ’ d im e n s io n a l s u b s p a c e s a re
is o m o rp h ic , W e h rli h a s s u g g e s te d in (8) th a t- th e a x io m b e a lte r e d to
Axiom 9 '
of E .
L e t E,,G b e o b je c ts in ^ , -and E1 b e a o n e - d im e n s io n a l s u b s p a c e
L e t L , w h e re L:K -*Ej, b e a n a lg e b r a ic lin e a r is o m o rp h ic m ap p in g
from K to E1 a n d r £ ^ ( E , G ) .
T hen r ° L i s a re m a in d e r from K to G .
The v e r if ic a tio n o f a x io m 9 ' i s s im ila r to th a t fo r a x io m ■9 .
c o n tin u ity in o n e d ir e c tio n fo r L:K
The
E^ is o f c o u rs e a c o n s e q u e n c e of -
s c a l a r m u ltip lic a tio n . .
Axiom 8
i s tru e a ls o ,f o r our re m a in d e rs
7^o(E A ,F a ' ) .
If r I s ^ i n
26
^ o (Ey2 /F yl ) , th e n rjU = 0 , U a vl o p e n n e ig h b o rh o o d o f z e r o .
i s a vl o p e n n e ig h b o rh o o d of z e ro in G a n d c le a r ly r|UT]G = 0 .
But th e n U/1G
T h erefo re
,F y ).
N ow c o n s id e r Axiom 2.
jK. o(Ea
If r-^l
T his to o c a n b e v e rifie d fo r o u r re m a in d e rs
/FylO . W e ta k e o f c o u r s e (rj. + r 2)(x) = r ^ (x) + r 2 (x ), r ^ , ^ /^ (E yl/F^,)A .
= 0 a n d r gl Ng = ^ w h e re U 1 a n d U 2 a re A o p e n n e ig h b o rh o o d s o f z e ro y;
th e n c le a r ly r 1+ \r2 I U ^ U 2 = 0 , a n d s i n c e U^fl U 2 is a /I o p e n n e ig h b o rh o o d .:
of 0 ,r ]; + r 2£ ^fo CEyl , F y ) . 'I f ^ K 7 th e n c o n s id e r Xr ,■ r £
(Eyl , F y ) .
If
•r I U = 0 , U a i o p e n n e ig h b o rh o o d of z e r o , c le a r ly xr | U = 0 a n d Xr6/f (E ,.FzlO
fo r xeK.
The re m a in in g p ro p e rtie s of a lin e a r s p a c e fo llo w e a s i l y .
Axiom 7 c a n
a l s o b e sh o w n fo r ^f0(Eyl7F y ) s in c e s c a la r m u ltip lic a tio n
i s c o n tin u o u s .
L e t r ],U = 0 , w h e re U i s a A o p e n n e ig h b o rh o o d o f z e r o .
'
I
T hen a ls o . ^ U ,
I
y
% £ K, is a A o p e n n e ig h b o rh o o d of z e r o ; b y th e
'.
-
c o n tin u ity o f s c a la r m u ltip lic a tio n .
. - i
I t fo llo w s th a t r Xj X U = O a n d r x is in
iL® a
I
W e now s e e th a t a x io m s 2 , 7 , 8 , 9 , 1 0 , 11 a re a ll tru e fo r our
re m a in d e rs ^ ,(E yl , F y ) .
I t s e e m s r e a s o n a b le th e n th a t w e s h o u ld re q u ire
t h a t t h e s e a x io m s h o ld fo r re m a in d e rs in g e n e r a l.
m a k e th is r e q u ir e m e n t.
From th i s p o in t on w e
.
W e note, th a t Axiom 0 fo llo w s s in c e w e to o k /f a s th e s e t o f a ll
lin e a r s p a c e s o v e r K.
'
27
Axiom I i s tr u e s i n c e ^ ( E yl , F yl/ ) w a s t a k e n a s th e s e t of all. l i n e a r
c o n tin u o u s f u n c t i o n s from Eyi to
Axiom 3
d e a l s w ith u n i q u e n e s s of d i f f e r e n t i a t i o n a n d so m e e x p a n s io n
i s i n o r d e r . ■ W h e n u n i q u e n e s s of d i f f e r e n t i a t i o n i s r e q u i r e d , t h i s m e a n s
t h a t for a n y d i f f e r e n t i a b l e f u n c tio n f(x) from Eyl to F / a u n iq u e d e r i v a t i v e
a n d r e m a in d e r p a ir A a n d r e x i s t s u c h t h a t
'
f(x + h ) — f(x) = Ah + r ( h ) , h e E.
T h is im p lie s t h a t if
f u n c tio n s from Eyl to F y ,
Ey , Fy , 5
(Eyl , Fy , ) i s t h e s e t- o f a l l l i n e a r c o n tin u o u s
^ (Eyl , F yl/ ) Pl /^(Ey , Fy / ) i s z e ro for a ll
To v e r if y th is , s u p p o s e t h a t t h e r e i s a n A£ ^(E yl./-F^Jsuch
t h a t A £ ^./i^and A ^ 0 .
-A £ ^(Ey , F y ) .
S i n c e ^((Eyl , f y ) i s a l i n e a r s p a c e , t h e n
\
•
'
I t f o llo w s th e n t h a t A(h) + (-A) (h) =A (h) -A (h) = 0 .
\
But t h e n th e z e r o m a p p in g h a s d e r i v a t i v e s 0 a n d A a n d th u s d if f e r e n tia tio n ,
i s n o t u n i q u e . W ith t h i s n o tio n th e n c o n s i d e r Axiom 3 u n d e r t h e r e q u ire m e n t
th a t d ifferen tia tio n be u n iq u e .
Let r £ /{(E^ , F y
to a o n e d im e n s io n a l, s u b s p a c e L SE i s l i n e a r .
im p lie s t h a t r | L £ ^(Ly , F y ) b y Axiom 8 , w h e r e
i n d u c e d o n L by A .
But ^(Ey
,F y
) be such th a t r re stric te d
Of c o u r s e
r £ "^(Ey , F y )
i s t h e lim it a ti o n
T hen r i s a l i n e a r c o n tin u o u s m a p p in g a n d r£ ^(.L ,, , F . - )
zlL 71
) H ^(L yl ,F y / ) c o n t a i n s o n ly t h e z e ro m a p p in g .
T h e re fo re
r I L = 0.
I t s e e m s n a tu r a l for u s to r e q u ir e t h e c h a i n r u le to h o ld fo r our
■
28
d e riv a tiv e s.
Some e x p an sio n , i s in o rd e r to d e te r m in e w h a t i m p lic a tio n s
a r e in v o lv e d in th e c h a in r u l e . ■
, F^/ , Q
Let
.
a t a p o in t x<=E.
,
-'
arid l e t f : E
, b e a d i f f e r e n t i a b l e f u n c tio n
Then b y o u r d e f in itio n t h e r e e x i s t
^ ( E a , Fyl/ ) s u c h t h a t
•
Q(Eyl , F yl/ ) a n d r^
in
.
• f(x + .h) - f(x)
= A1 h + T1 (h ),
h € E.
L e t a l s o g:F -»G b e a d i f f e r e n t i a b l e f u n c t i o n a t th e p o in t y = f(x)
in F .
T hen t h e r e e x i s t
,G a " ) and
a , , G A» ) s u c h
'
th a t
g (y + k) - g(y) = A2 k + r2 (k ), k e F .
S e ttin g y + k = f(x + h) a n d y = f(x) i t
f o llo w s t h a t
g (f(x '+ h-)) - g(f(x)) = Agk + r 2 (k).
S in c e k = f(x +j.h.) - f(x ),
Agk + r
(k) = A2 (f(x + h) - f(x)) + r 2 (f(x + h) - f(x))
=A2 '(A1h +
(h)) + r 2 (A1 h + T1 (H))
= A2 (A1A) + A2 ^ 1 (h))+ r 2 (A1A + r 1 (h))
The c h a i n r u le of c o u r s e s t a t e s t h a t A2A1 i s t h e d e r i v a t i v e of g (f (x )),
s o A2 Cr1 (A)) + r 2 .(A1 h '+ r 1 (h)) s h o u ld b e a r e m a i n d e r . N o t e h e r e t h a t if
w e t a k e t h e . f u n c t i o n g to b e s u c h t h a t g (y + k) = g(y) + r 2 (k ), th e n g
i s 's t i l l d i f f e r e n t i a b l e a t y a n d g(y + k) - g(y) = Ag k + r 2 (k) fo r k £ F ,
'
29
r e d u c e s to g W + r g ( W - g f y ^ f A g k + ^ .f k ) ) a n d th e r e f o r e A2 (k) - 0 , k £ F.
Then Ag(r^(h)) + r2 (A^h +r^(h)) r e d u c e s to r 2 (Ah + r(h)) w h ic h in t h e c a s e
m u s t b e a r e m a in d e r .' SincefR(Ea , Fyl') is a l i n e a r s p a c e , t h e re q u ir e m e n t
t h a t r 2 (Ah+r(h)) b e a re m a in d e r, u n d e r c e r t a i n c i r c u m s t a n c e s to g e th e r w ith
t h e re q u ir e m e n t t h a t A'2 (rj(h))+r2 (Ajh+R-|(h)) b e a r e m a in d e r , g iv e s m o tiv a ­
tio n for r e q u ir in g t h a t Axioms 4 , 5 , a n d 6 for a d i f f e r e n t i a l c a l c u l u s h o ld
a n d t o g e th e r r e p r e s e n t th e im p l i c a t i o n s of th e c h a i n r u l e . W e a s s u m e th e n
t h a t th e c h a i n r u le h o l d s , a n d th e r e f o r e t h a t Axioms 4 , 5 a n d 6 a r e v a l i d .
W e n o te a t t h i s p o in t t h a t a n a l t e r n a t i v e a rg u m e n t for t h e v a l i d i t y of
Axiom 7 c a n now b e g i v e n .
It is s im p ly a c o n s e q u e n c e of t h e f a c t t h a t
s c a l a r m u l t i p l i c a t i o n i s a l i n e a r c o n tin u o u s m a p p in g , a n d o f Axiom 5.
Thus th e r e q u ir e m e n ts s p e c i f i e d for d e r i v a t i v e s a n d r e m a in d e r s a g r e e
e s s e n t i a l l y . w i t h t h e s i x t e e n a x io m s , a n d in d e e d th e a x io m a tic
c h a ra c te r­
i z a t i o n of a d i f f e r e n t i a l c a l c u l u s s e e m s q u ite s a t i s f a c t o r y in th e c a s e of
l i n e a r lim it s p a c e s .
An im p o r ta n t c o n c e p t to b e c o n s i d e r e d in c o n n e c t i o n w ith re m a in d e r s
i s t h a t of th e U - s e t .
D e f in itio n
C o n s id e r !the fo llo w in g d e fin itio n :
\
A s u b s e t U of a n o b j e c t E in a c a l c u l u s ^ i s c a l l e d a U - s e t
w ith r e s p e c t to o b j e c t G yl" , w h e re G y1" h a s n o n z e ro d i m e n s i o n , if
w h e n e v e r t h e r e e x i s t r & /^'(Ey i, G yl" ) a n d D E ^ G yl" , s u c h t h a t
-i
f I U = r I U,
30
t h e n f £ "f (Eyi , G yi, ) .
.G a " of c o u r s e i s a n y o b j e c t i n ^ , a n d t h e d e f in itio n i n no w a y
d e p e n d s on a s p e c i a l o b j e c t .
The q u e s t i o n n o w a r i s e s — w h a t a r e t h e U - s e t s for t h e s tr u c tu r e
ju s t c o n sid e re d ?
Theorem 2 . 3
A s e t U £ E i s a U - s e t i f a n d o n ly if th e r e e x i s t s an
r e ^ ( E yi , G ^ - ) s u c h t h a t r | C U = c , a n o n z e r o c o n s t a n t .
Proof
F ir s t of a l l l e t r 6?{(EA , G a " ) s u c h t h a t r | CU = c , a n o n z e ro
c o n s t a n t , a n d l e t f b e a m a p p in g from 'E A to G a - s u c h t h a t f | U =
w h e r e j z e i£(E A , G A" ) .
%(E^, Gy1") s u c h t h a t
Axiom 12 im p lie s t h a t th e r e e x i s t s a
[C U .= f .
s i n c e f e i t h e r a g r e e s w ith J 0 \ OT
If a g a i n
on a
(Eyl ,G A "
U is a U -s e t
n e ig h b o rh o o d of z e r o .;
.in
Then Axiom 11 im p lie s t h a t fe f{ (E A , Gy)
2’
) i s th e s e t of r e m a in d e r s w h ic h v a n i s h
/I o p e n n e ig h b o rh o o d of z e ro j i t i s c l e a r ) t h a t :
Theorem 2 . 4
|U
i f a n d o n ly if i t c o n t a i n s a A o p e n
C h a p te r - i n
The Binz R em ain d e r
In a r e c e n t p a p e r ( I ) , Binz i n tr o d u c e d t h e fo llo w in g , d e f i n i t i o n for a
re m a in d e r r d e f in e d o n t h e c l a s s o f a l l lim i t s p a c e s
(Fi A 1
o v e r th e r e a l n u m b er f ie ld IR. .
[ (E, A ) ,
This i s a g o o d e x a m p le of
a r e m a in d e r a s d e s c r i b e d in c h a p t e r I I , a n d i t w ill now b e sh o w n t h a t t h i s
d e f i n i t i o n s a t i s f i e s th e s i x t e e n a x io m s u s e d to c h a r a c t e r i z e a d i f f e r e n tia l
c a lc u lu s.
- '
N o te t h a t i f 0 i s a f i l t e r i n A(O), ^ i n t h e fo llo w in g d e f i n i t i o n , w h e re
U i s a A o p e n s e t c o n ta i n in g z e ro in E, w i l l ' m e a n t h e f i l t e r g e n e r a t e d
b y t h e i n t e r s e c t i o n of t h e f i l t e r s e t s In^A (O ) w ith U ./L (0) w ill r e p r e s e n t
u
t h e A - i d e a l g e n e r a t e d ,by t h e i n t e r s e c t i o n o f U w ith th e s e t s of th e
f i l t e r s in A(O).
D e f n iitio n (Binz)
L et U £.
m a p p in g n U ^ F ^ /
, w h ic h i s c o n tin u o u s a t 0 <£ U i s c a l l e d a r e m a in d e r
w h e n for e v e r y 0
£ A
u
u
b e a A o p e n s e t c o n ta i n in g z e r o .
(0) t h e r e e x i s t s a ^A (O ) s a t i s f y i n g t h e fo llo w in g
c o n d itio n :
,
(B)
The
•
For e v e r y Ne ^ th e r e i s a n M £<p a n d a c r s u c h t h a t
u
■ .
'
r ( ' X M) £. ^ ( /i)N
d efin itio n
for a l l % in t h e d o m a in of
' o fcr,
c r • h e r e r e p r e s e n t s a r e m a i n d e r , i n t h e u s u a l s e n s e of a n a l y s i s , o v er
th e c a l c u l u s of r e a l n u m b e r s ,
jo-(x)j
i e . w h e re Iim —y - = 0 , a n d cr (0) = 0 .
x->ol " ^ 1
■
32
• From' (B) r(0) = 0 , a n d t h e z e ro m a p p in g i s a r e m a in d e r .
E arly in (I) , Binz p ro v e s t h e ,follow! ng theorem :
Theorem 3 . 1
Let U £ E b e a A o p e n s e t c o n ta i n in g z e r o .
A m a p p in g ,
r : E p - F y i s a r e m a in d e r i f a n d o n ly i f r | U i s a r e m a in d e r .
W ith t h i s , t h e a b o v e d e f in itio n c a n b e a l t e r e d b y s i m p l y r e p l a c i n g
U b y E, a n d t h i s w ill b e t h e form for t h e d e f in itio n c o n s i d e r e d
i n th e
p resen t paper.
The p ro b lem n o w i s to s h o w t h a t th e Binz r e m a in d e r s s a t i s f y t h e a x io m s
a s s o c i a t e d w ith a d if f e r e n t i a l c a l c u l u s .
Eyl /Fyl^
L e t, a s b e f o r e , <^(EA , F yv/ ) ,
.
c o n s i s t of th e s e t o f a l l l i n e a r c o n tin u o u s m a p p in g s from Eyl
to F y , a n d l e t ^ ( E z , F z/ ) b e m a d e u p of t h e f u n c tio n s w h ic h a.re re m a in d e r s
in t h e s e n s e of B i n z .
F ir s t of a l l w e n o te t h a t Axioms 1 2 , 1 3 , 1 4 , a n d 15 fo llo w a s
c o n s e q u e n c e s o f Theorem 2 . 2 s i n c e t h e z e ro m a p p in g i s a r e m a in d e r a n d
a n y f u n c tio n v a n i s h i n g o n avlopen n e ig h b o rh o o d of z e ro i s a r e m a in d e r , (of. .
p roof of Theorem 3 . 2 ) .
6.
In h i s
p a p e r Binz v e r i f i e s Axiom s 2 , 3 , 4 , 5 , a n d
C l e a r l y Axioms 0 a n d I h o ld b y d e f in itio n of
re sp e c tiv e ly .
a n d <^(EA , F y )
Axioms 7 , 8 , 9 . 1 0 , a n d 11 th e n re m a in to b e v e r if ie d for
th e s e re m a in d ers.
Axiom 7
of c o u r s e f o llo w s s i n c e r e a l n u m b e r m u ltip l ic a tio n i s l i n e a r a n d
c o n tin u o u s, and th e c h a in ru le h o ld s .
33'
Axiom 8
c a n b e v e r if ie d in t h e fo llo w in g w ay:
The c o n tin u ity r e q u ir e m e n t a t z e ro i s c e r t a i n l y c l e a r .
It m u s t now
b e sh o w n t h a t fo r a n y ^/1(0) , th e r e e x i s t s a n J1W(O) s u c h t h a t for a n y
N £ / 't h e r e i s a n M ^ a n d a <r s u c h t h a t
r( ^ M ) £<r( ^ ) N , for ^
The f i l t e r s i n A(O) a r e of c o u r s e t h o s e in d u c e d b y A ' (0).
f i l t e r i n A (0).
,£.1 .
L et 0 b e a n y
The s e t s of 0 a r e g e n e r a t e d b y t h e i n t e r s e c t i o n of th e
s e t s of a f i l t e r 0 ^ ( 0 ) w ith E.
But r £ ^ ( F y j' , G ylA .
.Therefore t h e r e e x i s t s
an$W (0) c o r r e s p o n d i n g to 0 s o t h a t for a n y NE ^ t h e r e i s a s e t W e ^ a n d a
such th a t
.
But W A E £ 0 .
r( X W )c c r( ^ ) N ,
^ C [-6 ,6 ] .
L e t then- M = WAE a n d c l e a r l y
r(& M )'s<r(% .)N ,
.
Thus r l E ^ ( E A , G z )o
From K o e th e 1s b o o k on l i n e a r t o p o l o g i e s ^ ) w e k n o w t h a t a l l o n e
d im e n s io n a l s u b s p a c e s o v e r t h e r e a l n u m b e rs a r e is o m o r p h i c .
t h e n Axiom 9 .
C o n s id e r
D e fin e
'C
■
-
(Jr i L)
(x), x £ F 1 G F
/
JD, o t h e r w i s e .
a n d s h o w t h a t (Fyl' , G y 1") <,
It i s n o te d f i r s t of a ll t h a t c o n t i n u i t y a t
34
N ow s u p p o s e t h a t <fi i s a n y f i l t e r i n T^(O) s u c h '
z e ro i s c l e a r f o r ( x )
Then (J) g e n e r a t e s a f i l t e r j) 1
t h a t j6 h a s n o n e m p ty i n t e r s e c t i o n w ith F1 .
i n F 1 ,.which c o n v e r g e s to z e ro in F j .
S in c e L i s a l i n e a r is o m o r p h is m "
L ( ^ ) i s a f i l t e r 'B 1 c o n v e r g in g to z e ro in. E1 .
T his m e a n s o f c o u r s e th a t
th e r e i s a f i l t e r QeA(O) w h ic h g e n e r a t e s &
S in c e r £
e x i s t s a n J^izf(Q) c o r r e s p o n d i n g to
.
(Eyl , G vl- ) th e r e
Q e A (0) s u c h t h a t for a n y N <£ A t h e r e
.
e x ists an
a n d a cr so t h a t
. r ( ^ . M ) £ c r ( %) N, UL-s.^1.
C le arly th en
r [ X(M A E1)] £ cr( X )N , UU.
L et W 1 £ 9 1 b e th e p r e im a g e .o f M flE 1 u n d e r L.
-
. ' .
Then
r [M L (W 1))] £ t r ( a ) N , a n d
i
\
s in c e L is lin e a r
r [ L ( X W 1 )] ^ c r ( X ) N .
A lso b e c a u s e W,^£^ 1, t h e r e e x i s t s a . W .
in W 1 .
s u c h th a t W fl F 1 i s c o n t a i n e d
T hen
for I f
f o llo w s d i r e c t l y from t h e d e f in itio n of J>
x e F ^s . F , a n d z e ro o t h e r w i s e .
t h a t r • L=y3 on F ..
sin c e
Ther e f or e y 5
A = (r • L) (x) for
, G a -) w ith t h e p ro p e rty
.
Now i f p h a s e m p ty i n t e r s e c t i o n w ith F , c o n d itio n (B) f o llo w s t r i v a l l y ,
35 •
i . e = i f f i s a n y f i l t e r in A (O) , c h o o s e fo r a n y N e /', a n a r b itr a r y s e t
a n d a n a r b itr a r y o~ .
T hen b y d e f i n i t i o n o f P
P (%M) = 0 ,
a n d th e r e f o r e s a t i s f i e s t h e r e q u ir e m e n t t h a t
£ cr ( ^,) n .
To v e rify Axiom IQ, f i r s t of a l l c o n s i d e r th e q u e s t i o n of c o n tin u ity
at zero.
L e t A , A' , a n d A" a g a i n b e th e li m i t a t i o n s on E, F , a n d G
r e s p e c t i v e l y , a n d l e t ^/1(0).
If 0 h a s e m p ty i n t e r s e c t i o n w ith G , c l e a r l y
y ° (</ ) g e n e r a t e s th e u l t r a f i l t e r i n A(O).
If ^ h a s n o n e m p ty i n t e r s e c t i o n
w ith G a n d th e r e f o r e g e n e r a t e s a f i l t e r ^ ^ i n G i t is c l e a r from t h e
d e f in itio n ofj> t h a t
5 r ( ^ ) , and
g e n e r a t e s a f i l t e r i n A(O).
r ( ^ ) , s i n c e r6 ^ ( G yl- , FAZ) ,
Thus A i s c o n tin u o u s a t z e r o .
L e t 0 ^ b e ' a f i l t e r in A(O).
T hen ^ , is g e n e r a te d b y s o m e f i l te r ^fA(O).
S in c e r £ ^ ( G yl- , F a 1 )., t h e r e e x i s t s a n ft/fo ) s u c h t h a t for a n y N e / th e r e
e x i s t s a n M-^C (/> a n d a
such th at
r(A M 1) ^ c r(^ ) N ,
L et M /I G - M j ' M 6
^ ye X
M
N o te h e r e t h a t if ^ y f i M
To v e r if y t h i s l e t
y € G sin c e G is a lin e a r s p a c e .
now J > '(' X
M).
C l e a r l y J> (
c o n d itio n (B) i s s a t i s f i e d .
XM)
Xy (
XM0
a n d ^ y e G , th e n
Then y e
Ma n d
if X yeG ,
T h e re fo re y <£ M.^ a n d xy^xM .^ . C o n s id e r
= r( X M)ccr(x) N , XoC-^ci, a n d
If ^A (O ) w ith e m p ty i n t e r s e c t i o n w ith G ,
36
c o n d itio n (B) i s t r i v i a l a n d y 9 6 f (Ea
,F fl' ) .
B efore c o n s i d e r i n g Axiom 11 i t i s u s e f u l to in tr o d u c e t h e fo llo w in g
n o ta tio n u s e d b y Binz in ( I ) .
Let 6*be t h e n e ig h b o rh o o d f i l t e r of zero
%
in §( i w h e re w e th in k of ^ a s b e in g g e n e r a t e d
sy m m e tric i n t e r v a l s
I £>9} • N ow l e t
b y th e s y s t e m of c l o s e d
a n d 'd e f i n e
i s a n y f il te r in A , to b e t h e f i l t e r g e n e r a t e d b y [R1M | R 1=.
N o te t h a t if j>^A, t h e n
^
M6
.
j^j) i s t h e f i n e s t f i l t e r c o a r s e r t h a n b o th j
, a n d i t i s g e n e r a t e d by [M U [-£,£] M | f > 0 ,
now t h a t ^ A( O) .
w h e re (j>
M e (j)} .
and
Suppose
In (4) F i s c h e r h a s p ro v e n t h a t (^A(O) ^ A(O) , s q ^
(() ^ A( O) -
T h e re fo re s i n c e ^ lis a A i d e a l , ^ A P ^ f 6-A(O).
N o w c o n s i d e r Axiom 11.
The p roof h e r e i n v o lv e s e s s e n t i a l l y t h e
p r o c e d u re u s e d b y Binz in (1)^ w h e n h e v e r i f i e d Axiom 2 .
For r^
,rg
in
) , w e h a v e t h a t for a n y f i l t e r ^A (O ), th e r e e x i s t f i l t e r s ^ ^ a n d
(E^,
in A(O) s u c h t h a t for a n y N | ^ / j ', Ng^
<r2s u c h t h a t r ^ (^M ^)sc^(lL) N^
7-e E-^,£*■]. C o n s i d e r t h e f i l t e r
th e re . exis.ts M^ Mg 6 ^ a n d <r^,
a n d r 2 ,(T -M 2 ) 5 u^(T.)N^ ,
Z2 6A(O)Jrom t h e p r e v io u s d i s c u s s i o n
i t fo llo w s t h a t th e f i l t e r f a - (^feA(O)Is c o a r s e r th a n vK
A a ^ A.
Let K b e a n y s e t in
It c o n t a i n s a s e t of. t h e form K' U C-f/g'lK', for K' £ f , a n d 0<C I .
N ow for any. f i l t e r <f) c o n s t r u c t a - c o r r e s p o n d in g f i l t e r in t h e a b o v e
w ay.
Then a n y s e t K in A ^ Y c o n t a i n s a s e t in the- form K1U [-.^V] K1,
K '6 A , i . e .
K' 2. N 1 C N 2 T N 16
,N 2 ^ A 2 .
S e l e c t t h e n , c o r r e s p o n d in g
37
to N i a n d Ng w ith r e s p e c t to r 1 a n d r 2 , t h e s e t s M 1 a n d Mg in $ ,
a n d form M = M^ A Mg'.
C l e a r l y M 1 A M g^ ($> .
.
T1 (AM)C
and
C h o o s in g
a r e s a t i s f i e d for AeL- E ,O
C ^ ( A ) K f AC-C1 , £ ]]"
’ r2( A M ) c
[ - f , C] = M i n [[-E1 , £ j]
.
Then i t f o llo w s t h a t
cr ^
,
A )K,
[_-£ 2 ,
A 6 [-Eg
C '2 3}
N ow for n o n z e r o y<
£ g] .
, th e above
such th a t I
we have th a t
' ( K 'U [ - < E 'J K ') S ( K 'U [ - < 6 ] K ') ,
a n d t h e r e f o r e ytK S y t'K .
' N ow for
[-e.f], d e f in e
V 1(A )Z e'
1Z
I^
^ ( A ) Z e'
and U V ( A ) I
C le a rly lim L V - (A )Z A l
A-»0
> Max-
=0,
, W h e n l c r 2( A ) I ^
O e r 2( A )I
,
| CT2 ( A ) |
or (A )K .£ c r ( 'A ) K7 a n d
I
I c 1( A ) I .
cr
z
r( AM) c ■cr ( A)K, A£[-^ a ] .
r ( 0 )6 A z(O).
C le arly r ( 0 ) c
A z(O).
S in c e
i s a lim it s p a c e w e know
C ertain ly th en r g e n e ra te s a filter and .
r(^) > Zy1 U Ag. . T h e re fo re r ( 0') 6 vl (0).
/
'i
CA(O).
(j V, ' if ^ 1 a n d ■ Zz g a r e t h e im a g e s
of ^ u n d e r T1 a n d Tg r e s p e c t i v e l y .
/' A
).
( a )K s O-(^)K
It r e m a in s to s h o w t h a t r i s c o n tin u o u s a t z e r o , i 0e . t h a t for
th a t
( JL) |
V (A ) = <
.
T h e re fo re
,W h e n Ic r1( J L ) I ^
/
38
Thus th e Binz r e m a in d e r s s a t i s f y t h e a x io m s w h ic h c h a r a c t e r i z e
a d ifferen tia l c a l c u l u s .
C o n s id e r n o w th e fo llo w in g th eo rem :
Theorem 3 . 2
A s e t U ^ Ea
6 ^ z", ^s , a U - s e t w ith r e s p e c t to th e Binz
,
re m a in d e r if a n d o n ly if i t c o n t a i n s a A o p e n n e ig h b o rh o o d of z e r o .
Proof
F ir s t of a l l , s u p p o s e t h a t U is a U - s e t w ith r e s p e c t to th e Binz
r e m a i n d e r s S i n c e t h e s e r e m a in d e r s s a t i s f y t h e a x io m s for a d if f e r e n tia l
• c a lc u lu s ,
b y Theorem 2 . 3 t h e r e e x i s t s a n
t h a t rJC U = c , a nonz,ero c o n s t a n t .
s u c h t h a t c jf V.
r e ^(E yl , G yl" ) ,Ga- £
such
S e l e c t t h e n a A o p e n s e t V in G
It fo llo w s e a s i l y from t h e f a c t t h a t r i s c o n tin u o u s a t
z e r o t h a t r -1 (V) is a n A o p e n n e ig h b o rh o o d of z e ro in Eyl a n d t h a t r - 1 (V) £ U .
C o n v e r s e l y , i t m u s t b e s h o w n t h a t a n y s e t W ^ E c o n ta i n in g a A o p e n
\
n e ig h b o rh o o d o f z e ro i s , a U - s e t .
h o o d of z e ro w h e re U £ W .
L et t h e n
U £ E b e a A o p e n n e ig h b o r ­
D e fin e t h e m a p p in g r from Ey to F^so t h a t
r I W = O •
and r
| CW = c , w h e r e c is a n o n z e r o c o n s t a n t .
show: t h a t r 6 ^ ( E y
,F y - ) .
W e now w o u ld li k e to
C o n tin u ity a t z e ro is c l e a r , i . e . l e t <p b e
a n y f il te r in A,(O)lj S in c e U c ^ a n d r (U) = 0 , r (^) g e n e r a t e s 't h e u l t r a ­
f i l t e r 6 6 A '(O)J N ow l e t N b e a n y s e t in 0 .
It m u s t b e s h o w n t h a t th e r e
e x i s t s a n M e 0 a n d a re m a in d e r cr s u c h t h a t r( % M )c o ' ( X )N , for Le C-6/ CL
Let 0 b e t h e f i l t e r g e n e r a t e d b y th e A o p e n s e t s c o n ta i n in g 0 £ E .
39
C e r ta i n ly 6 ^Aq (O) , w h e re
vl
i s th e a s s o c i a t e d p r in c ip le i d e a l l i m i t a t i o n .
Then i f 0 ° i s t h e n e ig h b o rh o o d f i l t e r of z e ro in t h e b a s e f i e l d IR , th e n
c l e a r l y (j)-Q ^vlo (O),!. e. s i n c e s c a l a r m u l t i p l i c a t i o n i s c o n tin u o u s a n d
c o n v e r g e s to (0 ,0 ) with, r e s p e c t to t h e p ro d u c t l im it a ti o n on % x E,
(J)0-9
€ A (0).
But © i s t h e c o a r s e s t f i l t e r in A q (0) s o
(f>°-Q > Q.
T h e re fo re for th e A o p e n 's e t U in 0 , th e r e e x i s t s a A o p e n s e t Z £© a n d
a n o p e n n e ig h b o rh o o d
' such th a t V • Z £ U.
n e ig h b o rh o o d of z e ro in E, Z £
a n d s i n c e V i s a n o p e n n e ig h b o rh o o d
o f z e ro in /R , [ - £ , £ ] £ V, for s o m e £ .
0
a n d th e i n t e r v a l [ - £ ,£ ] -
But s i n c e Z i s a /I o p e n
C h o o s e t h e n t h e s e t M = Z in
c o n t a i n e d in V, an d ' i t f o llo w s t h a t
■ r f l M ) £ O- ( I ) N,
sin ce
?-M
<£ U S W a n d r ( W ) = O.
Thus r e ^ ( E yl ,Fy l' ) a n d b y
\
Theorem 2 . 3 W i s a U - s e t .
I t m u s t b e n o te d t h a t t h e l a s t p a r t of t h e p ro o f of Theorem 3 . 2 d o e s
n o t d e p e n d o n t h e s p e c i a l p r o p e r tie s of t h e r e a l n u m b e r s .
a u s e f u l f a c t in c h a p t e r jv'.
■
T his w ill b e
C h a p t e r IV
The Strong Binz R em ain d er
A g e n e r a l i z a t i o n o f t h e Binz r e m a in d e r i s p ro p o s e d a s f o l l o w s .
Let
K b e a s e p a r a t e d f ie ld a n d s u p p o s e t h a t t h e r e i s so m e c a l c u l u s g iv e n
o n K.
= ((E , yl ) , (F, A ) , (G, I i ' ) , . . . . j
Now d e f in e
c l a s s of a l l l i n e a r lim it s p a c e s o v e r K, a n d l e t
d (E^
s e t of a ll l i n e a r c o n tin u o u s m a p p in g s from E ^ to Fyl'
in
to b e th e
, F 1- ) ■b e th e
, for a l l E ,F^,
. At t h i s p o in t w e p o s t u l a t e t h e fo llo w in g d e f in itio n for a Strong
Binz r e m a i n d e r :
D e f in itio n
. The m a p p in g of r: E^ — F^, , , w h ic h is c o n tin u o u s a t OG E
w ith t h e p ro p e rty t h a t
r (O) = O, i s c a l l e d a r e m a in d e r w h e n for e v e ry
f i l t e r 0 GVl(O), th e r e e x i s t s a f i l t e r /^6/1(0)such t h a t
(B) for e v e r y N £
th e r e i s a n M.£.pa n d a cr s u c h t h a t
r(
for a l l
M ) e « "( X)N- . ,
in a U s e t a b o u t z e ro in K.
H e re of c o u r s e <r r e p r e s e n t s a re m a in d e r from t h e c a l c u l u s on th e
b a s e f i e l d K.
' If t h e s e t
(Eyl , F yl/ ) c o n s i s t s of a l l Strong Binz r e m a i n d e r s , th e
q u e s t i o n i s th is — d o e s s u c h a d e f i n i t i o n to g e th e r w ith t h e a b o v e
d e f in itio n of (X (Fa , F yl- ) d e f in e a C a lc u lu s in t h e s e n s e of th e p re v io u s
a x io m s ?
41
Axiom 0 is c l e a r l y tru e s i n c e
c o n s i s t s of a ll l i n e a r s p a c e s
o v e r K. .
Axiom I
i s tru e s i n c e
^.(Eyl , F yl' ) i s t h e s e t of a l l l i n e a r c o n tin u o u s
m a p p in g s from E A to Fyi^ .
Axioms 1 2 , 1 3 , 1 4 , a n d 15
fo llo w a s c o n s e q u e n c e s of th e .
c o n tin u ity of r € "/[(E A , F y j a t z e r o , a n d o f t h e f a c t t h a t a n y f u n c tio n
w h ic h i s z e ro in a' A o p e n n e ig h b o rh o o d of zero, i s a r e m a in d e r , i . e .
t h a t a f u n c tio n w h ic h i s z e ro in a-A o p e n n e ig h b o rh o o d of z e ro is a r e m a in d e r
-follow s from th e l a s t ,p a rt of th e p roof to th e o re m 3 . 2 .
The p ro o fs h e re
a r e e s s e n t i a l l y th e s a m e a s t h e p ro o fs of t h e a x io m s in c h a p t e r I I .
Axiom 3
f o llo w s a l s o in th e s a m e m a n n e r a s in c h a p t e r II w h e n
'
u n iq u e n e ss of d ifferen ta tio n is req u ired .
\
■Axiom 4 ' i s t r i v i a l for th e s e t
In Axiom 6
y , F y ).
a re m a in d e r r e "^(E A , F y
) a n d a d e r i v a t i v e A6 # ( F y , G y )
a r e g iv e n a n d i t m u s t b e s h o w n t h a t A re ^ ( E yl ,'G y ) .
■is c o n tin u o u s a t z e r o .
C l e a r l y Ar
S in c e r e ^ ( E yl , F yl ) for a n y filter^< A (0 ), th e re
e x i s t s a f i l t e r J^A(O) s u c h t h a t for a n y N e/'t h e r e i s a n M £ / a n d a <r
s u c h t h a t r('%M jQ v(^)N , for a l l
in a U - s e t a b o u t O e K . -
But A i s a l i n e a r c o n tin u o u s m a p p in g from F y
in
/ ' ( O ) , and Ar ( % M) G A( <r ( %. )N) =
{ % )
to G y
, s o A(/') i s
A (N ). Thus t h e
f i l t e r s p a n d A' ( t ) s a t i s f y c o n d itio n (B) in th e d e f in itio n o f a re m a in d e r ,
42
a n d Ar & ^ ( E yl ,G A"' ) .
Axiom 8
Axiom 9
c a n b e v e r if ie d j u s t a s i t w a s a in c h a p t e r I I I <•
c a n b e p ro v e n a s fo llo w s :
L et E^ a n d
b e o n e d im e n s io n a l
s u b s p a c e of E a n d F r e s p e c t i v e l y , a n d a s s u m e t h a t a ll s u c h o n e
d im e n s io n a l s u b s p a c e s a r e
is o m o r p h ic .
b e a l i n e a r is o m o rp h is m from F^ to E^ .
is af 6
L e t r &^ ( E a ,G a " ) a n d l e t L
It m u s t b e sh o w n t h a t th e r e
ji / G yl'/ ) s u c h t h a t r -L = y=> I F ^ .
As in t h e v e r i f i c a t i o n of
Axiom 9 in c h a p t e r I I I , l e t
(r-L)
(x),
x 6 F1 a F
o th erw ise
a n d th e p ro o f i s t h e s a m e .
'
.
Axiom 10 ' c a n a l s o b e p ro v e n a s in c h a p t e r .'IIIz.
At t h i s p o in t Axioms 2 , 5 , 7 , a n d 11 r e m a in .
Axiom 7 i s c l e a r l y no
d if f ic u l ty o n c e Axiom 5 i s s h o w n s i n c e Axiom 7 i s a c o n s e q u e n c e of th e
c h a i n r u le a n d t h e c o n t i n u i t y o f s c a l a r m u l t i p l i c a t i o n .
H o w e v e r Axioms 2 ,
5 , a n d 11 a r e t r o u b l e s o m e , a n d o n e n o te s t h a t th e i r v e r i f i c a t i o n in th e
c a s e , of t h e Binz r e m a in d e r s in c h a p t e r HI a n d in (I) d e p e n d s , v e ry
•
h e a v i l y o n t h e s p e c i a l p r o p e r tie s of t h e to p o lo g y of t h e r e a l n u m b e r s .
It '
i s q u ite d o u b tfu l t h a t t h e s e a x io m s c a n b e v e r if ie d w ith o u t a d d itio n a l
assumptions,-. A s .a n a l t e r n a t i v e to a d d in g r e s t r i c t i o n s in o rd e r to pro v e ' •
43
Axioms 2 , 5 , a n d 1 1 , o n e c a n c h o o s e to s im p ly a d d a d d i t i o n a l r e m a in d e r s
to t h e s e t ^
(Eyl , Fyl' ) u n t i l a c a l c u l u s i s form ed w h ic h c o n t a i n s th e
Strong Binz r e m a i n d e r s .
In th e fo llo w in g m e th o d o n e m u s t b e c a r e f u l to ■
e n s u r e t h a t in f a c t h e i s d e a l i n g w ith s e t s .
s o m e s e t S o f l i n e a r lim it s p a c e s
T h e re fo re w e r e s t r i c t 9( t o
[ ( E f A ) , (F, A ' ) , . . . j
f i e l d K, w ith t h e r e q u ir e m e n t t h a t K b e in S .
o v e r .the
C o n s id e r th e n t h e s e t of
a l l c a l c u l i t h a t h a v e t h e o b j e c t s o f S a n d w h ic h c o n t a i n t h e Strong Binz
re m a in d ers.
L e t ^ ( E yi , F yl' ) b e t h e s e t of a l l l i n e a r c o n tin u o u s m a p p in g s
from E^1 to F 1'
,
,F ^ < S .
Form now
O T K E ^ .F ,!' ) .
over a ll th e s e c a lc u li.
The r e m a in d e r s ' in t h i s s e t c l e a r l y s a t i s f y th e
a x io m s for a c a l c u l u s .
W e n o te h e r e t h a t t h e r e a r e c e r t a i n l y c a l c u l i
th a t co n tain
^ ^ (E a ,F y
) s i n c e , for i n s t a n c e , t h e s e t Ti (E, F) =
{ f : ^ r—*• F y I f(0) = 0 , a n d f I s c o n tin u o u s a t z e ro ] to g e th e r
w ith t h e s e t o f a ll l i n e a r c o n tin u o u s m a p p in g s ^ (E a , F y
th e a x io m s for a c a l c u l u s a n d c l e a r l y
for a ll E a , Fyl- £
), sa tisfie s
(E a , F a ' ) ^ ^ (Ea , F y
)
' Thus w e a re n o t d e a l i n g w ith a v a c u o u s s e t .
U n iq u e n e s s o f d i f f e r e n t i a t i o n h e r e of c o u r s e c a n b e c la i m e d s i n c e w h e n
a n y s i n g l e c a l c u l u s c o n ta i n in g th e Strong Binz re m a in d e r s h a s
t h i s p r o p e r ty , t h e i n t e r s e c t i o n of a ll re m a in d e r s h a s i t .
.A m ore c o n s t r u c t i v e p r o c e s s f o r .o b ta in i n g t h i s c a l c u l u s c o n ta in in g
44
th e Strong Binz r e m a in d e r s i s s u g g e s t e d in th e fo llo w in g d i s c u s s i o n .
H e re
w e w ill d rop th e r e q u ir e m e n t of u n iq u e d i f f e r e n t i a t i o n .
C o n s id e r a g a i n a s e t o f l i n e a r s p a c e s ^ o v e r th e f i e l d K, a n d l e t
(^(Eyl , Fa ' ) b e th e s e t of a l l li n e a r c o n tin u o u s fu n c tio n s from
a n y E^ , F^e.^". As b e fo re l e t ■/^1^ (Ezl,
to F y ),
) b e th e s e t of Strong Binz r e m a i n d e r s .
As p o in te d o u t p r e v i o u s l y a l l a x io m s h o ld for t h e Strong Binz re m a in d e r s
e x c e p t Axioms 2 , 5 , 7 , a n d 1 1 .
S in c e Axiom 7 f o llo w s w h e n t h e c h a i n r u le
ho ld s,, in f a c t w e a r e .only c o n c e r n e d w ith Axioms 2 , 5 , a n d 1 1 .
w ith ^ (E^ , F y ) a n d c o n s t r u c t t h e l i n e a r h u ll of t h i s s e t .
C le arly
,F a ' ) ’ s a t i s f i e s Axiom 2 .
S tp rt t h e n
C a l l i t ^ ( E a ,F y )
N e x t r e q u ir e t h a t for r e /^ (F y , G y ) ,
A e^(EA / F y ) an d y ? 6 ^ ( E A , F a - ) ,th e n r(A+/>) £
• T his of c o u r s e m e a n s
t h a t m ore r e m a in d e r s m u s t in g e n e r a l b e a d d e d to t h o s e in s e t ^ ( E a , F y ) .
C all th is new s e t ^ ( E a , F y ) .
C e r t a i n l y Axiom 5 h o ld s for /{g(EA , F y ) . It
m u s t b e n o te d a t t h i s p o in t t h a t Axiom 2 m ay no lo n g e r h o ld for ^ ( E a , E y ) •
■
C o n tin u e n o w a n d r e q u i r e t h a t w h e n r^ , ^ e /fg(EA , Fa ' ) , a n d r: E y ^ F y
is
s u c h t h a t w h e n r(x) ^ r ^ ( x ) , t h e n r(x) = ryCx) for a l l x £ E', i t f o llo w s t h a t
r &^ (E A , F y ) . T h is m ay r e q u i r e t h a t s t i l l m ore r e m a in d e r s b e . a d d e d to
^(E a
,F a ' ).
C a l l th e n e w s e t
h o ld s for /f^(E A , F yi' ) .
^ ( E A , F a ' ) , a n d c l e a r l y Axiom 11
At t h i s p o in t a l l a x io m s e x c e p t p e r h a p s
Axioms 2 , 5 , a n d 7 s t i l l h o ld s o r e p e a t t h e
p ro cess.
C o n tin u in g
t h i s i n d e f i n i t e l y l e a d s to a s e q u e n c e of s e t s
I
45
t x^ A
^ 2 (E a , F a ' J S f 3 C , , , F y ) . . .
.
.
Form n o w t h e u n io n
'
o o
.
- f A' ) = .U I t = A - f A ' ) , '
a n d 'K.{^ a , F yl- ) to g e t h e r w ith ^ ( E yl , F a ' ) c e r t a i n l y form s a c a l c u l u s
w h ic h c o n t a i n s th e Strong Binz r e m a i n d e r s .
K
Thus t h e Stro n g Binz
-
re m a in d e r s s e e m i n a c e r t a i n s e n s e to g e n e r a l i z e th e d e f in itio n of
a r e m a in d e r o r ig i n a lly in tr o d u c e d b y Binz in t h e c a s e of t h e r e a l fie ld •
An .in te r e s t in g q u e s t i o n w h ic h c o u ld b e p r o p o s e d a t t h i s p o in t i s —
w h e re d o e s t h e p r o c e s s j u s t d e s c r i b e d fo r t h e c o n s t r u c t i o n of a c a l c u l u s
c o n ta i n in g th e Strong Binz r e m a in d e r s e n d ?
o n 1t h e '. s p a c e s c o n t a i n e d in s e t
8
.
It seem s c le a r th a t it d epends
46
L ite r a tu r e C ite d
1.
B in z , E. Ein D i f f e r e n z i e r b a k e i t s b e r g r i f f in L im itie rte V e k to rra u m e n ,
C o m m e n t. M a t h . - H e l v . ,
41 (1966) , 1 3 7 -1 5 6 .
2.
B o u r b a k i, N . T o p o lo q ie g e n e r a l e , H e rm a n n a n d C i e . , P a ris ,
1951. .
'
3.
D ie u d o n n e , j . F o u n d a tio n s of m o d e rn a n a l y s i s . A c a d e m ic P r e s s ,
N ew York a n d L o n d o n , I 9 6 0 .
4.
F isc h e r, H .
5.
F r o l i c h e r , A. a n d W . B u c h e r. C a l c u l u s i n v e c t o r s p a c e s w ith o u t
n o rm , S p r in g e r - V e r la g , B erlin -H e id e lb * e rg -N e w Y ork, 1 9 6 6 .
.
K e lle r, H . H . D i f f e r e n z i e r b a r k e i t in t o p o l o g i s c h e n V e k to rra u m e n ,
C om m ent ; M a th , ' H e l v . 3 8 7 (1 9 6 4 ).
7.
K o e th e , G . T o p o l o g is c h e l i n e a r e R a u m e , S p r in g e r - V e r la g , B e r lin G o ttin g e n -H e id e lb e rg , 1960.
8.
W e h r l i , M . D if f e r e n tie lr e c h u n g i n a llg e m e in e n l i n e a r R aum en I ,
C ir c o lo m a te m a tic o d i P a le rm o , to a p p e a r .
9.
W e h r l i , M . D if f e r e n tie lr e c h u n g in a llg e m e in e n l i n e a r R aum en II,
C irc o lo m a te m a tic o d i P a r le r m o , to a p p e a r ; ,• ■
6
L i m e s r a u m e , M a t h . A n n ." (1 9 5 9 ), 2 6 9 -3 0 3 .
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