Bilinear forms on vector spaces of countable dimensions in the case characteristic of k equals two
by Robert Dean Engle
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 Robert Dean Engle (1965)
Abstract:
For k-vector spaces of a denumerable dimension supplied with a symmetric bilinear form, Kaplansky
proved that in the case where characteristic k is 2 and O(ge)=1 there are, up to orthagonal isomorphism,
precisely four non-isomorphic types. Kaplansky has raised the question of how this result is altered
when characteristic k is2 and O(gk)>1 . The question is answered in this thesis for the case O(gn)=2.
This thesis is divided into two parts, the first part is an investigation of properties of fields of
characteristic 2. In the second part the classification of non-isomorphic spaces is carried out. The major
tool used in the classification is a certain theorem of automorphisms (2.4) for fields of characteristic 2. BILINEAR FORMS ON VECTOR SPACES OF COUNTABLE DIMENSION
IN THE CASE CHARACTERISTIC OF k EQUALS TWO
"by
ROBERT D. ' ENGLE
A t h e s i s s u b m i t t e d t o t h e G r a d u a te F a c u l t y i n p a r t i a l
f u l f i l l m e n t o f th e re q u ire m e n ts f o r th e d e g re e
of
DOCTOR OF PHILOSOPHY
in
M a th e m a tic s
A p p ro v e d :
■ \
’M
H e a d , M a jo r D e p a rtm e n t
C h a irm a n , E x a m in in g C om m ittee
\ •
\
D ean , G r a d u a te D i v i s i o n
MONTANA STATE UNIVERSITY
B ozem an, M o n ta n a
A u g u s t, 1965
iv25. r.: -!"O STACK
(ii)
VHA
The a u t h o r , R o b e r t D ean E n g l e , w as b o m I n B i l l i n g s , M o n ta n a ,
O c to b e r 3 1 , 1935 t o M r. a n d M rs . R o ss E n g l e . He r e c e i v e d h i s s e c o n d a r y
e d u c a t i o n a t B i l l i n g s S e n i o r H ig h S c h o o l , B i l l i n g s , M o n ta n a g r a d u a t i n g
i n 1 9 5 5 . He a t t e n d e d M o n ta n a S t a t e U n i v e r s i t y a t M i s s o u l a , M o n ta n a a n d
r e c e i v e d a B a c h e lo r o f A r t s d e g r e e s i n M a th e m a tic s a n d P h y s i c s i n 1 9 5 7 .
I n 1 9 6 1 he r e c e i v e d a M a s t e r o f A r t s d e g r e e i n M a th e m a tic s fro m M o n tan a
S t a t e U n iv e r s ity a t M is s o u la .
/
I 11H
I
\
(iii)
ACOTOWLEHMnM1
The a u t h o r i s d e e p l y i n d e b t e d t o D r. H e r b e r t G ro s s f o r t h e v a l u a b l e
s u g g e s t i o n s a n d t h e m any h o u r s he. s p e n t d i r e c t i n g t h i s t h e s i s *
Z
/
sS
/
I
(I v )
TABLE OF CONTENTS
CHAPTER
I.
PAGE
I
INTRODUCTION
II.
NOTATIONS AND RESULTS
III.
-
2
FIELDS
IV .
CLASSIFICATION OF
5
k -^ A G E S ( E j <j? ) JN TEOE
CASE OF COUNTABLE DIMENSION
.
9
APPENDIX
20
LITERATURE CITED
.
/
I
22
(V)
■ L IST OF TABLES
TABLE
I.
PAGE
MAJOR UQN-ISOMORPHIC SPACES
y
21
%
;/
i
r
(v i)
ABSTRACT
F o r k - v e c t o r s p a c e s o f a d e n u m e ra b le d im e n s io n s u p p l i e d 'w i t h a
s y m m e tric b i l i n e a r f o rm , K a p la n s k y p r o v e d t h a t i n t h e c a s e w h e re
c h a r a c t e r i s t i c k i s 2 a n d OCgt3) - t t h e r e a r e , u p t o o r t h a g o n a l
is o m o r p h is m , p r e c i s e l y f o u r n o n - is o m o r p h ic t y p e s . K a p la n s k y h a s r a i s e d
t h e q u e s t i o n o f how t h i s r e s u l t i s a l t e r e d w hen c h a r a c t e r i s t i c k i s . 2
and
O (ejh ) ^
. The q u e s t i o n i s a n s w e re d i n t h i s t h e s i s f o r th e" b a s e "
O ( Q t.)
—
d.
•
T h is t h e s i s i s d i v i d e d i n t o tw o p a r t s , t h e f i r s t p a r t i s a n i n v e s t i ­
g a tio n o f p r o p e r tie s o f f ie l d s o f c h a r a c te r is t ic 2 . In th e seco n d p a r t
t h e c l a s s i f i c a t i o n o f n o n - is o m o r p h i c s p a c e s i s c a r r i e d o u t . The m a jo r
t o o l u s e d i n t h e c l a s s i f i c a t i o n i s a c e r t a i n th e o r e m o f a u ta m o rp h ie m s .. .. I
( 2 .4 ) f o r f i e l d s o f c h a r a c t e r i s t i c . 2 *
X
\
\
CHAPTER I
INTRODUCTION
The c l a s s i f i c a t i o n , u p t o m e t r i c ( o r t h o g o n a l ) is o m o r p h is m , o f f i n i t e
d im e n s io n a l
te - v e c t o r s p a c e s F , s u p p l i e d w i t h a b i l i n e a r fo rm (fr; F*/=-* k
i s a v e r y d i f f i c u l t p r o b le m .
I t h a s b e e n s o l v e d f o r p a r t i c u l a r f i e l d s /?
,
s u c h a s t h e f i e l d o f r a t i o n a l e , r e a l s , p - a d i c n u m b ers a n d f u n c t i o n f i e l d s
o f one v a r i a b l e o v e r a f i n i t e
F or v e c to r sp aces £
c o n s ta n t f i e l d
( V-) a n d ( a ) ) .
(se e , e .g .
o f a d e n u m e ra b le ( a l g e b r a i c ) d i m e n s io n , t h e d i f f i ­
c u l t i e s v a n i s h i n a l a r g e n um ber o f c a s e s b y sh o w in g t h a t
o r th o n o r m a l b a s i s f o r a l a r g e c l a s s o f u n d e r l y i n g f i e l d s
5
(
a d m its a n
cA m
- M V= a .
)
( < 9 .) ./ I n t h e d e n u m e ra b le c a s e , a n e x c e p t i o n a l r o l e i s o n c e m ore p l a y e d b y
t h e u n d e r l y i n g f i e l d s o f c h a r a c t e r i s t i c 2 . ' E v en u n d e r t h e a s s u m p tio n t h a t
b e p e r f e c t , i . e . , e v e r y e le m e n t o f
b
i s a s q u a r e , K a p la n s k y p r o v e d
t h a t t h e r e a r e , u p t o o r t h o g o n a l is o m o r p h is m , p r e c i s e l y f o u r n o n - i s o m o r p h i c 1
t y p e s o f d e n u m e ra b le v e c t o r s p a c e s s u p p l i e d w i t h a s y m m e tric b i l i n e a r
f o rm .
w hen
K a p la n s k y h a s r a i s e d t h e q u e s t i o n o f how t h i s r e s u l t i s a l t e r e d " ■
k is n o t p e rfe c t.
I n t h e f o l l o w i n g , we w i l l a n s w e r t h i s q u e s t i o n
f o r a l a r g e n u m b er o f n o n - p e r f e c t f i e l d s .
\
v
CHAPTER I I
NOTATIONS AND RESULTS
2 .1 :
L et £
Le a
k - v e c t o r s p a c e o v e r t h e c o m m u ta tiv e f i e l d
v i t h a s y m m e tr ic , n o n - d e g e n e r a t e b i l i n e a r fo rm
i s n o n -d e g e n e ra te , i . e . ,
if
0 (x, y) = o
i s a ls o c a lle d s e m i-s im p le .
v e c to rs
-TC-^o- v i t h
As u s u a l v e v r i t e
\\x u
'X
£
(ji- £ * e ->
fo r a l l
-Xe E
.
, s u p p lie d
B e c a u se
, th e n
y = p. ,
may t h u s c o n t a i n i s o t r o p i c v e c t o r s , i . e . ,
(J)Qxj fX) — o > "but n o v e c t o r s
Iiv it
k
x =/• O
f o r t h e " l e n g t h " (J)Cxi fX)
v ith
o f a v e c to r
-X Jl
.
e
x e E
i s a l s o c a l l e d t h e norm a n d v e v i l l o c c a s i o n a l l y r e f e r t o t h e .map.,.-..
■x — ^ //%"Il a s t h e norm f o r m .
d e n o te d b y
Cxy) if
c le a r ly th e space
.( £ j 4 ) •
The v a l u e s
(J)C x,^)
of
(jj> v ti- ll o f t e n b e
th e r e i s no r i s k o f c o n fu s io n .
E
a n d t h e fo rm " c a r r i e d " b y
T h u s , i f CEi (I)) a n d ( E j (D) a r e
o r th o g o n a lly is o m o rp h ic (
fc
E
I f ve v is h to in d ic a te
ve r e f e r t o th e space
s p a c e s , th e n th e y a re s a id to be
& ( E j (D) Or s im p ly
E =
^
i f th e r e i s no
r i s k o f c o n f u s i o n ) i f a n d o n l y i f t h e r e e x i s t s a v e c t o r s p a c e is o m o rp h is m
V: E
E
such t h a t
(J)Cx j y ) -
(J)(VCx) JHj Cy ) )
fo r a l l
Ofj y
e
E
•
.
F o r b r e v i t y , v e v i l l u s e is o m o rp h is m a s synonym ous v i t h o r t h o g o n a l
is o m o r p h is m .
If
v e c to rs X e E
F
i s a sub sp ace o f E
such t h a t
r a d i c a l ( r a d ( F) )
r a d . Cf ) ~ 0 •
If
t h a t f o r a l l fX a F
th a t
E
F
is
, and
and f o r a l l
G
•
F
E
of
o su ch t h a t
F = F sx .
if
F + G- i s
O ~ c) D ,
H ence
i s c a lle d th e
f
/= ,c
in t h i s case ve say
F
u m Zl
subspace
F
■
of
i s c l o s e d a n d <? i s a f i n i t e
c lo s e d .
E
\
.
Etor a f i x e d b a s i s
deH U (j>( s c y S6) Il .
a n d D = (->)
k ) v h ic h i s an in v a r ia n t o f
F .
Qxj y ) -=•£).
We a l v a y s h a v e
f=r , l e t D =
F
m
E . , t h e n F J j. & m eans
b e a f i n i t e d im e n s io n a l s u b s p a c e o f
i s any o th e r b a s is o f
p h is m s o f
Lj & G - 1
, th e n
F A P
F i s s e m i-s im p le i f a n d o n ly i f
F and Q a re subspaces o f
s a id t o be c lo s e d i f
\ 5 ti . . . j
^
F
i s o rth o g o n a l t o
L et
/= ^ i s th e sp a ce o f a l l
( x , </) = £ f o r . a l l y a / = -
of
d im e n s io n a l su b sp a c e o f
2.2:
, th e n
If
- - •, $ '* }
// (J) ( s i , S j J U , t h e n t h e r e e x i s t s
CD d e f i n e s a. c l a s s m o d u lo s q u a r e s i n
F , /.,e,, i n v a r i a n t u n d e r o r t h o g o n a l is o m o r ­
T h is c l a s s i s c a l l e d t h e d i s c r i m i n a n t o f
P
( f o r e x a m p le .
- 3 P
i s s e m i-s im p le i f an d o n ly i f i t s
, 2.3:
d-nn E
cha ri-e.,
^
m eans
th e n
f
O
h a s a d e n u m e ra b le a l g e b r a i c b a s i s .
h a s a n o rth o g o n a l b a s i s ( £
a b a s is 4'd/,
£■'j £ c )
£
d is c rim in a n t i s n o t th e z e ro c l a s s ) .
• j
•
If,
such t h a t
(6^ 6 ^)
case
£
a lw a y s s e m i - s i m p l e ) ,
— O '
h o w e v e r, char* k = =2
n e c e s s a r i l y p o s s e s s a n o rth o g o n a l b a s i s .
If
if
, th e n
zV y
£
and
does n o t
But a s in th e f i n i t e
d i m e n s io n a l
i s a n o r t h o g o n a l sum o f s u b s p a c e s o f d im e n s io n a t m o s t 2 .
The
i r r e d u c i b l e 2 - d i m e n s i o n a l summands a r e h y p e r b o l i c p l a n e s
TV , i . e . , p l a n e s
sp an n ed b y a b a s is
e.c , <2/ w i t h
a n d </>
I n o th e r w o rd s,
d e c o m p o se s i n t o a n ,o r th o g o n a l sum E - Fr & (BPt.-
w h e re
p
f
h a s a n o rth o g o n a l b a s is an d th e
a l s o s a y t h a t t h e s u b s p a c e .69 TLI n p a r t i c u l a r , i f t h e fo rm
a ll
UGd H = U e l n
1X e J=
} th e n
on
E =JSjfc
c o u n ta b le c a s e , s e e e . g .
of £
£
L et
f
be a
<i\M ( E/ v )
v
(3 )).
is a lte rn a te , i . e . ,
E
w ith
Env
fo u n d i n
(3.).
Zr
—
and
E
£
p o s s e s s e s a n o rth o g o n a l b a s i s .
£
^
fo rm
such t h a t W x s t
a re f in i te
y
v* -1 -
.
11/
F u r th e r ­
(O)
and
d i m e n s i o n a l is o m o r p h ic ■
E n V = I o ) } t h e n e v e r y o r t h o g o n a l is o m o r ­
I t w as p r o v e d i n ( I ) .
T~ o f E
A p r o o f can a l s o be-
The a b o v e f o r m u l a t i o n i s m o re g e n e r a l t h a n t h e f o r m u l a t i o n s
( I ) and ( . 2 ) .
H o w e v er, i n s p e c t i o n o f t h e p r o o f y i e l d s t h e m ore g e n e r a l
r e s u l t q u o te d .
M
•
T h is th e o r e m w i l l p l a y a n i m p o r t a n t r o l e ' i n o u r c l a s s i ­
f i c a t i o n b e lo w ( s e c t i o n 4 ) .
2 .5 :
= O fo r
H as- p .E s p c a n b e e x te n d e d , t o a n o r t h o g o n a l a u to m o rp h is m
w ith T ( v ) = v .
in
y
On t h e o t h e r h a n d we s e e t h a t i f
17 b e s u b s p a c e s o f
If
We
i s sp an n ed b y a s y m p le c tic b a s i s .
- v e c t o r s p a c e o v e r a c o m m u ta tiv e f i e l d o f a n y
= d\m C e /P t) •
subspaces o f
p h is m
and
.
,
a re h y p e rb o lic p la n e s .
c h a r a c t e r i s t i c s u p p lie d w ith a n o n -d e g e n e ra te a l t e r n a t e
m o re ,, l e t
) = I
( f o r a p r o o f o f t h e s e th e o r e m s i n t h e
. c o n ta in s no i s o t r o p i c v e c to r s , th e n
2.4:
TV'
=- O
'■
I n t h e f o l l o w i n g i n v e s t i g a t i o n s we w i l l a ssu m e t h a t
- v e c t o r s p a c e w i t h c .h a /' k = 3. a n d dun E = cVi ..
w i l l f a c i l i t a t e th e re a d in g :
" EffiJ
t*= j <fi) i s a
x
The f o l l o w i n g n o t a t i o n s
w i l l s ta n d f o r a sp ace spanned by a
v.
- 4 d e n u m e ra b le s y m p l e t i c b a s i s , i . e
S im ila rly , "
, £ (o) - 3 Po
, T i' a h y p e r b o l i c p l a n e .
" w ill s t a n d f o r a s p a c e s p a n n e d b y a d e n u m e r a b ly i n f i n i t e
o r t h o g o n a l b a s i s , e a c h b a s i s v e c t o r o f w hich, h a s t h e sam e l e n g t h
F i n a ll y a 1 -d im e n s io n a l sp ace
as
A
m
. and s im ila r ly
WCxU = /3 O a )
/
as
h. Ce)
k C e ,,^ )
w i t h He I/
w ith
oi. =£ O .
a) w i l l b e r e f e r r e d t o
, / I 6lll =»
>
Afcp-).
Z
I
;
11 /I
CHAPTER H I
ITELDS
I n t h i s s h o r t s e c t i o n we l i s t ' a few e l e m e n t a r y f a c t s a b o u t f i e l d s o f
c h a ra c te ris tic 2.
F irs t le t
m u l t i p l i c a t i v e g ro u p o f
&
&
b e a n y c o m m u ta tiv e f i e l d ,
, and
k * 2' t h e s u b g ro u p o f s q u a r e s .
d e n o te t h e m u l t i p l i c a t i v e g ro u p
If
, /3 e A , we w r i t e
The o r d e r o f
in fin ite ,
fc* b e t h e ' '
§7,
* o f n o n ~ z e ro e le m e n t s m o d u lo s q u a r e s
" i f th e re , e x i s t s
w i l l be re p re s e n te d by O( c j h )
'Y & h
such t h a t
.
can be f i n i t e
h
= rVjG .
or
i f i t i s f i n i t e , i t i s a p o w e r o f 2 s i n c e e v e r y e le m e n t o f
o f o rd er 2.
P r o v id e d t h a t
f ie ld s s a tis fy in g
O C gU = S h.
n u m b ers, r e s p e c t i v e l y .
in fin ite .)
(E x a m p le s f o r »
=
£ ? , /
,
3 a re :
For
k
th e f i e l d o f r a tio n s
on t h e o t h e r h a n d , chav h = Z ., t h e n t h e s u b s e t
s u b fie ld o f
e le m e n ts o f
Cjk
k
; i . e .,
Ik
is a
Cjk. i s a l r e a d y
If, .
o f sq u a re s ( in c lu d in g
' te 3, v e c t o r s p a c e .
The
c o rre s p o n d t o th e s t r a i g h t l i n e s th ro u g h th e o r ig i n o f
• th is v e c to r sp a c e .
M ore p r e c i s e l y , we' h a v e t h e
" ■
ctnan k = 3. t h e n e i t h e r O
Lemma 3 . 1 :
Tf
P ro o f:
OCg.te) ? I
, th e re e x is ts
L et
i f a n d o n l y i f ( x - i j ) 2^ =
dL.
-
^
s in c e
cL a n d
/3
.
such t h a t
.
(T ^ i
Thus
o V ^ /C T
A =
a.
In p a rtic u la r,
x a n d =K=
c o r d C s)
^
fo r fin ite
b
).
x
O( Sk) = i
O (gw) = J
■.
.
a n d h e n c e .th e a s s e r t i o n .
The f i e l d s w i t h
.
F o r^ y -y ^ s
char
) w i t h / X * £ ■, t h e n
w h ic h c o n t r a d i c t s t h e c h o ic e o f
'
/
o r e l s e O ( C jic ) = (iar-u
a n d /3 e ^
; i.e.,
q
1A
Cs ) = c ovd? C/?)
, th e n
S u p p o se
te2
The
, 2 -a d ic
A d e ta i le d i n v e s t i g a t i o n o f su c h f i e l d s i s g iv e n i n ( l ) .
z e ro ) i s a
is
ClnaK k j= & t h e r e e x i s t s f o r e v e r y n a t u r a l >i
f i e l d s o f co m p le x n u m b e r s , r e a l n u m b e rs , p - a d i c n u m b ers w i t h
If
V ill
(chan k = 3.) a r e p r e c i s e l y t h e p e r f e c t f i e l d s .
fo r fin ite
Thq d e g r e e
: fe1}
As ' ( i f
dUv k ^ a
m ay b e f i n i t e
f i n i t e , i t n e c e s s a r i ly i s a pow er o f 2 .
th e n
O(Qu)= A
or in f in ite ,
I f i t is
- 6' Lemma 3 » 2 :
L e t Char k = A t h e n f o r some n a t u r a l n u m b er 7) ,[fe.' Zr7J = A77
p r o v i d e d t h a t Jk : JkyJ
P ro o f:
L et S ~
e le m e n t s o ve>
is
fin ite .
■ ■■: i (9-n^ b e a f i n i t e
Z?2- s u c h t h a t
irre d u c ib le o ver
Iki ( Si ) : h 2' J
K = k* ( Qlj ■■■ (p^) .
6>i%-
= <9- •
The f a c t t h a t
s e t o f in d e p e n d e n t a lg e b r a ic
Cx--O1J i
and
S in c e ' x * - o ?
Q i d k 7'
is
) u e have
S u c c e s s i v e a d j u n c t i o n s g i v e / 7 ? / /?2J •= A?1
D? /
.
i s in v a r ia n t u n d er a f i n i t e a lg e b ra ic
e x te n s io n i s g iv e n i n :
Lemma.3 . 3 :
of
k
I f Cbam k = A ,
, t h e n J j t k ' J = jk
C k i k 7-J
f i n i t e , and
~k
a f i n i t e e x te n s io n
IfJ .
/
P ro o f:
I t is
s u f f i c i e n t t o g iv e t h e p r o o f f o r a s im p le e x t n e s i o n
L e t [ h Cs ) ; Jej= »■/•'/, t h e n i , ©, Gl^ ^
a b a s is o f
hence
A ("<£>) o v e r
k
•
jCF a r e l i n e a r l y in d e p e n d e n t a n d fo rm
We h a v e
(k C o ) : k (Q)CJ [ k C s ) \ ‘ k f j =
Kz-J - Jk( O) I k j
— Ac -V- Am S
hence
•
Of e M CO) z
+ -*« « "VAvT (Q2,11 y
/? C<$ ) 2=
K C t Q 2:)
in d e p e n d e n t o v e r
k 3, :
.
<^o a K
F o r su p p o se
e le m e n t s o v e r
P ro o f:
If
k
clici^h =r A.
, and
L e t [k : kzJ = Jl
h ( S a)
in d e p e n d e n t o v e r
T hus
Y = CAo IlA 1S / - -- -V-AtSn)
Cs ) z e. K
C Q 7-)
* ' ‘j
Qu&J
a re lin e a r ly
J(C O * - t ------ ^ A m S 1”-
S
i s an in f in ite
K ( S c ) 3- .
O
O
,
•
i s f i n i t e , th e n
s e t xo f a l g e b r a i c
f k C s ) : k ( s ) zJ ^JJk^k2-J
a n d s u p p o s e [ k ( s ) r K ( S ) zJ 7 J
. QJl j
and
,
,
l i n e a r l y i n d e p e n d e n t e le m e n t s o f
C o n s id e r
,■ t h e n
We. w i l l show t h a t
O a n d A o V-A1 G> V- - - • A m s ’1 =
Js c = O ^ i = i, Z j * .
C o ro lla ry 3«1:
•
I t i s c l e a r t h a t I j O 3"j
h e n c e ( AoV- A1S -v - • - V-A t
th e re fo re
k t C. k C o t c K ( O ) a n d h t c /k c. k ( a )
k j C k ; /?aJ
If
fc - b f # )
k(s)
, Oj . + 1
over
.
S e le c t a n y /v -J
k ( £ ) z , s a y 50 = £@ i, - ■■,Q x-ti}
a r e in d e p e n d e n t o v e r
B u t b y . Leinma 2 . 3
/=? Cs J 2 , t h u s
[k (So) Z k C s S ] =
w hence a c o n t r a d i c t i o n , t h e r e f o r e Jk ( s ) ; h ( S ) 7J ^ J
Jl
t
■>
F o r t h e a b o v e c o r o l l a r y , we re m a rk t h a t we c a n n o t h a v e . e q u a l i t y i n '
; 11Ji
- 7 g e n e r a l a s i s w i t n e s s e d b y t h e t r a n s i t i o n fro m
c lo s u re
k
( f o r w h ic h j]?l TffJ = O( ^ h ) — d
Lemma 3 « 2 y i e l d s
f a : k'J =2*.
if
k
to i t s a lg e b ra ic
),
f a I k iJ
How d o e s one a c t u a l l y o b t a i n s u c h a f i e l d ?
'-V
. is
f i n i t e and
.
T h is q u e s tio n , i s a n s w e re d i n
t h e f o l l o w i n g Lemma-:
Lemma 3 . 4 :
I f f a / k^J
is
tra n s c e n d e n ta ls o v er
Proof:
L et
k
f i n i t e f and
Xi
'X-n
, t h e n far/ k* ] = fa:kfaj A *
{ /30 » - • ^
be a b a s is o f
A
a r e in d e p e n d e n t
w h e re k = k CrKtj - - - ^
over
k x .
we w i l l show t h a t
i s a b a s is o f
e le m e n ts .
e n t over
/c
D e f in e
=
over
K 3"»
O
.
= I f ', ;
or
JL
F i r s t we n o t e t h a t t h e a b o v e s e t h a s yv> a ^
S e c o n d ly , we m u s t show t h a t t h e e le m e n t s a r e l i n e a r l y in d e p e n d ­
Hrz .
T h is w i l l b e e f f e c t e d b y i n d u c t i o n on t h e n u m b er o f
tra n s c e n d e n ta l v a ria b le s
a s s u m p tio n ■£/3,, . - -
Xi j
^ j Xn .
F o r z e r o v a r i a b l e s we h a v e b y
f i - n j a s a b a s i s , th u s th e y a r e l i n e a r l y in d e p e n d e n t.
Assum e t h e a s s e r t i o n t r u e
f o r t h e n u m b er o f v a r i a b l e s l e s s t h a n
Yi
.
S
S u p p o se 2 " SJCr Kh . . ■, X m ) / 3.4
L1Ki j * * ' j Xn ) Z=k O
•
- - X n6 " =
. I f
a ll St’ 6
case i s
Si
O
L et
p o s it i v e pow er o f an
O j th e n
a s s u m p t io n e a c h
5-c 4- h
m u s t o c c u r i n '5 V , s a y
.
SiK ( X i , j » - , ' Xn - I j O ) ^/3 c
Si ^ O
i s a c o n tra d ic tio n .
0 and an
f o r w h ic h
Sc
=
O
Xn
W ith o u t l o s s
have a f a c t o r
X-n
<3 „
y,6: ..
= ^
.
L et
and by th e in d u c tio n
—O .
T h is i m p l i e s t h a t
t h a t f o r some t e r m SV Cxh ’
, say
Xj . -.
H S x C rK u t - t KXn) / 3 $
Xn
Nows u p p o s e
€./. ■& O
, t h e n some,
a n d t h u s a common f a c t o r o f e a c h
O ■ h a v e a common f a c t o r ;
p a rts ; i .e .,
d iv id in g by
Sc 4= O
S c ’ ( X i j t - - J Xn~i j O ^ / Z X — 0
is a ro o t o f each
Sc
, th e n th e z e ro v a r ia b le
be such th a t
o f g e n e r a l i t y we m ay a ssu m e n o t a l l
Xn =
a n d t h a t some
I f 6 / = 6 j. ■= j - - = 6 n - O , t h e n we h a v e
2 St- ( X ij * ' ' j X n J z =■ <9
c o n tra d ic te d .
<9 ,
-Sc'
XV)
w h ic h
, x-n ) 2Jd / x ,6-'- X n tlj
A g a in we a ssu m e t h a t n o t a l l
Xn
.
5,f
X 16I - ,
we h a v e 2 T 5 , ( X, , » » ' j X n ) > 3
, t y th e in d u c tio n h y p o th e s is a l l
S p l i t t h e sum i n t o tw o
^
a11
X ,^
S iX
r , 6 . - .X n -T 'x n -i^
, th e n
> » Xn*7~Lo.
L et
Sc' CXt j « « . , Xn -1 , <3 ) = <3 ,
*» 3 **
and th u s
CYVi i s a f a c t o r o f a l l
sXz-H- O > t h e n l e t
Si
w h ic h i s a c o n t r a d i c t i o n #
?fn = 6 a n d we o b t a i n
S*. ( lX i i - - ■) zTfw--, o ) / 5^ x,6- -
S y t h e i n d u c t i o n a s s u m p t io n s -f'j. (CKij • » s Trl- , , o ) - O f o x a l l
im p lie s
th e n
%-n
If a ll
'Jf-rr')
« ■* *Al) -)
O
, t h e n 5 " St
^
t h e a s s u m p tio n t h a t t h e
=-O
%vi
sp an s th e
i s o f t h e fo rm ^ (.Xi >> "_>rx-n') /
•
/c , b u t
As
•
O
Hr O
} le t
Ey t h e same
But t h i s c o n tr a d ic ts
QC-n
, th u s a l l
F i n a l l y i t m u s t b e show n t h a t
space
(Xij ' *. , Pr-n J w h e re
Ac . '
A
— A g ( ( f ‘^
c o n s id e r p o ly n o m ia ls . ■ L e t f
a r b i t r a r y e le m e n t o f
5"/
S l ' s do n o t h a v e a common f a c t o r
. , T-n"" ^
w ith c o e f f ic ie n ts in
Si
' ^=- O .
a n d we h a v e l i n e a r i n d e p e n d e n c e .
t h e s e t ^ (bo
5 -j . w h ic h
Xi ** '
I f some
a fa c to r o f a l l
" -O
-Zfr1) - Pf-M Su (X'J"
^
* - Pr?,-, j <0) /3 ^
r e a s o n i n g a s a b o v e we h a v e
l
Av)
Si z x O t h e n we h a v e a - c o n t r a d i c t i o n .
TCn =
S
L e t $ 2. ( x , j » *
i s a f a c t o r o f e a c h 5 ' a.
.. 6-.
Sr)
S'i. ( Xt j -
I f some
and
g
a r e p o ly n o m ia ls
t h u s we o n l y n e e d t o
- j or?, I =
C o n s id e r
E v e r y e le m e n t o f Ar
^
. . ^ ri X i
^
be an
A v* .
L et
CXotl^ jotn » AJ1yS y Tt- . . - -f cA«/3rv, ’ a n d de com pose x / '- - - X-^h=- X,6- " Xn6Yx* ' ' '
such th a t
O £ € l <. d a n d
GiAi- - -Am Kf"- --'Xvf —
+5. IOr — H. l
/3J-Xi6 . - • X-n
(
"J
I t f o ll o w s t h a t
Xi - - - XvC )
.
As a n e x a m p le we m e n tio n a l g e b r a i c f u n c t i o n f i e l d s 'Ar o f f i n i t e l y
m any v a r i a b l e s , i . e . , f i n i t e e x t e n s i o n s o f r a t i o n a l f u n c t i o n f i e l d s
k C1Xij *'-jX-*) ( t h e
X t *5 . in d e p e n d e n t t r a n s c e n d e n t a l s o v e r
k ( Xi ^' - j X- n) ( o , j - - - j ( pn).
Ac
L et
k.
.
S in c e k o A"*- c . . Ar
-over
k
f b u t e a c h e le m e n t o f
of
'Tl
L e t /A f /?^J
v a ria b le s o v er
J k f A aJ =
'k
and
h
i s a l s o f i n i t e l y g e n e ra te d
i t fo llo w s t h a t
M* i s a l g e b r a i c o v e r
Ey. Lemma 3 « 3 we h a v e 5 - Z?2J = U?** & * ]
C o r o l l a r y 3«2: .
A
£ - A 2J - V = { k * /
in
From g e n e r a l f i e l d t h e o r y we know t h a t
Ar, b e tw e e n
over
is' f i n i t e .
) > t\ =
fcf d e n o te t h e a l g e b r a i c c l o s u r e o f k
( t h e ! 'f i e l d o f c o n s t a n t s " ) .
any in te rm e d ia te f i e l d
k
be f i n i t e ,
and
is
f i n i t e l y g e n e ra te d
k.
, hence Q t: n J .
T h is g iv e s u s :
/<■ a n a l g e b r a i c f u n c t i o n f i e l d
th e f i e l d o f c o n s ta n ts i n
k
.
T hen
2J V .
I : I I in
CHAPTER IV
CLASSIFICATION OF k-SPACES
, d>) IN THE
CASE OF COUNTABLE DIMENSION
L et
is a
'
........ ..
"be a n o n - d e g e n e r a t e b i l i n e a r f o rm , <jf>JE x E ' -—> M
k
- v e c t o r s p a c e o f d e n u m e ra b le d im e n s io n .
, w h ere E
I n t h e f o l l o w i n g we
w i l l c l a s s i f y a l l s u c h s p a c e s u p t o o r t h o g o n a l is o m o rp h is m s f o r c e r t a i n
fie ld s
k
.
I n t h e s p e c i a l c a s e w i t h O (qk) = / / ? « ' fc’J =
c a r r i e d o u t b y K a p la n s k y i n ( d ) :
F o r such
k
t h i s has been \
we h a v e u p t o o r th o g o n a l
is o m o rp h is m s p r e c i s e l y 4 n o n - is o m o r p h ic s p a c e s , n a m e ly
,
E( j.) >
E(o) ®
I)> a n ^*
The f i e l d s f o r w h ic h we s h a l l c a r r y o u t t h e c l a s s i f i c a t i o n a r e t h o s e
w i t h [k / k zJ E CL ( s e e t h e e x a m p le s i n s e c t i o n 3) •
( Ej d) )
L et
v e c to rs
rX G E
be a space o v er su ch a f i e l d .
o f le n g th
Uxli=
E if. o f a l l
The s e t
O i s a sub sp ace o f
t h e s u b s p a c e o f v e c t o r s w hose l e n g t h
c o n s i d e r e d a s e s q u i l i n e a r fo rm .
,• . •
H
( th is is p re c is e ly
Il X Il i s a " t r a c e " i f
For th is
is
c o n c e p t s e e , f o r e x a m p le ,
La g e o m e tr ic d e s g r o u p s c l a s s i q u e s b y J . D ie u d o n n e , B e r l i n 1 9 5 5 , p a g e 1 9 ^
C le a rly
£?* i s a n i n v a r i a n t s u b s p a c e o f
a u to m o rp h is m s o f £F ) .
be c a lle d
L
E'
( i n v a r i a n t u n d e r o r th o g o n a l
An a l g e b r a i c co m p lem en t o f f r* i n . . E
w i l l a lw a y s
in th e s e q u e l:
/
E = £ * (B L
S in c e di ™ L = d i m
, t h e d im e n s io n o f
L
is in v a ria n t a ls o .
L-
c o n ta in s no i s o t r o p i c v e c to r s , hence i t p o s s e s s e s an o r th o g o n a l b a s i s , an d
b y t h e same t o k e n , d i m L E J k S k iJ
L e t, f o r a fix e d
in
E* ,
1— ,
k
CL . .
.
b e a n a l g e b r a i c com plem ent o f
E#D
;
E = ( E* A L j-) ® k @ L
t
E x = (E* A L u-J @
(I)
The f o l l o w i n g i n v e s t i g a t i o n s w i l l b e b a s e d th r o u g h o u t o n s u c h a decom ­
p o s itio n ( l)
of
E
.
We s h a l l now c o n v in c e o u r s e l v e s , t h a t di m k
n o t d e p e n d on t h e p a r t i c u l a r
Lemma 4 . 1 :
and.
L
L et
L
does
c h o s e n , m ore p r e c i s e l y we p r o v e :
( e > d)) b e a n a r b i t r a r y s e m i- s im p le
a r e a s d e s c rib e d above th e n
k - v e c to r sp ace,
If k
(
- 10 cLirn H
P ro o f:
L et
G-
~b ' d/'/v) (
-h b i /
b. v,)
£ * in
.
t h i s i m p l i e s d/ori k ■+ dry*
a n d E.n C
£ _) z r cL/M l_
"be a n a l g e b r a i c co m p le m e n t o f
a l g e b r a i c co m p lem en t o f
< ^/ m H = <Lnv
# -h £ n /
Erx 4- yS^ .
£*■ -+£>?i n
E
> and
H
H iu s E =* E * & H ® & a n d
I t i s s u f f i c i e n t t o show dVw /< = ch'm G— <£i-m & -t- ck-*™ a = c4'w
3 we h a v e E^ Yi
an
/v)^
^ )
£* /) W-Vl
»
fo r
S in c e L - G&H
E * /1 G-^ .
Now
u s i n g t h e f a c t £ * n L1 = £ * /I GX, we o b t a i n t h e c h a i n o f e q u a l i t i e s :
cL.Vi k =
E-v. 4 - G x
duC-m ( ^ V S x . ALj") —
is
cLc yn ( E * / t= ^ (\ q.x ') ■= C^ctyb (E x + C - / ^ , ^
s
c l o s e d ( s e e s e c t i o n 2 . 1 ) a n d Ex' f \ G —f o) b y c o n s t r u c t i o n ,
h e n c e one o b t a i n s E x i - G d = ( £ * + Gu)
( e £ /)
( E^ / )
=■ E r .
6o)
H iu s
cCc'ym k
=
c l t-Yn ( £ .> 4- G xZ
q
j -)
—
<£ty*” ( I = Z g j -) —
Prom ( l ) we s e e t h a t ( ExKL1) f\ / i a d E * i s o r t h o g o n a l t o
s e m i - s i m p l e if
a lw a y s b e t h e s p a c e
E x A LTi
a l g e b r a i c c o m p lem en t o f
V
.
H i th e f o llo w in g d is c u s s io n
( f o r some c h o ic e o f
in
, hence, as £
i_ ) a n d
ro d e*.
a re in v a r ia n t sub sp aces o f
B
o u r in v e s tig a tio n in to c a se s i s a s fo llo w s :
a c c o r d i n g t o t h e CLdw L ; i . e . ,
k
X
w ill
w i l l be th e
jzM .
I n t h e f o l l o w i n g c l a s s i f i c a t i o n , we u s e t h e f a c t t h a t
and
is
I n o t h e r w o r d s , we c a n a lw a y s c h o o s e fz
Cl L""\) / I A a J E* = (o).
i n s u c h a w ay t h a t r a d E* <L K
CP
.
Ex ,
£*
,
The schem e f o r d i v i d i n g
The f i r s t d i v i s i o n i s
t h r e e c a s e s : cLt-m L ■= o , H , ci 3 ..
T hen i n
e a c h o f t h e s e c a s e s we a p p l y Lemma 4 . 1 a n d f u r t h e r s u b d i v i d e b y t h e
a llo w a b le , v a l u e s o f d-tl-™
e
Z ~ a n d <£I y« C ^ a d £■») .
d i f f e r e n t c a s e s ; f o r a . sum m ary, s e e p a g e «2. 1
T h i s y i e l d s 10 ; ■ .
'''
'•
We now p r o c e e d w i t h t h e c l a s s i f i c a t i o n :
C ase I :
E
(it!™ '£ /£ %
— O -
We h a v e
i s o f c o u n t a b l e d i m e n s io n ,
E
E = E x and
^zf i s a l t e r n a t e .
S in c e
p o s s e s s e s a s y m p le tic b a s i s , th u s
E = E ("o).
C ase 2 . 1 :
do-™ E/j=^ =■ I
;
don™
Ex
— O
3
cLoyn
( rad ExZ = O •
'
i I in
- 11 L et
J— "be a n y a l g e b r a i c co m p lem en t o f
K
some a l g e b r a i c c o m p lem en t of" V
f e - V & H (b L •
L et
L
Ey Lemma 4 . 1
E> i n
in
£
/S V ,
Ji
IZ = E
th e n
fc = k ( u ) a n d
be spanned by th e v e c to r
,
=
Co H
V
is
L et
fo)
and
^
.
S u p p o se t h e r e i s a n o n - z e r o
f t =k 6
be su ch t h a t
, a c o n tra d ic tio n to
S in c e
d.i'm h -
cZl'w
and
V Q H
and
s i n c e V'ad- £ » = (o) .
«
"We n e e d t h e f e e t t h a t t h e o r t h o g o n a l c o m p lem en t
/r
/ I Z. ^
IZ
y e E
& -hftQ , c o ) =
j.
^
of
Iz
in
s u c h t h a t ^ -i
O
.
Vz
.
T hus (»<j? v-yfl ^
£ ■J'= ( o ) .
i - t h e r a n g e o f t h e n orm fo rm i s a s i n g l e c l a s s i n
We now show t h a t a n y tw o s p a c e s f a l l i n g i n o u r
c a s e a r e is o m o r p h ic
o n l y i f t h e r a n g e o f t h e r e s p e c t i v e norm fo rm s
c o in c id e .
One h a l f o f t h e
a s s e rtio n is t r i v i a l .
£
a re spaces
L e t u s t h e n a ssu m e t h a t
and
E
i f and •
f a l l i n g i n o u r c a s e , a n d t h a t t h e r a n g e s o f t h e norm fo rm s c o i n c i d e .
if
Zv
and X
T hus
a r e a n a lo g o u s o b j e c t s a s a b o v e , we may a ssu m e t h a t i\ i H~ IlTll
i n th e d e c o m p o s itio n :
E = V S
M u>) & h a ) ,
Jl-L V
> and
E = V ® k ( C o ) ® k ( 7 )>
FJ-Vr
'
I n g e n e r a l we s h a l l h a v e (Jljco')
b y Z -if- y t
w h e re
a s we h a v e s e e n ) .
■b e c a u s e
IItII=O
h a v e Cu
v
-£•
(I)
( H ilCo) >
i s some v e c t o r i n ~v
We t h e r e f o r e r e p l a c e
w i t h C i j Co) H C
jZ
( 7 " ^ » =,Yo)
T hus we c a n a c h i e v e t h a t ( H j Co) = ( d j o d ) -vriLth/lHll=IStIl
.
; i.e.,
L et
co
V. —
/I h ( l ) .
S in c e
( Jtj Co)
s t i l l s p a n s a c o m p le m e n t o f
7
in
=J= O
Z= x- .
we
We now
h a v e t h e f o l l o w i n g d e c o m p o s i ti o n :
■E— V
fc ( t o) (b fc ( Tl)
U Jl H =Jl F U
H Jtf
and E*
a re t r i v i a l l y
jmF * - (o) a s we h a v e s e e n ."
>T-LV
> ( I j Co) = (Qj L)) •
s e m i-s im p le i n o u r c a s e .
^
F u r th e r m o r e
H ence b y 2 . 4 t h e r e e x i s t s a n is o m o rp h is m
T * : E * ---- >
w ith Ty ( v ) = V
we c a n e x t e n d T
t o a n is o m o rp h is m 1 T / E r —> E
ZT . .
\
f and
, TC( O) = ZT
.
Ih view o f ( 2 )
s im p ly b y s e n d in g
Z
in to
T h is p ro v e s th e a s s e r t i o n .
I f /I'// H 111 — 6/ , t h e n t h e s p a c e
v e rifie d .
£ (^)
f a l l s in to our case as i s r e a d ily
We h a v e t h e r e f o r e f o u n d a l l n o n - i s o m o r p h i c .s p a c e s w h ic h f a l l
v
in to th e p r e s e n t c a s e :
If
[k * k J - ^ t h e r e i s o n l y one t y p e a n d £(£) i s a r e p r e s e n t a t i v e «
I f [R ‘ k zJ > d t h e s p a c e s £Y„g v h e r e
'
re p re s e n ta tiv e s o f ^
^
ru n s th ro u g h a s e t o f
a r e n o n - is o m o r p h ic a n d t h e y r e p r e s e n t a l l
s p a c e s u p t o is o m o r p h is m .
C ase 2 . 2 :
In th is
c le m
E/
e
^ ■"* i
> cLom Eix~ —
c a s e v e s im p ly h a v e B - H * Q
sp an s
f
-
E ^
, th e n
£Yo) 0 k. U) »
a n d o n ly i f
If
Jh :
-
If
j k ; te J >
E
is
Two s p a c e s
dLcm
( r a c t £ %) ' — O
i s t h e r e f o r e s e m i-s im p le
a n d p o s s e s s e s a s y m p le tic b a s i s , th u s
.£
;
^
=r E<0) .
If
^
= U jI U
c o m p l e te l y d e te r m i n e d b y t h e o r t h o g o n a l sum
/=
and
S r i n t h i s c a s e a r e is o m o r p h ic i f
and
E ^
I
t h e r e i s o n l y one t y p e , n a m e ly E f 0)' S
a r e is o m o rp h ic .
th e sp aces
re p re s e n ta tiv e s o f
v h e re
T hus:
v h e re
.
ru n s th ro u g h a s e t o f
<0 /, a r e n o n - is o m o r p h i c a n d t h e y r e p r e s e n t a i I
s p a c e s u p t o is o m o r p h is m .
. .
'X
C ase 2 . 5 »
cillm
= X
Upon d e c o m p o s i ti o n v e
= k(4)
- X C*o) e
? and w
V
have E =
J- E v
su ch t h a t
^
XJ
.
cLcyn Ex
Vz i s
O su ch t h a t
cL = D £ H
.
E.
I/ =
cLi-'m (Vat#
) — jL
•
fc = kCu>)
,
s e m i-s im p le , f o r i f n o t , th e r e e x i s t s
V , b u t t h e n -y -I
s y m p le tic b a s i s , t h e r e f o r e
,
V & B : &L. v h e r e b y Lemma 4 . 1
T h is c o n t r a d i c t s t h e a s s u m p t io n t h a t
fin d
=■ ^
k @> L v h i c h i m p l i e s -X J- E .
E- i s s e m i - s i m p l e .
(Z
has a
E( 0 ) . . u 0v C oJ> J ) E O , h o v e v e r v e c a n '
( &> JL-+ \ <J>) ~
O , th u s
n o v h a s t h e fo rm E -
l< €B L
=
A
v h e re
£ f0) <3> A ( ^ j - )
A g a in i n t h i s c a s e t h e r a n g e o f t h e norm fo rm i s a s i n g l e c l a s s i n
.
We s h o v t h a t a n y t v o s p a c e s
£T a n d
E
in th is
c a s e a r e is o m o r ­
p h i c i f a n d o n l y i f t h e r a n g e s o f t h e i r r e s p e c t i v e norm fo rm s c o i n c i d e .
I f t h e r a n g e s o f t h e norm fo rm s a g r e e v e s im p l y map b a s i s v e c t o r s .
and
B
a r e i s o m o r p h ic v i t h d e c o m p o s i ti o n s
E M &&(£>£>)> t h e n c l e a r l y
ot
/3
.
E -E c^
o ji
)
and
B =
T hus:
I f JJh ° h 2J = X t h e r e i s o n l y one t y p e , n a m e ly
I f [k « /=? J > X t h e s p a c e s
® A
IfE
& A ^-=U) v h e r e
E f 0) Q A (j. i.)
•
ru n s th ro u g h a s e t
;
- 13 \
o f r e p r e s e n t a t i v e s o f <0 k a r e n o n - is o m o r p h i c a n d t h e y r e p r e s e n t a l l
s p a c e s up. t o is o m o r p h is m .
-Ca s e 3 « 1 -
cl<-m
D e c o m p o sin g /=
^
j
of
/c
, g .)
or C & o ) e /<
we h a v e
h
.
In th is
such t h a t
- k (jXj Cj) .
c o n t r a d i c t s cL^-m E #. =
such th a t
= O
>
(o) .
^
If
C 1Xj y j
c o n tra d ic tin g
"
.
case
1
X-L-V
If
C r , o r ) =r <3 a n d c h o o s e
( c ' t ^ , 1X^ -X E v
an y o rth o g o n a l b a s is o f
1
th e re e x is ts
L
, th e n
O } le t
cXX-m
^ rx*1" =
oo/e
iXo e V w i t h
A ( w /) in
( c o / J tX0) 4= O
w h ic h ( Co, CO, ) ~
O
i.e ./L
.
.
If
.
(
m u ltip ly in g
CoL b y s u i t a b l e s c a l a r s
LLh j LO3. ) —
and
( Jl xj LO, )
—
-X
Zo
and
.
v ith
i I !'I
'
O
S i n c e code'™ h a l f
.
L e t ' ''/? ( co) b e a n
th e n s e le c t
Xu —
.- f ■A 1X0 f o r
ajJ.
L c o L l Jla. ) = O •
Xi
C le a n ly
L e t JL,' a n d Jl3L b e
u3 ) = 6 j th e n I e t x 05 -
LO L — u)L -b
•
'
A we fo rm
s e l e c t yiA. s u c h t h a t
LO!
) .=- q
I f ' (CU,', Cu) ^ O ,
F or s u ita b le
b a s is
b e a v e c t o r i n i_
I
= h i J > ! j JiL )
K
p o r su p p o se
w h ic h
.
.
e
O •
Vi t h
to a
.
Ar s u c h t h a t ^ oj L j JiL ) =
a l g e b r a i c c o m p lem en t o f
&
Co)
.
x
'X -L
f o r w h ic h f Z > <3x ,
,
j
V <b H ® L-
C o m p le tin g
^ l
A
( J ^ clcA J?*. ) ~
V z ? /v ~ (o) .
.
We now p r o c e e d t o g i v e a n o r m a l fo rm f o r
=
cLi yn
a n d a p p l y i n g Lemma 4 . 1 we o b t a i n . E -
K = hLu>jU>*) a n d L =
th e re e x is ts
(tcm
and
S
} if not
How ^ -----
we o b t a i n /
— <5^ Ce)/ a n d c o L '— V* t o /
.
D ro p p in g t h e p r im e s a n d s u m m a riz in g we h a v e :
E - V ® k
Q H Ce*, - ^ ) , l l ( o o i . = l i c o ± u = o , i i J l o i = 4 L * ~ o T ~ ' ~ ^ o ' c ~
D l O l = P (.*0) } <k x x p J Cut ,J LO3.) =
= X u l, I
,
Ov CUitJld)= C(Di1At) ^
I n t h i s c a s e we h a v e t h a t a n y tw o s p a c e s a r e i s o m o r p h i c .
we p r o c e e d a s f o l l o w s :
L et
n o r m a l fo rm ( 3 ) J i . e . ^
£"
0 - v js> JL & L > h
2r
and
H e x t we show t h a t
B = X
(sq r) .
a 7, a
su ch t h a t
• ~ 0 , Jh -t B , Jlz.
chosen so t h a t
Cf1V
Lr
h a s t h e d e c o m p o s itio n s
k CL , T i )
3 II F, Il -
S C * o)
,
a n d t h e b a s i s e le m e n t s s a t i s f y i n g t h e a n a lo g u e s o f
c o n d itio n s (3 ) •
Q
b e a n y tw o s p a c e s o f t h i s c a s e i n
i s a s in (3 ) an d
= / Wi v, , L u , ) , L =
HLI I = r ( * 0 ) 3 S--XxXf
& h
E
To s e e t h i s
Lr
S in c e / k J
-+
/b
and
C J l / , JlL )
=
=
j=
c a n b e w r i t t e n i n n o r m a l fo rm a s
~
S
; th e re e x is ts
and
^
-t
CL JL > ba, ^ 2. +/< w
O
.
..
. V
O-, , b 1 j Cl3.
b f /3 =
> w h e re
^
. L e t
is
xV
— ll). —
C a l l L.'~ k ($.! ,SlL )
• and
V 1=
L uZ l E i f f t h e n
/Er.=-. I /
'<3> L . '
t h e p r e c e d i n g , . we m ay p u t t h i s i n n o r m a l fo rm w i t h o u t c h a n g in g
I n o r d e r t o e s t a b l i s h t h e iso m o rp h ism , we n e e d t h a t
and
V ueii =
(o').
s e m i-s im p le i s
s in c e
s im p le .
K =
s e m i-s im p le a n d
se en by n o tin g t h a t
VacL E if. -
th e re e x is ts
( H*
Co) we. h a v e ■ W
=
rad
mE+
c
I / .' .
V
Xo & IZ s u c h t h a t yTo L. iZ a n d X o J - f <
F u r t h e r s u p p o s e rX 0
62. Tt-
Xe w i t h
oo,
( T i j CO1) -
V &■ k (ooZ) w h ic h i m p l i e s
E* is
case.
^ v =£
s in c e
, th e n f o r a s u ita b le
O .
H t' a n d £* .
T hat
in t h is
S u p p o se
Iy
i s s e m i- s im p le
(o') ) .
E^ — (o)
.
A ls o
O
} th e n '
£ ■» i s s e m i­
^
we w i l l h a v e
L e t U = k U t i h ) , t h e n U u Zl E if=
cLom k. E ±
,
T h is i s a c o n t r a d i c t i o n f o r t h i s
case./
S in c e
5%. a n d
c o n c lu d e
E if. a r e s e m i- s im p le a n d h a v e s y m p l e t i c b a s e s , we
= E c o a n d EV =
E f 6)
w h ic h i m p l i e s
t h e r e e x i s t s a n is o m o rp h is m T*. :
a n d TC( Oa ) ~
.
— >El
.
Ey ( 2 . 4 ) .
s u c h t h a t T C v ) — V f TC(Zi)=Co,
Xn v ie w o f t h e n o r m a l fo rm ( 3 ) we c a n e x t e n d T V t o
a n is o m o rp h is m T S E —V e
b y T / ^ 5t =-
T UU) —
E Er W
.
IEif ^
T h e refo re
and
T ( £ f ) = jT,
, ,
\
One c a n r e a d i l y v e r i f y t h a t t h e s p a c e E -
® E (p )
, Z ^ /3
o f t h i s c a se a n d th u s can b e c o n s id e re d a s a r e p r e s e n t a t i v e .
C ase 3 . 2 :
dc-im E / Bif ~
D e c o m p o s itio n o f
k ( £>} U ) .
I /
•
> dijn
E =
E^
=' J-
V 0 H & L
V/
a d m i ts a n o r t h o g o n a l b a s i s ,
„
M o re o v e r , m u l t i p l i c a t i o n o f t u
w ith
oo‘=
.
z
> dt'** C r o d E ■* ) =
w ith
L
y i e l d C c o 'j Ji2.) = X
p rim e .
H
y ie ld s
H =
k ( 00)
Dy a p p l i c a t i o n o f Lemma 4 . 1 we h a v e . d , J-
S in c e
d ic u la r to
£
E *
O •
and L =
fo r s u ita b le
I 1 can be c h o sen p e rp e n ­
by a s u ita b le s c a la r w i l l
I n t h e f o l l o w i n g we w i l l d e l e t e t h e
'
In th is
c a s e we w i l l show t h a t tw o s p a c e s E =
V ® k (£ L a r e i s o m o r p h ic i f a n d o n l y i f ' i . ^
L et T
b e a n is o m o r p h is m
T
j e
------ > E
V ® E <& L. a n d E
E v - ) E = V S k C u o ) &■ kCST, j £ l ) f a n d
~
ZT
o f tw o s p a c e s i n t h i s c a s e
w i t h d e c o m p o s itio n s ( n o r m a l i z e d a s a b o v e ) E = V S k (u>) (B k ( I / f d a )
iit I
is
X , 4- B if
.
S in c e
,
- 15 T C G ■><.) — S.
Er/ " -
( 4 >)
ve have T ( B ym )
and '
- klT,)' .
— E/
} th u s T
^1/ "because
I t fo llo w s t h a t Il i , U
J-JL
A lso T C
; ~
E.x- 0 kCV<) •
A gain th e ra n g e o f th e norm form m ust he e q u a l on
__£
E Z j - and we f in d
- U T* H .
r e s p e c t i v e l y hence J)
jl
pr i._ i
Er* © k ( 0,) and
/I Al /I
S in c e
A =
a r e o r t h o g o n a l b a s e s , we h a v e t h e r e f o r e
E xjl. J-'^ and
27
A z , and
A.
-
ZT .
B e f o r e p r o c e e d i n g w i t h t h e s e c o n d h a l f o f t h e p r o o f , l e t u s show t h a t
f o r any sp ace
£
w h ic h f a l l s
in our c a se ,
oZTin C r o d S x ) - .O t h u s
m e n t - /r f ^>) o f
I/
hence. V
^ d
su ch t h a t
rX o J -
Ir?
I/
C C.j
S u p p o se
I /. .
I / ^
s in c e
CrXo }
) we c a n f i n d
C u ) — <9 .
cZcw s T
i s s e m i-s im p le .
C le a rly
Co).
Now, n o a l g e b r a i c c o m p le ­
£ ■* c a n b e p e r p e n d i c u l a r t o
.
w o u ld im p ly Xo-L S
£ > i s s e m i-s im p le a n d
I/
( o t h e r w i s e o> e i-W £.*)
( 0 ) , th e n th e r e e x is t s
uj)
=£ O ( o t h e r w i s e
d * O such t h a t
Ar2 - I Zr*
~ AL w h ic h i s a c o n t r a d i c t i o n t o
// .
rXoJ- S x. a n d Yo ± L
Tz
- J T - J - A rX 0
J.
a n d s in c e
Yo £
cCT-m s t
and
we w o u ld h a v e
~
J- •
Khus
.
I I I1I
V j "5 *- J= C o) .
L et
f
and
E
b e tw o s p a c e s o f t h i s t y p e w i t h n o r m a l iz e d decom ­
p o s i t i o n s : B = I/
and
i/
£ •*
Coo) B k ( i n JU.-) , E = V & M Leo) & k
,
JLk J. B ^
(T i K
Tl
)
> [ i n t o ) — ( T i j Co ) - J . ) C l i j l i ) = (£< j £,.)
q
IlAlH= I t r i Ii j IMz.il '= DTz / / .
' £ v and
Ex a r e s e m i- s im p le a n d b o t h h a v e s y m p l e t i c b a s e s t h u s
E-x- - E Z o a n d E y ^
iso m o rp h is m .T V »
e v-
( 4 ) we c a n e x t e n d
T ( JL) =
T
S Co) w hence
— ^ £*
E
S x
. ,
w i t h T» ( v ) ~
V
TV t o a n is o m o rp h is m
and. T ' ( Jlz )
=
ZT
( 2 . 4 ) t h e r e e x i s t s - a n ......
a n d TV Cco) — <53
T : E — > W
by T I
.
is
E = Eu) ®
E (p)
A ls o we c a n g e n e r a t e a l l n o n - is o m o r p h ic
ty p e s be l e t t i n g
th ro u g h a s e t o f r e p r e s e n t a ti v e s o f
su ch t h a t
cU -m
=
J- >
Qb
cLc-m. s T
—
± ■t
Ey ( 4 .1 ) we have g = v & h ® L w ith E = k loo, j oj
U i t U -yJ* H A z / /
cCAm
•
^ =
TV
,
T
oL
-t* / i
,^
and
CU/ e /c
/3 ru n
\
.
dc'™ ( - r a d Ex ) = I •"
3.n& L = hz ( i , > A= )
I n t h i s case we have oCAvi C i/y ? & ) — d-
( v a d S x) ~ ± th e r e e x i s t s
if*om
\
An e x a m p le o f a . s p a c e o f t h i s t y p e
Case 3 .5 :
b
.
su ch t h a t C u / JL E*. .
.
s in c e
Let C u/
,
- 16 a n d (A)3. b e a b a s i s o f
if
M OJt
/c
th e n
Assume C t o ^ i to', )
For a s u ita b le
CoZ J-
we h a v e
E
JI3 & L
yZ Z
.
.
f £ a j t o a. ) — O
L e t, E ' . a n d
E
=
E'-xr ,
E - E
T
i f and o n ly i f
T ( r o d £ * ) =: r a d
e q u a l , h e n c e H £,11 ^
//
E
How
T hus
O 3
C i a i CO, } = /. .
U A, // ^
// ZT n
.
s ’* , a n d T f
-
-/?
, th e n
( r a c J E x. ) J~0.
^ ^ Ok(Jh)
and
The r a n g e s o f t h e n o im fo rm m u s t b e
a n d ']= a r e a s a b o v e w i t h JJ £,11 =- H ? , /y
I/ JL 11
, h o w e v e r we c a n r e w r i t e
k ( c o l , c o / ) <& fc CA/ , I / ) ,
= I l T //.
.
// ;
S e c o n d ly , s u p p o s e
S in c e Il A 1 // ^
CuJl i LAJa) =■ ( Jh > £ * ) =
i s a n is o m o rp h is m s u c h t h a t T s E ----- 5» T
C T a d E- *) ^ =J k C uJl ) M=r E *■ (& fe. ( IT).
V '
( J l s j Co/ j — O
a n d E = V O k (U)l i CD2 ) O k C £ i , J l )
From o u r n o r m a l fo rm we r e a d o f f ( T a o f f *
g e n e r a l U JL !I'd*
j_ ,
b e tw o s p a c e s o f t h i s t y p e i n n o r m a l f o r m , i . e . ,
F i r s t 's u p p o s e
T O * -)
cii'hr, ( i ra d E» ) =
M o re o v e r a s i n c a s e 5 . 1 we c a n a ssu m e
a n d I. Jh J COs. ) =r
E = V O k ( Ml i Ujs) Q k U li £ a )
We w i l l show t h a t
,
H ence b y d r o p p in g t h e p r im e s
t h a t t h e d e c o m p o s i ti o n i s n o r m a l i z e d , i . e . ,
( £i j CO') ~
IZ
be such t h a t
LoJ.
= £3 + <\u>l s u c h t h a t ( T Fs j C O / ) — O .
cCC-m E J l - J
and
S u p p o se
w h ic h c o n t r a d i c t s
Za
Za -I £■«. w h ic h c o n t r a d i c t s
we h a v e c u , . j .
hz ~ k CcoZi c o i ) •
th e n l e t
q
A
j i.e.,
CF
H Jls. 11 t h e r e e x i s t s
is
a
Q .
L et
such th a t
E
pn
—
i n n o r m a l fo rm a n d Il J l / // = - , / / Z l / / .
and
^
s u c h t h a t a / JI Jh U ■+Io IUhlJ
( OA< I JL1 ) J: O s o we c a n f i n d a
IoAa-I-A LOs a n d C£2 , Ji, ) =
£"
.
L
A
f o r w h ic h Z a ■= Cl J!/ +
~ M ( A i i £ / ) a n d decom pose IE -
.
V ; (B k ' & L ' a n d t h e n n o r m a l i z e .
From t h e a b o v e we may a ssu m e t h a t i n
a n d Il Jh.ll =
H JFz H '
E
Z
Zr
=
(0 ) .
* 6 LT a d E ) n
Z
-L E •
Z
T a d P
S e c o n d ly s u p p o s e X e
n/ e V
=J^ O
.
(if
K=J O
, th e n
V &> k Ccjs ) i s
C
k(tOs)
/=
,
, but
/
How C fX- + A ( Oai T ) CrV + A co 2. j cOx. )
=k O
q
1
/ =
ot J. 00
T h is i s a c o n t r a d i c t i o n s in c e
may a ssu m e
, // £, U -
UT
//
\
F i r s t we c la i m (VacC /=) /?
and
Zf
.
I n o r d e r t o c o n t i n u e we n e e d t h a t F =
and
and
E
is
C O z -J /
( 0) .
, oc J-
F o r supp o se
L.
s e m i-s im p le .
Z,
w h ic h i m p l i e s
T h e r e f o r e we
w h ic h I m p l i e s T a d F =Qo) .
, t h e n oc = v
im p lie s
s e m i- s im p le
+ A 00 ^
v h e re
A C c o i ,- iz ) -•=• Q a n d ( v p a
w e re O t h e n -iJ + At ox
a
ir a d P - ^ im p lie s
- 17 C 'V j W a l
£>
L e t %o C
e
, th u s
A - = O
I x - zZ= , t h e n s i n c e
f o u n d s u c h t h a t J l / =* I ,
£ = £ *< &
lJ
&Vo
T h e refo re
5.
/=
and
/=
T1 Z P - J
and T (v) - V ,
m
=■• ^
— ^ ^
h
S in c e A / j .
O
e »
can he
.
How c o n s i d e r
and
CO, J- ^ x-
we
/Er
.
( o j , b y ( 2 . 4 ) t h e r e e x i s t s a n is o m o rp h is m
by
.=
Tip.
= T lj
UJ3.
-
.
We e x t e n d
T,
t o an
f a , , ) = ZD},' T f j2, )
=
,
—
e x a m p le o f t h i s t y p e s p a c e i s
T y le ttin g
cA z>f / 3
U)* ) =
A
h a v e sa m p le t i c b a s e s , i n o t h e r w o rd s F ^
Iz ^
is o m o rp h is m T <
C
and
a , a s u ita b le
I /.
w h ic h i s a c o n t r a d i c t i o n .
M o re o v e r s i n c e
f= '
T h e r e f o r e i t f o l l o w s t h a t V zZr c
( OC0 , oo* )
w h e re L z= k ( € / , ^ 3 ) .
h a v e d Cnnn S * 2
and T ( A )
.
E =
,
(A ,ru n t h r o u g h a s e t o f r e p r e s e n t a t i v e s o f
o k - r /3
.
<&(. s u c h t h a t
we g e n e r a t e p r e c i s e l y a l l is o m o r p h is m t y p e s .
^se_3A:
Ly‘(4 .1)
cA bn
f
= 5. ,
eLc-m e / '
h a s t h e d e c o m p o s itio n
—
3-
,
E ,=• \ / <& L.
f r a cf
w ith
K=
E*
; =
O
•
and
*■ ■
'L = k ( £<) £*■) ’
yL
Ilia.//
i s n o n - is o tr o p ic , hence L =
and
can- b e w r i t t e n
cA -Ar / 3
B - E (d) i
.
F u r th e r m o r e s i n c e
i s o m o r p h ic i f a n d o n l y i f
S u p p o se T
T 6 frv) = ^
A
C o n v e rs e ly , l e t
T ( E 6) =
E 0
s e m i-s im p le f
J-
—
A (V yS )
) and B =
“ A
i s a n is o m o rp h is m s u c h t h a t T C E — > T "
£"(o) = A f ^ t h u s
T,
is
E - B (o) & A
w h ic h i m p l i e s T ( B d ) -
can e x te n d
y
, IU,!!=* ( J o ) ,
E c^p, ). ■
I n t h i s c a s e we f i n d t h a t tw o s p a c e s
'Sco) © T ^ a r e
c*#)
V /3 )
T/
A
CJ-TD
T
But A w
-
A ^ yO ) a n d . '
.
b e a n is o m o rp h is m T, S A zz/ ) ) -----B ( S r ) *
t o a n is o m o rp h is m
.
.
, th e n
I gra
T «r / E ----- > E ■ b y T / A( J ^
We
■-=
and
i s d e f i n e d b y m a p p in g b a s i s v e c t o r s i n t o
b a s is v e c to r s , .
R e p r e s e n t a t i v e s o f a l l n o n - is o m o r p h ic s p a c e s o f t h i s t y p e rem b e
o b ta in e d by l e t t i n g
a t i v e s o f g/%
and /3
( c J -^ jC , ) r u n t h r o u g h a s e t o f r e p r e s e n t ­
.
d - z'7" % * . =
a
,
e /
=r a .
,
C r a d f a r * ) -= ±
'
- 18 -
By (4 .1 )
L=
has th e decom position B -
B
k( . G>)
f
cU w ( v' ccI B
HMxIJ -
ol -A / 3
>
klco) e A? CMijM x )
Let J7l3 = JLx -4T hus
H Jl3. 1/
— -i i t fo llo w s th a t
k)
/3,
I I S 1II
,
, and
£,
Lo -L
E m
IJ io Il —
i s sem i-sim p le
w
is
g fc)
4
/3
o f t h e fo rm
L et £
4 A
We have H J2, // — cZ.
and
X a -L
A f/3/3) ®
J l z,
.
.
£
S in ce
-= O .
and
A
y E m E (<,)
CO')
\/
,
.
is'sem i-
t h u s h a s t h e fo rm
a n d %T h e tw o s p a c e s o f t h i s t y p e w i t h d e c o m p o s itio n s
II jU I - / ! A U
£ ( ,/ ® A / s ; )
we , f i n d t h a t
© A £*,)
E = E
Il Jls Il =
D Z n ■= H A
a n d ' H Ta H — r
i f a n d o n ly i f
A /^ )
One h a l f o f t h e a s s e r t i o n i s t r i v i a l .
a n is o m o rp h is m s u c h t h a t
E- X= £ to) ® A
T :
E
— > £? .
L e t u s a ssu m e
•'
th a t
/X (ss-) ® A i f i i f
S in c e
I/?
and
cl 1
A=1 / 3
Z3
•
and £ =
In t h i s ' case
e AA) = A
^
A /»
L e t u s a ssu m e t h e n t h a t "T
S in c e Zr* = A o )
- +T) and T (
B k 1 1 , + L ')O h ( M T )
is
•
Since
.
£ = £(.)
^ u.
.
E „
, and ( I , , -I2J -=* O .
O
s im p le a n d h a s a s y m p l e t i c b a s i s , e r g o
£=
J.
such th a t /< - k (cc) ,
( M x j t o ) =M O "because (
th en Il Hs /I
© k ( M. j M a )
V @ H: ® Z-
is
Q k (M, + <Lx)f
/
i t f o l l o w s t h a t k ( Z + h)&k(ij)
As b e f o r e t h i s i m p l i e s
]) Z 3 Il ■=
k x3
H-Jl3 Il
T hus weo n l y n e e d
~ 3- t h e r e e x i s t s
O s in c e
to
show t h a t A ^ ) < 9 /^ =
(\,, ^ ^ such t h a t ^ -
.
^
H A H j z.e,, yS
The d e s i r e d is o m o rp h is m
o b ta in e d b y th e fo llo w in g tr a n s f o r m a tio n :
A = ^
M1
4-
(Al-+ ^TSi.) A
T3 -
-A fU ls
& ±£ ( D1 4- j?,)
4
la .
=
d 1 J?a 4-
/
•
AxQa j
Ai oZ.
'
' We r e m a rk t h a t b y t h e a b o v e is o m o rp h is m we c a n c o n c lu d e t h a t t h e tw o
spaces
Zr
and
z=" a r e is o m o r p h ic i f a n d o n l y i f
A.(p)
=
.
Case 3 . 6 : (Jo yo Zr/^.^ = Q ,) cLL™ E J i" = 9 - 5 c(t-yv. ( r o d E x ) =■ 3 . °
A pplying ( 4 .2 ) t o E we ob ta in Zr = V © Ar. # Z_ w ith J c = k (uJ) j W x )
j.
Vz > a n d
L=
n o rm a liz e d , i . e . ,
I Mx , u>I) = !.♦
fo ( M. j Mx ) .
(JL; , &u, ) =
As i n c a s e . 3 • I we may a ssu m e
( Mx j U>*-)
K © Z-
is
= ( 5 / , - (W,,w,)=, 0 a n d ( j t i j u i x ) =
A ls o b y a p r e v i o u s c o n s t r u c t i o n we c a n w r i t e
k ( uj>,JL) = A ^ o )
- 19 and A
S in c e
(M t, £,)
I/
A (pp-)
.=
w h e re /I
til
/3
-=
Il J* Il = A
and
i s s e m i-s im p le a n d h a s a s y m p le tic b a s i s ,
s p a c e o f t h i s t y p e i s o f t h e fo rm ' E = B m <9 A
In th is
,
V ■=
>.
w A f z6y0)
m
oL s x / 3
,
A f ^ l ) & A ((!,&) •
©4 -?<• / 3
oAr + d
=
Ad/h £ F
•
©
L et
A css.) ~ k ( e‘j f J
/3
(n o te t h a t
F
)•
and
A
- A C ^3, e^) , t h e n
i s f i n i t e an d s e m i-s im p le ,
A f^ ) - F .
A (v.y3 j
A O s)
.
B ecause
A
is
<S
A f^
Of
and
r 4
3
^ 2-S
•
V
; c a l l T^i, = V Cj> -r y $ 2.
p o i n t we h a v e A A / y & A
d e t e r m i n a n t s a g a i n we o b t a i n
-L
A css) e
T hus A ' ^ / v )
=
A css)
T
A
•
is
s e m i- s im p le
=
Afrw)
<& A c * < r )
th e ir
T h is i m p l i e s A
and
=
s u c h t h a t <A =
w h e re H <g> Il =
A f s s ) <i A f ^
^
)
( h y p e rb o lic p l a n e ) ,
A
A g a in t h e r e e x i s t s
A f i-s ) — h ( y >, ^j z).
■A a n d '
s e m i- s im p le a n d o f d im e n s io n 2
d e te rm in a n ts m u st be e q u a l u p t o a s q u a re f a c t o r .
^
=
We s e e t h a t h ( o J , -+b J 3 j C A * ClMi,)=
t h e r e f o r e i t m u s t b e i s o m o r p h ic t o one o f
Afv-s)'
.
i s i s o m o r p h ic t o
t h e r e e x i s t s f i e l d e l e m e n t s a , b , c a n d d s u c h t h a t a?<r -h IcA S
or
T hus e v e r y
c a s e e v e r y s p a c e i s is o m o r p h ic t o E 1 ~ E m & A c ^ <&A (&>)'■ I t i s
s u f f i c i e n t t o show t h a t a n y s p a c e F =
th u s A
.
v*
A l Au A ) ,
,Hyi-Il=S',
^ d *
At th is
andxby a p p e a lin g t o
T hus A f o i - O
& A ( p/ s . ;
A C^
\ ■
\
,
-
20
-
APPEHDn
Iii/
\
V.
- 21 —
TABLE I
MAJOR RON-ISOMORPHIC SPACES
, .
CxLnm
Cass
*
dCyrt
e
I
O
0
A. I
i
0
a ,a
i
A .3
dLi'inft
/
trype s
r#c< 'e *
oS
0
Z ( 6)
0
£(v)
I
O
Bu) &
i
I.
I
E Co) © A ' f . w )
3.1
A
0
O
E u )
&
3. 2
3.
I
0
E
&
3,3
3.
I
/
3 . U-
A '
a
0
@ A
B(o) ® A w ) © A(Tj)
'
.
3.6
■ a
a
I
3.6
R
a
A
m
S p a c e s
&(v)
’
B f f i -)
i /
^
x /
^ ) © A riO/*J ; ^ "A /J @
B (
q)
© A U j .)
-V -^ x
a.
' U - M < yu
j
£ ( t ) OoA( Afi) G A M
t V
m /s
G> 2 s ( & M
A l l t h e s m s a r e o r t h o g o n a l , { <*, /2 $ i s some f i x e d ^ h a s i s o f M o v e r fc* ;
n / a n d >x r u n i n d e p e n d e n t l y t h r o u g h a f i x e d s e t o f r e p r e s e n t a t i v e s o f
(Jk ) s u b j e c t o n l y t o c o n d i t i o n s l i s t e d i n t h e t a b l e . A l l t h e s p a c e s
t h u s o b t a i n e d a r e m u tu a lly " n o n - is o m o r p h i c a n d , i n e a c h c a s e , t h e y / a r e
u p t o o r t h o g o n a l is o m o rp h is m s a l l
k - s p a c e s Cs j d>) w i t h
£ = cK
a n d t h o u k =<2 *
- 22 LITERATURE CITED
G r o s s ,, H . , E i a W i t t s c h e r S a t z i n E a l l e T on V e k to rra u m e n a L z a h T b a r e r
D im e n s io n j J . R ie n e Hf. A ngew , M a th .
(To a p p e a r s h o r t l y ) .
G r o s s , H ., a n d F i s c h e r , H. R . , Q u a d r a t i c Form s a n d L i n e a r T o p o lo g ie s I I ,
M ath* A t i m I e n .
( to ap p ear s h o rtly ) *
“
K a p la n s k y , I . , Form s on I n f i n i t e D im e n s io n a l S p a c e s , A n a i s , A c a d . B r a z i l .
C i e n c i a s XXXE ( 1 9 5 0 ) , 1 - 1 ? .
:
O 'M e a ra , 0 . T . , I n t r o d u c t i o n t o Q u a d r a t i c F o rm s, A cadem ic P r e s s , Rew Y o rk .
(1 9 6 5 ).
Z
~
:--------------
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