Linear topologies induced by bilinear forms
by Vinnie Hicks Miller
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 Vinnie Hicks Miller (1966)
Abstract:
The purpose of this paper is to identify some linear topologies associated in a natural way with a
continuous, bilinear form 0 on a vector space E over an arbitrary discrete field k. The finest topology
on E for which the canonical maps (Formula not captured by OCR) is continuous is denoted
by(Formula not captured by OCR) denotes the extension of these topologies by sums to the tensor
algebra and by quotients to the Clifford algebra.
For V a fixed totally isotropic subspace of E, (Formula not captured by OCR)the topology is defined by
taking a neighborhood basis at zero of subspaces (Formula not captured by OCR) a finite dimensional
subspace of E. It is shown that if(Formula not captured by OCR) is a linearly topologized space
with,nontrivial topology T then the following are equivalent: (Formula not captured by OCR) for some
totally isotropic V, (ii) (Formula not captured by OCR) is continuous, (iii) T has a zero neighborhood
basis of sets (Formula not captured by OCR) with (Formula not captured by OCR)(iv)(Formula not
captured by OCR) and (Formula not captured by OCR)is Hausdorff. If the case where dim E = and V is
orthogonally closed, it is proved that(Formula not captured by OCR) is a topological algebra.
An investigation of the completion(Formula not captured by OCR) of a space E with(Formula not
captured by OCR) topology and bilinear form (Formula not captured by OCR) , shows that E can be
decomposed as follows: (Formula not captured by OCR), where (Formula not captured by OCR) is the
algebraic dual of (Formula not captured by OCR) and H2 are totally isotropic for (Formula not
captured by OCR) (Formula not captured by OCR) for (Formula not captured by OCR) and(Formula
not captured by OCR) is nondegenerate iff V is closed (Formula not captured by OCR) These
completions coincide with the locally linearly(Formula not captured by OCR)compact spaces on which
the form (Formula not captured by OCR) is continuous. LINEAR TOPOLOGIES INDUCED BY BILINEAR FORMS
by
VINNIE HICKS MILLER
A thesis submitted to the Graduate Faculty in partial
fulfillment of the requirements for the degree
of
DOCTOR OF PHILOSOPHY
in
Mathematics
Approved:
ead, Major Department
MONTANA STATE UNIVERSITY
Bozeman, Montana
June, 1966
(ill)
ACKNOWLEDGMENT
■ Professor Herbert Gross has contributed to every phase of this
research project from suggesting lines of investigation to reviewing
the manuscript in detail.
He has taken every opportunity to help me
develop research skills and has provided perspective on the problem's
significance.
unfailing.
His enthusiasm has been infectious and his consideration
I wish to take this opportunity to express my sincere
appreciation of his guidance.
LIST OF FIGURES
FIGURE
I.
PAGE
Relations Among a Vector Space, Its Tensor and
Clifford Algebras, and Various Completions.
44
TABLE OF CONTENTS
CHAPTER
VITA
II.
III.
IV.
ii
ACKNOWLEDGMENT '
iii
LIST OF FIGURES,
v
ABSTRACT
I.
PAGE
vi
INTRODUCTION
I
DEFINITIONS AND GENERALREMARKS
3
CANONICAL TOPOLOGIES ON TBECLIFFORD ALGEBRA
10
THE COMPLETION OF (E,7^1/)
33
LITERATURE CITED
50
(vi)
ABSTRACT
The purpose of this paper is to identify some linear topologies
associated in a natural way with a continuous, bilinear form 0 on a
vector space E over an arbitrary discrete field k. The finest topology
on E for which the canonical maps ty lfe — ^/T
is continuous is
denoted by S i
; ®Z
denotes the extension of these topologies by
sums to the tensor algebra and by quotients to the Clifford algebra.
For V a fixed totally isotropic subspace of E , the
topology is
defined by taking a neighborhood basis at zero of subspaces VD Fx9
F
a finite dimensional subspace of E. It is shown that if (E,T,^ ) is a
linearly topologized space with,nontrivial topology T
then the follow­
ing are equivalent:
(i) "O T^V
for some totally isotropic V, (ii)
(j) :E x E — > k is continuous, (iii) T has a zero neighborhood basis of sets
4, with E- IJ Ujm
(iv) QTlg -T
and (v) (C.(E),0T) is
Hausdorff. If the case where dim E =
and V is orthogonally closed,
it is proved that (C(E) , @1^1/) is a topological algebra.
An investigation of the completion (E,-?, $ ) of a space E with Z^V
topology and bilinear form 4> $ shows that E can be decomposed as
follows: E - H'£ ® H3i Q H,
, where V-H^
is the algebraic dual
of Hz ,
H f and H2 are totally isotropic for 0,
!Ji(JllJ0Jv1) JvJ (Jb3)
for J l2 G H£
and -ZvzG Hz 0 H1-L ( HJ + H3.) ‘ $
is nondegenerate iff V is closed ' iC -Z fiV =Vp H j *
These completions
coincide with the locally linearly
-compact spaces on which the
form 0 is continuous.
CHAPTER I
INTRODUCTION
The present paper is part of a larger program concerned with an
algebraic theory of quadratic forms on infinite dimensional vector spaces
over arbitrary fields, IE51, K63 and
gonal group of such forms.
K71, and in particular of the ortho­
For the finite dimensional case a substantial
literature exists, (see Dieudonnd:
La gdomdtrie des groupes classiques,
1141, which also contains a comprehensive bibliography).
The techniques
used there do not extend however to the infinite dimensional case.
In
the hope that topological techniques may replace the finite methods, a
natural topology, compatible in a certain sense with the quadratic form,
is introduced.
More specifically, by means of canonical topologies
TT11
certain subgroups of the full orthogonal group would be singled out for
investigation, namely the subgroups of T -continuous orthogonal auto­
morphisms.
These topologies may also be of service in the classification
of spaces into classes unique up to orthogonal isomorphism.
In the finite dimensional case, consideration of the Clifford
algebra has been very fruitful.
For both finite and infinite dimensional
spaces there is an intimate relationship between the orthogonal group
and the group of Clifford algebra isomorphisms.
So it seems reasonable
to require that the topology on the vector space extend first to a
suitable topology on the tensor product, and then, by the usual sum
and quotient operations, to a Hausdorff topology on the Clifford algebra;
finally, the Clifford algebra topology must induce the initial topology
,
1
on the underlying vector space.
This leads to consideration of the
topologies, and their completions which are of the same type.
~C^V
CHAPTER II
DEFINITIONS ^xND GENERAL REMARKS
In this chapter the definitions and notational conventions to be
used in the remainder of the work are collected.
Known results which
will be needed later are stated here without proof.
When no reference
is cited for the proof the reader should consult the excellent summary
in Ktithe E9I for further details,.
.>
'
Sections are numbered for later reference.
I.
The discussion in Chapters III and IV will be concerned with an
infinite dimensional vector space E over a (commutative) field k with
characteristic unequal to 2.
V=
//2£
] / a/subspace of E the notation
is intended to convey that
indexed by o( in I.
expression becomes
^ K / </>2,
V^ ^
For
V has
as a basis the
/V^
When the index set is denumerably infinite this
V~
and for I/ of finite dimension
an^ similar expressions will be employed.
If the
fl£ are generators of I/ but not necessarily linearly independent we
i — -7«
write
2. It is assumed that E is equipped with a bilinear form
(that is
and
Further
=
0 will be assumed to be symmetric (
nondegenerate (for every
//C^O
(j)(Mj')
there is a ^
subspace F is called semisimple when the form
FX F.
If the length of /%/ is zero,
( 0
with
"
^
})
(f)(/)C^ 6/ZZZ) ).
and
A
(}) is nondegenerate on
/X .)-0).;
±s called an
- 4 —
isotropic vector.
A subspace with no isotropic vectors except the
zero vector is called anisotropic.while a subspace in which every vector
is isotropic is totally isotropic.
iff
0
A subspace F is totally isotropic
is identically zero on FX F since
C ft (
Ob-j- Lj*) * (pi/X/j /X ) ~ 0 (
(p f
^ ^"
yU. By Zorn’s lemma each
totally isotropic subspace is contained in a maximal totally isotropic
subspace.
3.
Two vectors
(j)({)C/y L j)-O c
write
If fX is orthogonal to every vector in a subspace
/TzJ-Ft
(F T G
F=
F~*~~JJfX/EF j XOJ Fj-1
codim F j
£21. For F and Q-
By definition
finite dimensional, dim
E,
1%/ and Ljr are orthogonal, and we write /Xj± Ijz
-
F^"1 then
F =
F) CJl
F
It is always the case that
is said to be
J L -closed.
When
F
we
F
is
, iff
subspaces of
F F F
t
If
Every finite dimensional
subspace of E is J.-closed, £21.
4.
For a direct sum
FOG-
with
fX /J -j/ for every fX ^ F
and
JL
we write
F (B & .
If E has denumerably infinite dimension and
totally isotropic
J s i is
J -closed subspace of E then there is a totally iso­
tropic closed subspace
with
IX~
V ~ ^ .(fL jJ J > l
such that
0 C XJJl ^ / ^ yV - (Sjj (i.e. , zero if zL'=fj
F- ~ ( 1/0 \X *) (B Q-
and one if .J —
j
).
For
a proof see £81; the proof given there carries over to our case.
5.
If
-J-i ( F 1(J) ~* (
)
preserves the form, that is if
I
~ 5 —
j } ' ( ^ ( QC)^ ^ ( ^ ) J ~ (JlfC1y )
metric (orthogonal) map.
for ,ill#/ and
^
in E, then
is called a
A metric automorphism of a vector space
(E (j))
is called an isometry and the group of all isometries is called the
orthogonal group.
6.
A quadratic form is a function
Q j E—
of the vectorspace E
into the groundfield k with the following properties:
and (ii) 0
defined by
a bilinear form.
If
^
(J)(
)
~ -Jj-[ Q
~
is a bilinear form on E and
(i)
Q Q l/Z)—/^ Ql^C)
1(C)—Q 1 ^ )3
Q
is
is defined by
Q f/X )= (JloLjfl-) them. (Q is a quadratic form called the quadratic form
(J , and J-(J) . Throughout this, paper when quadratic
associated with
forms are introduced it is understood that they are associated with the
bilinear form under discussion.
7.
With the tensor algebra T(E) defined as usual over the vector
space (E,^>), (for an English reference see 131), let
tic form associated with
is by definition T(E)/I.
0
cV
C , then for S =
.
/%,
The
3 ~ ~A(
J
i -'** ■)(Cm )
with
0,*, 0 ^ % ,
J
'h
In particular if
linearly independent elements of E, then
will
asymmetrically
cC3C jtX cJll let
Ec together with the scalar /
C(E) (for a proof see 122).
If f:E-—>k
Then the Clifford algebra C(E)
The equivalence class of
0CCm -
be denoted by
ordered by
J) and let I be the two-sided ideal generated
Q(CC) in T(E).
by the elements
Q be the quadra­
Zflj
are a basis for
Zfll j
are
° Zfli0t ^ 0,
is a linear map of a k-vector space (E,0) into a
6
k-algebra A, then f extends to
a. unique algebra homomorphism g mapping
T(E)— > A which is the identity on k.
If in addition f (x)2 = Q(x) then
f extends uniquely to an algebra homomorphism h:C(E)->A which is the
identity on k.
A Clifford algebra over a finite dimensional vector space is simple
that is there are no proper nontrivial two-sided ideals, O l .
For E
of infinite dimension, C(E) is the union of the Clifford algebras over
the finite dimensional subspaces of E so C(E) is again simple.
8.
A linear form is a linear function (i.e., vector space
j.
homomorphism) into the ground field, u:E— >k.
forms on E is denoted by E*.
The set of all linear
Ert becomes a vector space over k, called
the dual space of E, under the usual definition of addition and scalar
multiplication of functions.
each /%/
If F and G are subspaces of (E, ^ ) then
in F induces a map
( p / (2/,,
with
maps are elements of G* because of the bilinearity of
F
{/Ztf
then the maps
P D Q F ( O ) t For .ZZy;
there exist elements
74).
In particular if
I* c w i t h
9.
y
(ZL
These
If
are linearly independent if f
linearly independent functions from-G*,
of G with
^
PaSe
^
then there exist
'
A linear topology on a k-vector space E is a topology with a
neighborhood basis at zero of linear subspaces"
hood basis at
Cb
Lj^ of E and a neighbor­
composed of the linear manifolds
fXj-hU^e
A linear
7
topology is always a uniform topology.
O
U0^ ~ (O)1
It is Hausdorff if and only if
A vector space together with a Hausdorff linear topo­
logy is called a linearly topologized space.
In the sequel we shall
always assume that the field k carries the discrete topology.
If
is
a linear topology on a vector space E then the operations of addition
and scalar multiplication.are continuous (the product topology being
If f is a linear function Ey—> Ez with linear
taken on EX E and kXE).
topologies on Ey and Ei then f is continuous provided it is continuous
at zero.
If Ey and Ei are linearly topologized spaces and
function EyX Ey— > Ei then
(p is continuous iff it is continuous at
(0,0) and the partial functions
are continuous at zero.
^ (fcbf L
is a bilinear
lPfp > ^
^
—>
and ^
)
This follows immediately from the identity
j
j
o
^
Z -/ 5
W
Mo? Z ' ^
In particular a multiplication defined on a linearly topologized space
is continuous iff it is continuous at (0,0) are separately continuous
at 0.
10.
If the
sum topology
(Eyi )Ou ) are linearly topologized spaces then the direct
on
basis the spaces
neighborhood basis for
is defined by taking as a zero neighborhood
with
TOy•
Uy running through the spaces of a zero
The topological direct sum
is once more a linearly topologized space.
((BEc^ , (B Ly)
The direct sum
E ~ F Q Q-
8
is a topological direct sum iff the projection p :F($> G — ^F, with
p (x+y)=x, is continuous (iE91,. page 95).
In the case of a linearly topologized space the definition of the
quotient topology
Let
CF
can be given more easily than in the general case.
be the canonical homomorphism E — >E/F, then the quotient topo­
logy is defined by taking for open sets in E/F the images under
the open sets in E.
Under this definition CF
Cf of
is a continuous function.
Ty. is Hausdorff iff F is topologically closed (equal to its closure).
11.
Two vector spaces E, , and E 2. form a dual pair with respect to
a bilinear form ^
:E, X
Ei— > k if for each #
in E, there is a
in
i
The vector spaces E and E* are a dual pair with respect to the natural
bilinear form
(C/j ^ Mj ) — JUsf fh )
for
E and
MB E*.
For-(E1T )
a linearly topologized space we denote by E' the subspace of E* con-?
sisting of
~C -continuous linear forms.
dual space to E.
^E, E'^
E' is called the topological
is a dual pair for the natural bilinear form
•mentioned above, O I .
A particularly important linear topology is defined in terms of dual
pairs.
If
is a dual pair with respect to
topology on E, with respect to
ft
then the weak
E2 has a neighborhood basis at zero of
F running
- 9 -
through all finite dimensional subspaces of E 2. .
denote this topology by
in KBthe is
Tgs (E ^) ).
(p then Ejz =
We follow Bourbaki and
0~(E; , Ez ) ; (for reference the notation used
If <(% , E2)
is a dual pair with respect to
Ez where E^ consists of the
linear forms on E, , (191, page 90).
Cf (E1 , E2 )- continuous
CHAPTER III
CANONICAL TOPOLOGIES ON THE CLIFFORD ALGEBRA
Let E be an infinite,dimensional vector space over a commutative
field k, characteristic k unequal 2, and ^
bilinear form E x E — >-k.
a nondegenerate, symmetric
Our eventual aim, is to study continuous ortho­
gonal groups, that is the groups of isometries of E which are continuous
for some natural linear topology on E.
In this chapter the meaning of
"natural" linear topology is delimited and topologies with the specified
properties are
investigated.
is an isometi
Jv : C(E)-+C(E)
rphism
and
with
.Jv the identity on k.
This follows from section 7 in Chapter LI, taking
A = T(E) then A = C(E), since
Further, let E have basis
for E.
Basis elements
o ^
complete set of basis elements
is bijective.
of E induces an algebra isomorphism
Thus
of C(E).
is an algebra isomorphism of C(E) which
to E
maps E onto E and is the identity on k then
is an isometry, for
(J)(JbfrfC)^ JLfft.))
Because of this canonical relation
between the isometries of E and algebra isomorphisms of C(E), we shift
our attention to the problem of topologizing the Clifford Algebra.
Starting with a linearly topologized space (E,C), there are many
-z?
ways of constructing linear topologies on the tensor products
'
11
Here we shall consider two tensor product topologies,.the
corresponding to the
topology,
£" -product of Schwartz and the "projective" topo­
logical tensor product topology corresponding to that of Grothendieck.
These topologies have been studied in 171.
A linear topology on the tensor products extends canonically by
taking the direct sum topology on the tensor algebra and then the
quotient topology to a linear topology on the Clifford algebra.
If this
extension is to be useful it must induce the initial topology when
restricted to E.
the
We now investigate whether this is the case for either
£ -product or the projective tensor product extensions.
Since it will quickly become apparent that the
£T -product topology
is not suitable in the sense just mentioned, we shall describe it only
briefly.
For further detail the reader is referred to I?!. The TTe
v?
topology is the finest linear topology on (&E for which the canonical
multilinear map
7/^
—^ ®Ei
is uniformly continuous.
For each
~Cq has a neighborhood basis at zero of sets
a
each summand containing .y? factors and the
Qp running through a zero
neighborhood basis for the topology ZT on E.
-A
A zero neighborhood basis
oo
A
LL,
and a zero neighP~l ' ■ A
borhood basis for the Clifford algebra of the sets (TO • where (J~ is
for the tensor algebra consists of sets
the canonical map T(E)— *C(E).
U~
These extensions as well as the
duct topologies oh the tensor products will be denoted by
TZq .
g
-pro­
12
Theorem I:
If (E»Z7) is discrete then (C(E).,
) is discrete.
If
(E,U) is not discrete then (C(E) , ~C
Q ) is trivial.
Proof:
If (E,C) is discrete then (0) is in the zero neighborhood
basis for ZT .
taking
Up^ (o )
In the expression for
for every
-jo gives
in the preceding paragraph
U - ( o ) and < r(0 )—(O) . Since (0)
is thus in the zero neighborhood basis for (C(E), )
* the latter is
discrete in this case.
On the other hand if (E,Z" ) is not discrete and
G~(Ut "h
E. -I- IE® Uz •/- U j ® E & E V-,,,/)
zero neighborhood basis of (C(E),
(J~(U ) "
is an arbitrary set in the
) then every element of the form
A
0^1°/Xs^i •>,
element
is in
(fu . . For since ZT is not discrete, there is an
^ i zO in U/n+z > and since E is semisimple, there is a
^ ^ ' f 0' ^
with
■Xj0rtj'd
C(U)
°
^
and 80
0/X-Z^i -t~Xj6■XJ'0XXje<>>0/ % % /
^ i0
6"
^,<5' E
0 X X rf-
cru.
is thus seen to be a subspace of C(E) containing a set of
generators of C(E), hence
Q-(U) - C(E). In this case
is the
trivial topology on C(E).
So requiring that the
topology
~C
topology on C(E) induce the initial
on E would leave for consideration only the uninteresting
cases where ZT
is discrete or trivial.
For this reason the
topology
will not be discussed further.
We now turn our attention to the projective, tensor product topology
(p)7J on the tensor product
SE
.
In 171 it is shown that there is
— 13 —
a unique linear topology on E
8) E with the following properties: (I)
COz : E X E — >E ® E is continuous and (2) if
the canonical bilinear map
f is a bilinear continuous map of E X E into a linearly topologized kvector space G then the induced linear map E &
The proof extends to
logy..
Clearly
(S)E .
is by definition this unique topo­
(^ZT is the finest linear topology on
$ E
OJp : 'IT E
^zr
E — » G is continuous.
is continuous.
(§)E
If Z7 is Hausdorff so
for which
Is (£) E
(for details see EC?!).
A neighborhood basis at zero for the
the subspaces
with
U1 ~ Uz ^)
U2J^ and
~t~ Y
j
a.
(E)E
C fiE ] <8 U2^
topology is given by
-A
running through a zero neighborhood basis of
This is so since
~C.
is continuous if and only if it is continuous at
(0,0) and is separately continuous at (-#,0) and (0,$.) for every /&£" E.
If
GJz
in the
is to be continuous for a topology "ZT
E
zero neighborhood basis must contain a space of the form
ZX
A
Uz . Conversely, the spaces Uz
which
on E(§) E then every set
GJz is continuous.
define a linear topology on E for
The same reasoning applies for any p
, so &
V3
zero neighborhood basis for
-h ,,, •h
subscripted
Ij
(p~C consists of the sets
C ( E / ] < S , , , ^EC/Cp-J]^
CL
Jp -!j
E
-PfCl Itx "• ftp-1
with the
U 's running through a zero neighborhood basis for ZT
and
- 14 -
C/faJ <& Llpi^ 0 ,,,
Upft
®
toe J QUpzt S) >*>
<&) Up^ &,i) $o LJp/%, -/•
Up Zt
®
meanings for the other
>+ Up^® Upl^(Z),,,
symbols.
E rfU
and
with similar
Henceforth in this chapter
will be abbreviated by ^ 7
and
piMrid/
U p ® ..,® Up + EU, C W J ® Upzp ®
simply
®
Up
will be written
® Up tt +
S UXzJ®E(fE®Upzt ^ Q ... ® UppLj-}-»>» > xakinS sums and quotients, the
projective tensor product topologies induce linear topologies on T(E)
and C(E) , both denoted by
0E •
We now determine for which topologies %T
on E the induced topology
GQtJg is equal to ~C .
Theorem 2:
If E has a zero neighborhood basis of subspaces no one
® t/g
of which is totally isotropic then
is trivial.
Proof:
-
4-
U-s®U^ Tl16, J)
basis for
®X,
on C(E).
is an element of
exists a
be an arbitrary space from the zero neighborhood
Uar
We claim that an arbitrary element
Q~{(j) . For Uf^
with
for arbitrary
of E
is not totally isotropic so there
Q ({^ ) t 0 ,
>\
ZV3- C
^
^
0 ~ (U ),
/fc — ~j—
==
^
But this implies
<r(U) f) E ~ E
CT((J) » hence the assertion of the theorem.
On the other hand, if the conditions of Theorem 2 are not met then
some linear neighborhood V of zero is totally isotropic.
Intersecting
- V.
_K •
15
V with the spaces of the zero neighborhood basis gives a zero neighbor­
hood basis of totally isotropic subspaces.
Theorem 3:
In this case we have
Let (E,75 ) have a zero neighborhood basis £
totally isotropic subspaces.
StT-j^ ~ ~C
Then
of
if and only if
Z -U U f.
Proof of necessity:
Let
Qb€.lz
„
U U ,.
We shall show
OL
OC Q
O
this conclusion is immediate so suppose
«
For
O
l
Since
is
a
Hausdorff there
±s a.
f]0(fL LJ^ *
with
- ZT
U
so there is a
(T(U)DEcUoi ; U= U^UzQUz+ZEocJ®U20,+ U3QU5QU3
with
+,
contradiction that
U3
Suppose by way of
Then there is a
with
f ' 'ZOt-OL
; therefore
which, since
OCeU
OtGE , implies OiG U
a contradiction.
We conclude
JL
'
Proof of sufficiency:
JU j
j
the ZT
®~C
Since each
7J, .
Now suppose
zero neighborhood basis.
0~(U )D E Cl U jt
for every
/Zr
m
Ul D f l U ^
note that
First take
there is a 66,
.
and
C T ((J )O )E - (
U1
such that
(
)DE
is an arbitrary space in
We shall construct
UmCl Ut
f
for all /Tb .
U
such that
By hypothesis
/ZrJ. 64, . Take 6/
v
™
a
U=
0~ ( Z / C Z j &
+
0 LZmJ 0
® Uz^
(it, <» Otmb
> We first
® 44 .y....4 /_ ) ~
- 16 -
c r ! Czpe>„, CtcmI ® Um r ,
A le Un
<JT(Q)
....
s>
zm >
A L o /%£
/X^ 0 L L j
is of the form
Ab - JE^
-JAntLts
then
element of
,
by way of contradiction
LOG C f(U ) b ) S
be a basis for the vectorspace
6/
•
/ <f_Z <r/ %
a~(U) t
by the remarks above
Si
Kt
& ±i
/x r
^
J-=I
/ X ^ U 1 . Let
but
The set
ffX^ Qoi
6^
C jo <$,<>»„, o A
is totally isotropic.
^
being
Since
Ms
,
Suppose
is off the form
*(FAv e^y^
plying through by
So every
tLZZv , AAtI ^
linearly independent can be extended to a basis for E.
Ot6
because i£
Af 4 “%'
gives
Ot0Bs o,,, o Q
Multisince
But this is not possible since
OC0BnI oVif oB i is an element of a basis of the Clifford Algebra.
“7
°m-
(TlOinEcUr
/.
Hence
®zjE
As an immediate consequence of the construction in the proof of
Theorem 3 we have for future reference the
Corollary:
If (E,ZT) has a zero neighborhood basis of totally
isotropic subspaces
basis of sets
with the
Pic
Qj
. 0
Ij
fU j
such that
then (T(E),
jb
in
) has a zero neighborhood
<!"(U ) implies Ab
°^
linearly independent elements from a single totally isotro-
— 17 —
Next we shall describe the topologies for which the conditions of
Theorem 3 are realized.
Definition:
U
The
'
U
Let
be a fixed totally isotropic subspace of E.
topology is defined by taking as a neighborhood basis at
zero the spaces
UF
F
,
a finite dimensional subspace of E, £51.
This gives a linear topology which is Hausdorff since
is assumed
to be nondegenerate.
Theorem 4:
If (E,ZJ) is a linearly topologized space with a zero
neighborhood basis of totally isotropic subspaces
then
Z > ZflUoi
for every
topologized space with
I/
of E then
subspaces
o( . Conversely, if (EsZT) is a linearly,
Z ^ Zq
V
F = U
ak ftn
1=1
for some totally isotropic subspace
To show
,
Z'^Z(pUt^g
F=
Z ^Z q V
be a set in the
'
^
with
Ml
u.
. c Lin
Fx
o
otI
Conversely suppose
IJ0^ H F j-
let
zero -neighborhood basis,
there exist zero neighborhoods
m,
F = U U g^
has a zero neighborhood basis of totally isotropic
and
Proof:
Uci0
Z
and
^
J.
Since
J. U. •
cX
proving the contention.
^
-jL
. The spaces K d ^
with
F^
a finite dimensional subspace of E are by hypothesis part of a zero
neighborhood basis
pic the
for C
Since
V F f^
may be chosen totally isotropic.
already cover E since
F= U F
/3
fore
.
E - i / U x.
P
an^
is totally isotro­
But the
f VF
F F F ' L'L CZ ( V O F~jF')J~.
P
p
r ''
)
There-
— 18 —
It is interesting to note that when ZT-Z^ K
dimension and codimension, the
finer than the
topology.
,
\ / of infinite
topology on E 0 E is strictly
This will be proved at the end of this
chapter at which time certain lemmas and theorems will be available to
make the proof easy.
V
The
the form
topologies are related in the following natural way to
j)
Theorem 5:
For (E,%T) a linearly topologized space with symmetric,
nondegenerate, bilinear form
Q
(i)
, the associated quadratic form, is C
(ii)
(ill)
(f) , the following are equivalent:
K
V
for some totally isotropic subspace
of E.
j) : E X E —>k is continuous (for the product topology on E X E ) .
Proof:
If Q
are satisfied.
is ZT -continuous then the conditions of Theorem 4
For the continuity of
totally isotropic zero neighborhood
I/
zero of totally isotropic subspaces
at
-continuous.
OC there exists a
with
Q
is totally isotropic this implies
Q
at zero implies
~C has a
, hence a neighborhood basis at
.
And if
%£. E, by continuity
U0^ ) —
fit
Since
Therefore
U0^
EZ— LJLjc^ f
and applying Theorem 4 we have (i) implies (ii).
If
'C'^'CpV , V totally isotropic, Theorem 4 can be used to show
j) : E X E — >k is continuous at (<£^).
exists a totally isotropic
$1
}
U0^
with
For
Ez - U U oi
ZpCzL UtJ
and
, so there
J- Uc{ 4
So (ii) imPlies (i:ii) •
- 19
It is apparent that the continuity of
(fi
implies that of
Q ,
completing the proof.
The topology on C(E) can only be considered admissible if continuous
orthogonal automorphisms of E induce continuous algebra isomorphisms of
C(E) and conversely;
The projective tensor product topology has this
essential property as the following theorem shows.
Theorem 6:
Let
^
be an orthogonal automorphism of (E,^ ),
the
corresponding algebra isomorphism of T(E) (with
-Jjs (ZjLl ) ® ,c. 0
), and let
isomorphism of C(E) (with
Jb
If
^
JL
Proof:
J ~ OxJ 8) I
a t,t o Jfrffu) *=
is
Suppose
V- »,v
J
is
-continuous.
Clearly
)
is ZT -continuous if
(J —U1
Let
) there, is a
(resp.
% a, % ^
® U2.
f
"
U
For
)
(resp.
^
^
0 ^
U1 V- Uz ® Uz +• U j CJ((%)1 ®
tinuity of ^
0^ jj Z / ^ )
be a set in the T(E) zero neighborhood basis.
^J(Vn i )C. Utfi
f
,,
(%)C -continuous.
e v e r y (^sp.
such that
be the corresponding algebra
V totally isotropic, then ^
(Z >
and only if
Af
.
O
establishing the con-
.
JLfTa* ( T ^
30
is likewise ^ZT -Continuous.
J b T ( V ) — CT^ ( / j (Z ( T ( U ) i
and
J
- 20 -
Conversely if
and
=■
Ji,
is an algebra isomorphism of C(E) with
then we already know
morphism of E.
Since ZT^
V 0 ~C~ 0 ~ C jg y
implies the continuity of
only if J
is an orthogonal auto­
so the continuity of
.
Applying the theorem to ^
Corollary:
-Jojg ”
/ and
.Jy
gives the
With the hypotheses of Theorem 6,
is open if and
is.
Although not essential, it would be desirable to have a Hausdorff
topology on C(E).
We first consider separate continuity of multiplica­
tion in T(E) since this result will be used in the proof of Hausdorff.
Later in the chapter the subject of continuity of multiplication will
be discussed in more detail.
Theorem 7:
Proof:
Multiplication is separately continuous in (T(E)9^)ZT)
First consider multiplication on the left by
S - X l (E)toe GPXj0 € (jpE,
with
( (Jsj 9 #»e7
^
^
For every
jj/ I
the map
•)"jp! Vi t f -J
is continuous and so induces a continuous map
Jpj €>.„
of
gup,®
E ).
Addition gives a
8)p y .
^ E —^ T T E — >1?p^E
^ /%j
^
Se -
0 ES
(by the definition
GDE continuous map T(E)— J-T(E) with
J s — -G SgfJy0
Now let
V
K-
S —Z-S:
■
with
be an arbitrary element of T(E),
= S> §£>',.(& S’ » Left multiplication by fj. is continuous at zero so
X
ajPI
^
for every
there is a j/ such that
®
LJt
Therefore
Lu
M A
*
J ® / I l/f C- i j i
Left multiplication by
and therefore continuous, since $£T
S ' is continuous at zero
is a linear topology.
The symmetric
argument proves right multiplication is also continuous.
Using Theorem 7 we can prove the
Corollary :
If
A
is a two sided ideal in (T (E) ,0C) then
the topological closure
of A
A
,
, i s also a two sided ideal.
/4 is a linear subspace of the linearly topologized space
Proof:
T(E), so
A
Now let
E T(E) and
is also; in particular
neighborhood of
,
/4
is closed under differences.
A
and let -£<8)S-h U
be an arbitrary basic
J/® $ . Since multiplication is separately continuous
A
at (i,0) there is a I/
A
such that
C JJ
I/ C
And
S ^n
/4 at some point SPW . Then Js (S-h W ) ~
y\
is in both /j and J j 0 3 + U*
So y 6 $ S £ J
implies S-/-1/ meets
Jj 0 /W t
from which we conclude
is a left ideal.
The proof of right ideal
is similar.
With the corollary above we are in a position to prove (C(E) ,0C )
is Hausdorff for ZT^
Theorem 8:
I/
TJ ^ TJfi V
:
for
J
some totally isotropic subspace of
E if and only if (C(E) ,fflTJ ) is Hausdorff.
Proof:
Using Theorem 4 it suffices to show that (C(E) , (E)TJ ) is
Hausdorff iff
TJ has a zero neighborhood basis of totally isotropic
- 22,-
subspaces
Uo.<
and
UU.±
=
The topology
Ot^
by quotients from the
topology on T(E).
on C(E) was obtained
Under these circumstances
it is well known (see for example 193) that (C(E)j^ZT) is Hausdorff
iff
I - I
(
I
the two sided ideal, in T(E) generated by the elements
/%0
or equally well by the elements
Suppose
0)V is
in particular
-Ip Z
X~I
Hausdorff.
_
.
-f-
is a proper ideal in T(E) so
/\
Therefore there exists
~
U U/~f~Uz®
the usual zero neighborhood basis for T(E)
with
~l-f-U
disjoint from
X
•
We claim
For if this were .not so there would be an
Put
I
fX>0 X
/2. H% H«
then
/fi— / &
J-I
is also in
(~IX U ) D
therefore assume that all the
is totally isotropic.
/X/€
IX
X1
G
and
A.
We may
are totally isotropic.
E, %%_/.
t^ ) ~ ! /^ 1
with
HZ-H 7^ Oe
contradiction.
We claim in addition that for each
would be a
*ith
pc®U'~bp 0 oc -I g (~I~hU ) X) X
X-Ju:.
U ^e
If not there
and then
as,before; contradiction.
* C
To prove the converse we assume
E ~U
LjJ~ for
<x
isotropic zero neighborhood basis
•
some totally
By the corollary to Theorem 3
proved earlier, T(E) has a zero neighborhood basis of sets
that if
J jG 0~( U )
then
J j- X , J jj 0
• with the 6?J
U
such
linearly
independent elements from a single totally isotropic zero neighborhood
- 23 -
U
We claim this implies /$£T
. For if
^
and so l-hCT(U)
meets (0) say in f-tjfc
.
°a
X
O-H-Jt
then
.
- I-h Z T Xj 0 (Bft
O - Gfo ,jj o
Multiplying by
V-/ /
Z
which is impossible since the
l-f-U
meets
We have
y,,, ^ ^
gives
are linearly indepen­
dent.
Thus it is clear that'
/^X
; in particular
C(E) = T (E)/I is a simple algebra and
JLClX
, so
X z^X(E)
X~ I
.
.
But
Thus (^ZT
is Hausdorff.
The discrete topology on E would give all the essential properties
thus far ascribed to the
topologies.
The next theorem guarantees
that we are not dealing with just the discrete case.
A
Theorem 9:
topology is discrete iff
V
is finite dimen­
sional.
Proof:
F
of E, dim
As noted in Chapter II, for a finite dimensional subspace
-
If
sub space
dim
F
codim
.
is discrete then
F
.
= codim
^
dim I/ .
On the other hand, if dim
That is
E
^
l/^(B F
for some finite dimensional
Vf F^
is direct, so
In particular dim I/ is finite.
V
— \/X) F^~
—
is finite the codim
for some finite dimensional space
(0) ~ F^m = l/^ F) F
■
I/X) F ^ — fO)
This implies that the sum
F^
F-
, which implies that
; \
‘:
Continuity of Multiplication
F
.
dim
V
.
Therefore
is discrete.
'
We turn our attention to the question of continuity of multiplication
- 24 -
in (C(E) ,
)•
For denumerable (E,^>) we shall establish the remark­
able fact that (C(E), QQVfiZ) is a topological algebra for closed I/ ,
I
(Theorem 10).
It is not clear whether a similar result holds in the
nondenumerable case.
For ZT strictly finer than
V , we shall give
an example of a denumerable (E,^>) for which multiplication fails to. be
/
continuous in (C(E),Q&Z ).' First we prove
Lemma I:
If
-
u
u
F =
€4
^ A/eZ
,
pZwru/
then the sets
+
+
form a zero neighborhood basis for
U^ u 3
U 's
®Z on T(E) when the subscripted
run through a zero neighborhood basis for ZT .
We shall again write
Proof:
Clearly each
Conversely, put
finite, put
«^4
C/X,? ®
to mean
® V
zero neighborhood contains such a
Vm =Uin and for
1 / ^ , . , ^ = ^
/X1? /Zz ?
Um e j , . . e J/M‘
® L e*/n J® U/nedl, „ e<m/ ® ■ " ®
® C vm ] ®
<$>
tains a 0%^
^
U
/j > ^
: Ihen
^
■?
50
U
COn"
zero neighborhood.
Theorem 10:
If dim
and
I/ is a closed totally isotropic
subspace of E then (T(E), (QZ^Z) and (C(E), QQZfi / ) are topological
algebras.
»
'
25
Proof:
Since dim
E = A-^6 and I/ is closed and totally isotropic
® Q-
there is a decomposition of E into
[Zz- ^
both totally isotropic and
with
(j)(/Vj_ ? /2^,' V -
topology has a zero neighborhood basis of sets
Zr
finite dimensional,
K
v n ^ n - A
+ $ /
^
j and
^
The
K
since for
~h C-
so
n e - ^ v n i
-)
|//7
- -4 f a h > m , ■
We shall need an enumerated basis for E , so let
F ~
^ >y
with
AJj.~ &Ai
) l >m are a Z y K
zero
for
/ .
neighborhood basis.
Then the sets
Uf^ ~
(They are not distinct.
Cg ~ i { ( AQi )u,>!
In fact
r
* etc‘) '• The advantage of this numbering is
& Ufyi
that it yields the following simple criterion:
and
Uj ~ ^
M yffL . The U * will be referred to as
iff
(= I/
-sets in the rest of
the proof.
To show multiplication in T(E) is continuous at (0,0) let
[ j ' - Uf
Uz® U2. Il -Zf U S il & U ze. ^
zero neighborhood basis.
suffices to find
We must find
V® I/ C (VCT LZ#
be a set in the (T(E) , $ Z )
v ® I/ C (_J •
Clearly it
somewhat, in order to make it more manageable, as follows.
inductively sets
/''I /"I
*</»-/ J1.,. j
U fyiff
t </ni
y
UMa
0
7'
Choose
Uyn which are -K -sets and such that U ^ U j F)
Um C U1 O U z Fl t ll O U m^ D Um U) Um
/,
U
With this, in mind we shrink
n o
for which
Denote by 6 / ^
and
the set
(i.e. , the intersection of all sets
r
:
\
ff/i is the largest Q -subscript). Define the
-
26
Umia
cVTTl by induction on /W to be ^ -sets contained in
m
sets
^
=VMM
^
""'tVW
(~) Ujfe U) U * Since the Um and U^e
tained in
are functions ^
™ith
Um- U ^ fo, and
#</71 <
-k
uP m r Unem C -U m
Also if
U
o<l<j
a
are
-set con-sets there
LLem ^ U f a t mQ1 43 a con-
sequence of the construction we have
and
meI
■"■
=
Uyfl CT ( J *
y
so
t J 0'
/9 % <T
so
then U ltI e j - U aiei C U
m
so
U*jpC U% u>
//, C
therefore
far
Take
U =.UpUzZiUi+ £ LeJaU2ei + U3SU^Ue -a
Z C e i J a U ^ U ssi + Z C e J 0 Cej J e>U 3 e ie . + ■ . ,
”lth
Ualg,
q
To define K
A A /V,
J[
either
AZ
,
-
"hare
).
we shall make use of a function used in enumerating
the nonnegative integers.
f /Tlf A
For
N Put
-p (Mji^ ZMs) ~
Then _^Y/%,,/*%J
/Tb, + /YTbl K J t z +/W ot /Tbl ^TTTLtj =YYl2 +/YYL1 and
our purposes it suffices that
pairs
U - max. (.U1,,.,, Xmj
iff
/YLx ,
For
have the property that for any two
(YYbjf YYYbf ) and (/Ylx ^TYYbJ1 ^(YYLj l YYYl)Znd ^,(/Ylx l TMjzTe
}
- 27 comparable, and for only finitely many ( /Tbt, Wlt ) is
JsfMsz , /Msz ).
prescribed
We now define the
Q.'S
5,
^
fMsl ?/TfLl )4
for our
|Z
.
For1
'
let
771*
.
n Cu
nHpiZtJ n<&
n U‘H'p+fy^Ms ) D
\L
y
J(-f>^Lm)4 -p(fyysj/ntj)
SC=I
J m = .MUl/Li (j/f, „»» s J J j ). Finally take the expression for
where
Vnj Qu ,,, £ •/ with J,m
T fa.
'fm
a
be the same as that for
out-
f f- f + f
^
~h
^
0
through-
is defined itaratively by
(f«n, ) =
^
replaced by
to
V'
8) Vx q ,
of each part of
V e's,,, P//
a” d
Put
0
as usual.
^
=
The reason for the choice
will become apparent in the cases we
zx
T fa "
■■
f/® )/ C (Jr
consider in showing that
. A
Let
Sf = G i' 8 .* ' 0 Gu (& G f/
^ ^ 4%'''' 6% / '
^jk
‘
9
be the subscripts
‘
the subscripts
4 VsflvKsLav^
0 £ ;/ C
IA3
with
Let
MV / /^ cA ^
TfitIi
sO
jLet
j^u vt
zZj 4ztzZ ^4
•t * 4
r
in their natural order and
•/
Jn+/? >•' ■}i^yk)
%n.+lG ^ e J ,,e j^
SLj4
< /»'
^
/
0
'
in their natural order.
^ is l ? Az,v-/ >
Since
giving the combined ordering
4 Js" Similarly ^et J1K ,,, 4Jm KJfln^ J ,,, 4 Jcjs
be the natural order of the^
.
Note that
*"A
p
i '"><§>
S
28 -
are all in
V
.
-A
SfQ jt' S U1 The general nature of the next steps in
We now show
the proof is this.
-L1,,<■»,
subscripts
^ ~^k/1
•
5
6
<
^
»<■»p
Then
we can always choose
^
Uil K
Let
;
Xs
JLs+ !^ 0^6
arranged in order.
^.s+l?
" so that A
% C% K '
be the
are
G Un
Let
-^n ]/ .
ri I ~
a
//?%. =
^
e^f- ® u^peit...<=,„c U^
^
}
t,
and
will not play symmetric roles in the sequel.
/MCyt
i i i J f-Mu ^
Swv-/^
aa^
,■
=»
We assume without loss of generality that ^
*
^
are also in
's',<9^ /e
^EiSe-iP ei"■ 0ceJJ0 Up*f
hence
We show that
by the definition of
^
^
4
or
*
)
and
^ 6
then
Cj
4
^n the
. G IjU
//
1
Sf (E)J jf G U*
*
the only other possibility is that
is
? SP /Yb,
S-pySeco S -^ftru S
Note that only
^
-Aw »
*
\ ^ mv+!
the immediate successor of some
Case B ;
^
Since as noted above
in these cases
.
*
Since
^
is the immediate predecessor of
ordered list of subscripts.
for
^a particular from the
second of these conditions we have
Case A:
^ vd,, Qj/^ we have
^fp+fyU.U > % + / ^ ^f-p+cpU)
^
Since
*"c ^ "zAsi^
-subscripts occur between
Ss_^
ca<>
and -Ay .
If
Ss-J^ ^
Unji™'! (by thebaslc deflnltlon o£ the,
^<-sets), and we're done.
Similarly if
U j. S
U U ^/) for any j t
- 29 -
S- A
with
t S
ET iJj?
then
, •
,
as desired.
^
hand none of these alternatives occurs then ^
-^S-J ^
^S -
etc*
So
these inequalities follow since
-rlrS < ^
Q rn I,
Case C:
Qrn j.
is nondecreasing.
^
^ ^
^ ^ /iv + i^ °°c
(resp.
-s O
the case when
-Zzzi
-"4
00e •
replacing
(resp.^a^- ^
$) * . e QQ QQ/ fc. Kp ® * <,o 0
(resp. (C/, 0 .,, 49
Ekpj0
In every instance
elements of I/
, , , 0
Sy
Vqii
But then
•
'/
^ nv
) take
) , and the proof goes through as above.
£Jv
and
^
^
proof is the same as for Case B but with ,Z^
In the case where
^
^ S -I ^ ^
-Ar ^
so as noted earlier
<ui,«
If on the other
The
throughout.
U7fl
This would be
\/po
)•
LJ • Now a product of two arbitrary
is a sum of terms of the form
S &)
hence also in
U ,
completing the proof that multiplication in (T(E),®~C ) is continuous at
,
( 0 , 0).
In Theorem
8
it was shown that multiplication in (T (E)1, ®~C) is
separately continuous.
Thus (T(E),®T. ) is a topological vector space
with continuous multiplication, hence a topological algebra.
We now prove that continuity of multiplication in (T(E).^ (^ZT )
implies continuity of multiplication in (C(E)s^ZT).
/IVLl (S jJ t)
ical map:
>
SQQJS
Let
■be the multiplication in T(E) and O' the canon­
T(E)-S-C(E) = T(E)/I.
Then
: T(E) X T(E)-^C(E) is ' -
continuous and constant on equivalence classes modulo I, so it induces
— 30 —
MVt ( CT(S)7 CF(^t))
a well defined map
multiplication in C(E).
Cfls)
there exist
Given
0~(S) 0Q~(i))
CT(S)0 C f(T)E
OO/} containing
and
x
0;
Since
CYyylcf(Cf(S))K (T (C (C t)))C lO '
6
"
which is in fact
Or
7O' open
and
T
in C(E) ,
respectively with
/ % '/fjfdry =
so YfYl is continuous.
and
Multiplication need not be continuous in (T(E)s^ZT) for
E = AJ 0
even when dim
and
T fC f^ V
I/ is a maximal (hence closed) totally
isotropic subspace as the example below will show.
The next lemma will
be used in the example and in the next theorem.
Lemma
2
.
Let
E- 1^9IV.with V- ^
and
/V-$f
have a topology for which there is a neighborhood basis at zero composed
~ (k (
of sets of the form
subsets
of I
.
Llrie.CT Um
L L ,. + T f e J a C e J e^ U ^ ®^ . . - ® ^ , , + , .
V e J ^® UneJ®*cf
Um
8) Urveu
0 C U mo O
,
from the zero neighborhood;.basis.
V*,® (® CofJ £ Qn?
Proof:
Uf1i_ (& B7
G y
^
Hhile
with /4
T e i l - ■> ceT
and
EoioQ tt ,0 )3 ^
O
and all subscripted
.
LJ^yl^
If
«
and
are of these types:
LS^J 0 C
6
with
UrriQ^_
^ J u e iu j-
either of the
or of the form
ol^oC/ . Since by hypothesis
A C F -C ^ a q ,
^ oM
Uej Ce G-Te1Q(Zr^-S)Gqn yjaJ.
(g
F 6 Q-
;
and
U 's
(J)ri then
^^
elements of the basis
containing a factor
or of the form
L)m._s>B
/»7^
^ nd ^
The summands of
IjS0( J 1
S )A
form
running through some of the
Let.
UJm,~U ?..* & Li.
with
L
■>
JnlC F@ C concluding the proof.
-
- 31 -
Example:
Let
I/-
with
j J-Vj'
both totally isotropic and
»
.Take ,for Z-
finer than
V
, the
UJ ~ J
basis of sets
(for example
1 0
Tp //
topology has a zero neighborhood
"
UJ*E) Uyyi
the
Ujl * ~ ^ ^
topology with neighborhood basis at zero of sets
As proved in Theorem
IV - ~A{ .Uty /j/^y
and
Um
Each
contains some
U
' yfrl
'*
i
/
') but not conversely so 77
is strictly
.
In the zero neighborhood basis for (T(E), 07T ) consider any set
U = U1-TUz^ U
EfVjl 0 Uznt^.
®
form given in lemma 2 and in particular with
Let
I/= Vj -h
$ | / £ w i t h
the subscripted
of the general
Um=U j*
V 's
UJ^D U •
and
JT
from the
neighborhood basis and suppose by way of contradiction that
Vj ~ U(j** ~ A (
>
/
for some
.
For
M
but /2% % #
<$EUfy+t, / 2 ^ ^
There is an odd
•
Therefore by the lemma
hand
/2^^
0 /V^o 0 0
And since
®
and dim I/^ A-^o
The state of affairs when
J
^
U.,
j/
, and a
/IUhL ^ J
®^ 0 $
0
^
'
/V/. E
, is odd,
/VJg £ \J® \/j
Examples can be given with
~CZ*LpV
i/ ^
Ei
V®]/
C U.
^
Va, ( L - f l U0 ,, !/iy o • «
that
odd,
zero
^
4^yL/ »
14.
such
U<$.+!*
On the other
contradiction.
closed and totally isotropic,
for which multiplication is not continuous.
L-T/pV
and dim
\/>^J0
is an open question.
In this chapter two topologies were considered on the tensor
product E(S)E.
It -is apparent from a comparison of the neighborhood basis
- 32 at zero that
< ZT(9£*„
is the weak topology.
On the other hand using several of our earlier
Ze
results it is now easy to show that
for
Z
a
Z<p V
E —Z<b H
Let
Proof:
Then
ZQZ
H - "A(
,
Z
totally
also of infinite
'CQEc
<
Since card
X
^ ,XJ0
there is a bijective function
mapping. I onto its finite subsets.
the
Ty "I/
zero for
' -
ZfiV
have topology
isotropic and of infinite dimension,
dimension.
is strictly coaser than
V of infinite dimension and codimension.
topology,
Theorem 11:
—T ® Z ■ when ZT
In E7]l it is shown that
There is a neighborhood basis at
topology
. - __ of- sets
is finite dimensional then
-JL
V f)h
U0
~ V D /4
% ^-
For if F
I.L
I / D ( V -h Vk(
^
^(o() ^
= v n - A ( \ ) f i e i U).
V= -k (n K f)y e ( t.
space in the
ZQZ
U3.= V 1
S V - ) - Z [n r^ J
zero neighborhood basis (see lemma I).
way of contradiction that
neighborhood basis at
dim
/% C
0
.
U2 3
®U
U 1 /2^ V
E ®U
U ®
t
U
in the
is a
Suppose by
Zfi V
Zfi Z is Hausdorff but not discrete.-since
A-J0 (Theorem 9), so there is a
is clearly in
Uti
Then
®
U0,^ with
,
~E U&E. but by lemma
i.e., some
~^k0
2
the basis for
it is not in
a
U2 contains no Eg zero neighborhood so Zg K Z Q Z t
U2^,
H
Thus
CHAPTER IV
THE COMPLETION OF (E, Zv/)
As in Chapter III assume that E is an infinite dimensional vector
space over a field k and that
form.
ft) is a nondegenerate, symmetric, bilinear
Let I/ be a totally isotropic subspace of E and equip E with the
V
topology
.
In several of the theorems which follow it will be
I / SHt ®
convenient to consider the' following decomposition;
with
V j - = H1B V ,
H = H, B Hz. Such a decomposition is of course always possible.
The symbol /V
will denote completion.
consideration is always
V
Thus
The topology
10 under
~C denotes the completion of the
topology.
The first theorem shows that the problem of completing E reduces to
that of completing K
E-
Theorem I:,
Proof:
then
U E) H =
is discrete so
Let
U
If
.
H
is continuous.
ZV
E i= V
ZV
b
t^ie discrete topology.
is any set in the zero neighborhood basis for
So (0 ) is a
zero neighborhood, hence
O pH
'CpVjj_f
is already complete.
be the projection mapping
a neighborhood of
/I/
V (B H .
( /IH-bHy)—/IO
Therefore
V oH
Since
into
I/.
■ Let /IO-b U
be
-^O, (flOV--Vl/ /- LJ) ~ HOV" LJ7
1/<BH is a topological direct sum and
/V
H
The completion is only of interest if
(j)
induces a continuous
bilinear form on EA • The next theorem guarantees that this will be the
case.
— 34 —
Theorem 2:
The quadratic form
<5/
continuous function
Q
Q '<•E~^/^ extends to a unique
K
is quadratic,
is totally iso-
tropic and, with respect to the associated bilinear form
~C }
V/
Proof:
for some totally isotropic subspace A/
Although ^
of elements of E which converge to
and
converging to
(X-&. E.
neighborhood,
C
Cauchy systems
in E the directed systems
be two directed systems in E both
equals some fixed -Vl
.
0(
for
is a 77
zero
sufficiently large and
And since both directed systems
AC , ^ /2^ /"
particular
v
Since both are Cauchy and
for sufficiently large
^
is also Cauchy so
; in
is Cauchy.
Now consider
//2^ Ti- J /<.
For OC
Q
and yf
sufficiently large,
~ °l4>(Md,-) k
Jv)-S-CjfMfi-ft)—
since we may assume /^.— /2^ <S 'A('^v)
Therefore
+
X
.
Similarly Iim
By computations similar to those above,
OL
E) ]/,
is a Cauchy system in the complete Hausdorff
space k so has unique limit
for
of
have one and the same limit in k (see E9T, page 17).
Let
converge to
,M t -L-V.
is not a uniformly continuous function it
zv
can still be extended to E- provided that for all
K Q f/X pf)
^
sufficiently large.
So
uniquely to a continuous function
A=A.
Q ‘„ E
Q
f/T/£/ - h
^
\ .
Q {/V^ + 'Azc^ ) — Q, (/V^f-h
Therefore
Q
extends
— *
If two continuous functions mapping the topological space X into
the Hausdorff space Y agree on a dense subset D of X then they are
^Q
35
identical.
Applying this well known result gives immediately that
V
is quadratic,
Since
is totally isotropic and
'Jvf)=O0
//2^
Q
Jof
y
Q
is continuous,
for some totally isotropic A/
by a theorem in Chapter III.
The results to this point are of an existential nature.
In the
next three theorems the form of the completion is made more precise and
(p
a computing formula is given for
(V 1
Theorem 3:
with the
0~
(
V
Proof:
F
dim
/"4
Is
J
) ~( ?
compact,
/ P
V
x
^
dim
hlJJ (& H
so
and the discrete topology
linearly bounded-, since for arbitrary
finite dimensional, dim IAf-
VFh
G~( H J ■) ^4 .) )
topology on
TjfrV
.
on H
.
VDF ,
( V F F l ) ^ V F F = dim V /V F )F '1' =
S / p M= dim
F
.
Therefore
V is
V * and
is topologically isomorphic to
77 linearly
= (T( V )
(see Ktithe C9I), page 101).
To complete the proof we show that
Z
is in fact
H2l , but first
we prove a useful
Lemma:
If
F ~ I/4& H^
with
PtTl/'^' and Z Z j
a dual pair for
(J) then ZM I//, ^ F lV rj H 2, I It is not necessary to assume that E is
semisimple.
Proof of the lemma:
sets
Q° - {J/IFF V
j
(TiVj
has a zero neighborhood basis of
(J(Jiz j TVj)= O
a finite dimensional subspace of
FfjfrVjy zero neighborhood basis.
Hz
for all
J(Ix FQ -JJ=VF)G r 9
The sets
VFQ
C-
are in the
In fact for an arbitrary set
VF F
36 -
Vj^
in the
contained in some
So the sets
F
zero neighborhood basis,
V + Q- so
being finite dimensional is
V O F ^ D V F) l / ^ f ) Q-1 ~ Z D Q~L,
ZFG -^ are even a zero neighborhood basis for
Z jy ,
Returning to the proof of the theorem we show the lemma applies.
^
is a dual pair for
then
W
^
so
~$£0 , and if (ft(
is also orthogonal, to
simplicity of E.
conditions
Pjh
0
for all
for all /^& ]/ then since
which implies
IZ^ Vjy ~ (T lZ^ Hx )
of
(/I/)=
(j)
is separately continuous.
HVijz * Z
uniquely to a linear continuous function
; lT ^ e
H1J = J-J2
so
t//# =
[/
namely
are linear, and they are
p j
/V
f
by the semi­
and under these
corresponds to a function on
■)OVj)
, The
even continuous since
l//=
/V=O
,
Each element
with
Jl i) - 0
\/^ flf'-LE
By the lemma,
Z ‘~
l
T^z
for if
H t.
extends
.
^ 1Tx »
We also have
/T ( J x )^
193, page 101).
Combining with the earlier result,
TT ^
0~( Z^ Z^) ~
\ / ( $ _I
V ^ I/
Zy2
f*y
with topology
(T IHxi H2\ This completes the proof of Theorem 3.
Applying the same proof technique to an arbitrary dual pair gives
the
Corollary:
If
FVj J x )
is a dual pair, the completion of
(Vll(TtVn V1)) is (VI, (TlVZj V2]I
Theorem 4:
P H ri
The unique bilinear form
Zl2 H /V j (H u 2 )
for
Z tFeZ
^
H *
(n
of Theorem 2 has
7^6
and
V
I
H1 ,
Z = Zfi
V.
37
(E,^>) is semisimple if and only if V
J. -closed.
is
If
K
maximal totally isotropic (resp. JL-closed) subspace of E then
is a
I/ is a
maximal totally isotropic (resp. _/_ -closed) subspace of E.
Proof:
)
suffices to specify
Theorem I.
<j)
+^
)
^
To show
E.
(J) (
^^^ ~
symmetrically and on sums b!linearly.
general formula
•/"
since other values are known from
We take as definition
(j)
define
0 of our bilinear form 0) it
To define the extension
Q
^ 4-
The associated (p
is continuous, let
A2 G ^
~ M j Z-A2 ) + '-M A A 2)'+ '
space in the
(Q Z - -M
has
3
Chen
-^ Z l(Z )2]+-
.
A2
4 Auj '+ A2 ) —
be arbitrary in
If
=
) ~ Q .(M -4 A j 4 A2) > And by Theorem 3, 1/g
77
Finally, (Q
and
This results in the
/^f= M V-)^ Tl
for some finite set
( ^ 2. )
zero neighborhood basis.
Q
So
agrees with (Q, on E,, for if
is a
is continuous.
/IrG ] / then
Q_(n/^4 Zij 4 A2 ) ^ S.Z irfA 2 ) + Q ( A j + A 2 ) - $ . ) ) Z/iTy A2 ) - 4 Q (At +A 2) -
Thus 5
is the unique function of Theorem 2, hence in particular
quadratic.
We turn our attention now to the completion topology.
tions of the lemma apply to (E, $6 ) .
if
For
(j)(nr4 Aj 4 A2 ^ -M/) --JJj (A2)-O
(H Z )- l P - H f (B H ,
identically in
t while from Theorem 2, f t ZZl++ A j j J J /)'— 0
■>
(
The condisince
JM
for all
then
M-G H 2 •
J,
— 38 —*
And
is a dual pair for
Applying the lemma,
O'( /~ ^\ /"4 )~
H
I / . Since the topology on
0
since
^ A z
$ ){M z ^ z ) ~
is the completion topology on
is discrete and the sum
topological, the completion topology is
)*
Z®
H
is
L ^ fZ .
We now determine the conditions under which ( E , ) will be semi­
simple.
If
First we prove that (A
Ht is
is nondegenerate iff
Hf is semisimple we must show that //C ~^Us-h ZLj "b
implies / ^ = <9,
If ^
were not zero there would be a
/=
,.with 1Z^7^ &
H , , with
exists an
Finally if
X L^O
so
there is an
f)^/ (A2 )= (J)(
/“
Mz=O.
with
Azz )<
.
with
if
then by the semisimplicity of /?/ , there
(j)(A r} A l' ) ^ (fifA lf 0 U&-+- Aj )
then there is an
A ^ € Hj
-J- E.
/7^ in Z
so
/Z ~JAj4- <’A>i
semisimple.
^ A?
Conversely if
A j X- Hj .
(Xj
L.et
For arbitrary
^
with
so
Aj =
~
is not semisimple then
(fij/ & f Z * with
J jr J A j V-A l
in E,
sojz)
is degenerate.
jTf
u) ' is nondegen­
The proof of the following lemma now shows that
erate if and only if
w
|/ is _Z -closed.
Lemma:
Zlj is semisimple iff
Proof:
I/"1 = j/<@ IXj
so
V is J_ -closed.
Z ^ - ( X-JIXj J
X/~ ZXj X) IX ^j (the last equality since
Z C Z ^ CT H j
—
D H j - (V+Hj ) Z) IAj
V~^ implies
and the subspaces of E form a modular lattice with
i
- 39
respect to -/*
rad
and /I
'J-I.
).
— !/© ‘ (rad /Vy ) , |/_ j/,11
Since K
Hi ~ (0).
To complete the proof of the theorem, sufficient conditions will be
given for J/
to be an orthogonally closed or a maximal totally isotro­
~ } / ( & Hj *
pic subspace. 'As observed above,
. -L -closed then
Hj is (p -semisimple, so H/ is
r*/
fore I/
For
V
is
-semisimple, there-
is J_ -closed.
V
Hj
a maximal totally isotropic subspace of E,
anisotropic.
- J l, - 0.
7
/V rJ
If
So if
Therefore
/Li-h iAtj G. V
V
then
must be
Q ( n Tz-V^f)
unless
is a maximal totally isotropic subspace of
(E,jp).
/V /V
A normal form for the decomposition of (E, (D ) is given by
Qj
Theorem 5:
!^/ = dim
and
(J) (
with
^
Ql i Q-^ totally isotropic,
^ 2. ^
^
for all
and
2 ^4.'
Proof:
and
From Theorem 3,
H j ~-Jt fALj^
As usual let
with (Jjl IJy1 j ^ (JlJu1 Jv2 )"
J l (j)
as specified.
the form
H1 , Qj and
E ~ A/, 69 Hz&
Put
Hf
H1- $
denote the function
G j= J ( 4 ^ " Pjiiu ^ .
The dimensions of
Every element
with
J Jvz
^ ( A ~ (J j) ®
have (0) intersection so
Qt
l(e /(
-- > k
and
and
are clearly
of E can be written in
J j lJ
and the spaces
E - H J J (D G1 ® Gf *
remaining relationships are verified by routine calculation.
The
Extending
— 40 —
each
■^kx -s .H/ :
by zero to all of E we obtain
is a topological isomorphism
A/^
-h ** .
J **J g
Q-^
^
In the next theorem we show that the completions of the (E5^ l / )
spaces coincide with the locally linearly compact spaces on which the
form
p
is continuous.
Theorem 6:
If (E,7" )' is a locally linearly compact space and if
the nondegenerate, bilinear form
(j)
is continuous (i.e. , Z
T
TZ -complete and ZT=
ZD
some totally isotropic I/ ) then E is
some linearly Z -compact
if dim
V /n
is finite.
is locally linearly
Proof:
Z
\/D U D F
U .
-compact, there is a linearly
\J is Z -closed so D —
And for finite dimensional
is a Z - zero neighborhood.
is a. linearly topologized space and
Since
D
is both a Z
is a topological sum and
gies (Ktithe E91, page 96).
To demonstrate that
Lemma:
If
isotropic space
Proof:
Let
Ij
Therefore
F , D D F
Therefore Z ^ Z ^ - Z A
topology is a linearly compact space, so
~DW D0
if and only
Zj1 V -compact.
-compact.
E91, page 98).
for
is complete then E
Since E is locally linearly Z
f-Z)? TjjDJjj)
Z^
TjfrD- Cji
Further
Conversely if (E,
-compact zero neighborhood
linearly Z
E"
DCZ IZ
for
and a
D
=
But
with the finer
IFjlD jj) ~ ~FIJ)
D jD
is
(Ktiethe
zero neighborhood,
is discrete for both topolo­
~C -L(J)D)'
T jD ~ Z j V we prove the more general
is a subspace of finite cpdimension in the totally
of E and if
69 /V,
[j
is
Since
J. -closed then Z^> I j ~ L j V2m•
Ij- is orthogonally closed and
" 41
H is finite dimensional we have tz, ~ ( 0 )~1~ ~ (Vj D H )
(V^n
K
IVj D H1Iq K
with
and
/L
— 1/^ 6
in the last equality,
= %
K
(\jMr)Hx)QS=Hj-
~ l/f^ ~f~ H
Taking orthogonal complements of the spaces
'JVj~ ( H Hj-)f)K **{Vf-}-H ) HlK~
/I A t
of H
is finite dimensional since it is an algebraic supplement
Vf, zero neighborhood is a
(i.e., V
fDF ~ F)(JV~hF)J). But we also have
So every basic
zero neighborhood
Vj
since
which completes the proof of the lemma.
Returning, to the proof of (iii), (E,"t^[9) is complete and
nondegenerate so, by Theorem 4, -Z)
T jD - Z f V
LjD
if
V
some space
D
is
-L -closed.
is of finite codimension in
D
VDF
then
V . Conversely if
IV7 for D
codimension, in V .
Conversely, if (E,£^A/) is complete, then since
lV=
VL
But jV
Hence
Vl=VI
U/ is
contains
Vj~ Closed5
Vf -bounded as we have already
is also linearly ^
shown in Theorem 3.
So by the lemma,
is of finite codimension in
which is of finite
<V> is
is linearly
shows that (E, TpVJ ) is locally linearly
I V -compact, which
Tjj IV -compact. '
The proof gives the following interesting corollaries.
The first
is an immediate consequence of Theorems 5 and 6.
Corollary I:
If (E,7T) is locally linearly compact and
continuous then E has a decomposition
form given in Theorem 5 with
Theorem 6.
JS j
( j j j <S_Z^ ) © FSf
(j) is
of the
the locally linearly compact
JS
of
- 42
Corollary 2:
TtyVJ-closed
~CpW)
If (E,
is complete and
subspace of E then either /4
A
is a semisimple
is discrete or d i m /4 ^
U f:
Proof:
1,0
By the previous theorem, (E,
IV-compact.
A / ) is locally linearly
/) is closed, /4 is locally linearly compact
Since
with respect to the induced topology.
Also
0
is continuous when
restricted to /4, so the previous corollary applies and
(3) (D 2 (BD j ))
/4 —
If dimZD2C AJfl
discrete.
then
d±mJ)T <f
Since
A
VDf
anc^ t^ie topology on
.
In this case
If on the other hand dim
dim
topology on
.Il-A
dim
Il^
Dn
is a
D2'Tl
And the topology on
T^.D)^
A
would be
is a linear zero neighborhood,
topological complement of
crete :as well.
Q
with the
is dis­
is the discrete topology.
HD 2. ^
^ (VJ0 then
^
ll'Jl.ll^ ^
so
0,
Applying known results.on locally linearly compact spaces we now
show that the dimensions in the decomposition of E given in Theorem 5
are unique in the following sense.
Theorem 7:
Hg
If dim
^ AJ,.,
and if E decomposes into
IV a linearly compact space isomorphic to -A
space then
CL- dim
Proof:
/V2
and dim ^
= dim
Ti
and
Q-
W ® Q-
with
a discrete
•
If (F, (T) is any locally linearly compact space which is
neither discrete nor linearly compact and if F decomposes into the sum
of subspaces
VJ
and
Q-
as described above, then dim
uniquely determined by F (0191, page 112).
Q-
and
(L
are
Now E is locally linearly
— 43 —
-compact since
V
is linearly compact as shown earlier.
On the
other hand E is not linearly ZT -compact as E is not linearly Z7 -bounded
dim
E+V / V —
dim
E /V ~
dim /7 ^
dim
^ A-J0 ,
And by
zV
Theorem 9 in Chapter III, E is not discrete.
The same theorem holds for E a locally linearly compact space
relative to the decomposition in part (ii) of Theorem 6.
Induced Maps
/V
/V
If (EjTT) is any linearly topologized space with completion (EjTT)
/V
then the tensor algebras T(E) and T(E) are linearly topologized spaces
when provided with the projective tensor product topologies.
completions denoted by T(E) and T(E).
tropic space
I/
If
They have
L -LS Vfor some totally iso­
then the projective tensor product topology on C(E) and
/V
C(E) is a Hausdorff linear topology.
Their completions are denoted by
/\J A/
/%/
C(E) and C(E).
We now consider various canonical maps between these
spaces.
Theorem 8:
For spaces as described in the previous paragraph, the
canonical maps of a vector space into its tensor and Clifford algebras,
of the tensor algebra onto the Clifford algebra, and of a vector space
into its completion induce the maps in figure I, and the diagram is
commutative.
Proof:
All the vector space under consideration are linearly
topologized.
By the definitions of the tensor algebra and projective tensor
— 44 —
Legend
y*
Imbedding (linear, bicon­
tinuous , and bijective
to the image)
Surjective, bicontinuous,
and linear
Zi Continuous and linear
C(B)
Figure I.
Relations Among a Vector Space, Its Tensor and Clifford
Algebras and Various Completions.
- 45
product topology, E is embedded in T(E) and E in T(E).
If
j/
totally isotropic, then as was proved in Chapter II the topology (0ZT on
C(E) induces the topology
ry
/v
It
V
ther ZT-
on E, so E — > C(E) is an embedding.
Fur-'
/v
so E — ^C(E) is also an embedding.
The quotient maps
T(E)— > C(E) and T(E)— ^C(E) are surjective, bicontinuous and linear
since the Clifford algebra's carry the quotient topologies.
E. :- - - - > TfE)
The diagrams
E
are commut ative.
Maps derived from
with Q on E).
f: E
-E.
The embedding f is metric (O' agrees
So reasoning as in Chapter III (cf. the introduction and
Theorem 6), f induces embeddings g: T(E)— ^T(E) and h : C(E)— »C(E) with
->E'
E
and
-4,(
V
v
T(E)
f
commutative.
and C£
>
+T(B)
'/
C(E)
D
v
-^C(E)
In addition g and h are algebra homomorphisms.
the usual quotient maps the combined diagram
^C(S)
So for
(fj
— 46 —
commutes.
therefore
GJ ~ /
For since
^
and
algebra T(E) , hence
J v Ti
and E^
~ ^ 2m }
^
agree on
J l Tj ~
Completion maps.
an^
(E) a
~^i ” ^ ^
1
set of generators of the
e
If u : E^— *
E
is a
continuous linear map and Ey
are linearly topologized spaces, then it is well known that
/V
/V
/XV
there exists a continuous linear map u: Ey-- f E^
E1
Ea
V
a
is commutative for
such that
A
the canonical map into the completion.
Further
/V
if u is an embedding, u is also
Applying this theorem the following diagrams are commutative:
-^C(E)
->T(E)
V
V
T(E)-
4* Cd)
v
TfEj-
xk
E
->gfE)
TH=)-
T(E)-
->T(E)
C(E)-
-+C(E)
c.X
S(E)
-=-OE)
\k
T(E)-C^--- »f(E) ' T(E) — ^ --- ,T(E)
' &t)
C(g)
The notations "e" and "c.l. 1 indicate embeddings and continuous linear
maps respectively.
- 47 Maps obtained by continuous extension from a. dense subset.
Another
well known extension theorem states.that if f and g are continuous
functions from a topological space X into a Hausdorff space Y and if f
and g agree on a dense subset of X then f = g.
From above we have the commutative diagrams
E —
^
—
» t (e )—
'
4/
F
c ■
# —
'4
'
T(E)
m e)
f
and
Xj/
> T(E)
»t
Since
the dense subset
f---- >fiE)
d-jL,
-j, <E)
-> TYEj
^1%
agrees with ^
of E;
->T(E)
V
ffE)
commutes.
A similar argument with T replaced by C shows that
commutes
^
48
Now consider
It is established that
@
,
(S) , (C) ,
Tracing through the diagram we find that
fore
(6) and
(E) commute.
~ ^
= ' $ 0 since they agree on the dense subset
/V
T(E) and the outer diagram commutes.
Applying the same sort of reasoning to
'
There­
(T(E)) of
“ 49 —
all the small sub diagrams are known to commute so ..
on
agrees with
^
UsI (E), and therefore
is commutative.
This concludes the proof of theorem 9.
If T(E) is a topological algebra then T(E), C(E) and C(E) are also.
This is the case for example when dim E = ,XJc and
closed totally isotropic subspace.
tient map
ZT^
^
I/ a
Under these circumstances the quo­
O' :T(E)<— > C(E) is an algebra homomorphism and induces an
algebra homomorphism
(T :T(E)— ^cf(E).
LITERATURE CITED
Bourbaki, N. Topologie generale, Ch. I, 2 and 3, ASI 1142, 1143, 3rd
ed., Herman, Paris, 1961.
_______ .
Algebre, Ch. 9, ASI 1272, Herman, Paris, 1959.
Chevalley, C. Fundamental concepts of algebra. Columbia University
Press, New York, 1956.
Dieudonne, J. La gdometrie des groupes classiques, 2nd ed., Springer,
Berlin-Gtittingen-Heidelberg, 1963.
Fischer, H. and H. Gross. Quadratic forms and linear topologies, I,
Math. Ann. 157 (1964), 296-325.
_______ , Uber eine Klasse topologischer Tensorprodukte, Math. Ann. 150
(1963), 242-258.
_______ , Quadfatische Formen und lineare Topologien, III, Math. Ann.
160 (1965.) , 1-40.
Kaplansky, I. Forms in infinite dimensional spaces, Ann-. Acad. Bras.
Ci., 22 (1950), 1-17.
Ktithe, G. • Topologische lineare Rtiume, I , Grundlehren, Springer, BerlinGtittingen-Heidelberg, 1960.
MONTANA STATE UNIVERSITY LIBRARIES
762
001 002 O
/
I
I
^op. 2
Miller, Vinnie (Hicks)
Linear topologies induced
by bilinear forms__________