Journal ol Geometry Voi.27 (1986)

0047-2468/86/020195-0751.50+0.20/0 1986 Birkh~user Verlag, Basel

ON A VECTOR-PRODUCT IN THE SEPARABLE, INFINITE-DIMENSIONAL

REAL HILBERTSPACE

Dedicated to Professor Martin Barner on the occasion of his

65th birthday

Rudolf Z. Domiaty

i. NOTATION.

H denotes a separable, infinite-dimensional real Hilbertspace,

( , ) denotes the inner product, and I] II the induced norm in

H. ~ e n I n s IN ~ be a fixed orthonormal basis in H. For every

A ~ H, [A] resp. - - A denotes the subspace generated by A resp.

the orthogonal complement of A. S = { a ~ H I I] a II = 1

denotes the unit sphere in H. If H = E ~ E l , then ~E : H ~ E

and ~EJ-: H - E i denote the canonicai projections.

2.INTRODUCTION AND RESULTS.

For every r e ~, H(r) denotes the Grassmann-manifold of all

oriented r-dimensional subspaces of H. H(r) is a connected C w-

manifold, and the manifold topology is metrizable [2, no. 5.2.6].

It can be shown that H(r) is locally homeomorphic to H.

DEFINITION. A continuous mapping n : H(r) -- S is called a normal

D El in H(r), if for every E e H(r), n(E) s . A normal in H(r) is

stric____~t, if n(E-) = - n(E) for every E e H(r) (E- denotes for

E a H(r) the same subspace, but provided with the reversed

orientation) .

PROPOSITION. There exists a normal in H(r) for every r e IN.

It is easy to see that the existence of a normal in H(r) is

196 Domiaty

equivalent to the existence of a certain r-vector,product in H.

A mapping V : H r ~ H is called a r-vector, product in H [%,

p.517], if the following properties are satisfied:

(i) V is continuous

(2) V(a I .... , a r) is perpendicular to al, a 2 .... and a r

(3) II v(a I ..... a r) II 2 = det((ai, aj)).

A r-vector-product V in H is called r-linear, if V is r-linear.

A r-vector-product V in H is called strict., if for ~ik a I~ and

a k a H, i, k=l ..... r, the condition

(4) V(~=ika k ..... ~rkak ) = det(~ij) V(a I ..... a r)

is satisfied. Trivially, every r-linear-vector-product in H is

strict. If V is a strict r-vector, product in H, and if we put

for every E e H(r) with the generators { a I ..... ar]

n(E) = V(a I ..... a r) II V(a I ..... a r II "I,

then n is a strict normal in H(r).

REMARKS. I. If H is finite-dimensional, dim H = d a IN, and if we

put G+(d,r) for H(r), then, via r-linear-vector-products in H,

the problem of the existence of a normal in G+(d,r) is com-

pletely solved by the following

THeOReM. ~C~NN [4] - ADAMS [I] - WHITEHEAD [11] - Z~N~ROWSKi

[12]. A normal n exists in G+(d,r) for exactly the following

cases :

(i) r = 1 and d = O [nod 2, (ii) r = d - i, (iii) r = 2 and

d = 7, and (iv) r = 3 and d = 8. In all these cases, strict

normals exist.

(See [3], [4], [8] - [12] as general references on this topic)~

2. An explicit formula for a strict normal n : H(1) -- S can

easily be given.

If we define

V : H ~ H; V(~" ~kek ) = ~2el - ~le2 + ~4e3 - ~3e4 + - .-,,

then V is a l-linear-vector-product in H [4, p.517]. Hence, if

we put for every E s H(1) with [a] = E,

n(E) = v ( a ) L l v ( a ) tL - 1

then n is a strict normal in H(1).

Domiaty 197

3. In the separable infinite-dimensional case, a r-linear-vec-

tor-product, r ~ 2, does not exist. By B. ECKMANN's reduction

theorem [4, p. 520], which indeed is independent of the dimen-

sion of H, it is sufficient to consider the case r = 2. Let us

assume that a 2-1inear-vector-product exists in H. Because of i { 1

H = [el] ( ~ ) e I = ~ e I + X I ~ x ~ e I ]

and

H ~ e I ,

there exists a 2-1inear-vector-product l i •

V : e I X e~ ~ e 1

By B. ECKMANN'S argument [3, p. 338], which doesn't need any

assumption of finite-dimensionality, a multiplication

�9 : H X H -- H

in H can be defined by

(~ e I + x).(~ e I + y) = (~ - (x,y))e I + =y + ~x + V(x,y).

It is obvious, that this multiplication is bilinear,

e I a H is a two-sided unit element, and the norm-product-rule

lla'b li= ilall llbjl is satisfied for a, b e H. But by a theorem of I. KAPLANSKY [5],

such a multiplication does not exist in H. Hence the assumption

that a 2-1inear-vector-product exists in H is false.

3. PROOF OF THE PROPOSITION.

For every E e H(r), there exists an open neighbourhood

U E =[ F e H(r) I F hE/t= {O~ , and ~EiF is orientation pre-

serving ] in H(r) .

u : UE- L(E'EI )' YE (F) = nEJL~

maps U E homeomorphical!y onto the separable Hilbertspace L(E,E L)

of all linear mappings of E into E ~. Here, L(E,E l) is provided

with the Hilbert-Schmidt inner product. The induced Hilbert-

Schmidt norm is in this case equivalent to the usual operator

norm in L(E,E ~) . i .

I E e H(r)~

is the disjoint sum, and : T- H(r)

198 Domiaty

(with ~-I(E) = E l~ H for every E s H(r)) is the natural pro~

jection. Now we consider the open covering ~U E I E s H(r)~ of

H(r). For every E ~ H(r) and F e U E we define

~EF = ~E ~ I F1 : FI ~ EL ,

each ~EF being a topological linear isomorphism. For every

E e H(r),

8 E : =-I(UE) UEX E i ; ~E(F, a) = (F,~EF(a))

is a bijection which commutes with the projection onto UEO For

every E, G s H(r) with UEG = U E~ U G ~ @ and F s UEG

-i L i ~GF r ~EF : E" ~ G

is a topological linear isomorphism. Finally,

~EG : UEG ~ L(EI ' GX ); WEG (F) = ~GF ~ ~E~

is a continuous mapping of UEG into the Banachspace of all con-

tinuous linear mappings of E • ~nto G • . Therefore, (T, ~, H(r))

is a Hilbertbundle [7, prop. 2, p. 43]. By a theorem of N.H.

KUIPER [6, ~ 5.2(i), p. 74] (T, ~, H(r)) is trivial, hence there

exists a continuous bundle isomorphism I of T onto the product

bundle (HX H(r), A, H(r)),

I : HXH(r) ~ T.

If h ~ s H, h ~ ~ O, then for every E s H(r), I(ho,E) s E i- {0].

Hence

n : H(r) -- S; n(E) = I(ho,E) li I(ho,E) iF 1

is a normal in H(r).

4. EXAMPLE: A STRICT 2-VECTOR-PRODUCT IN H.

The main result is the explicit construction of such a vector

product in H. Be ~ : I~X~ -~ a bijection with the properties

w(1, i) = 2 , w(2, 2) = 1

(5) ~(i,j) a ~ )~ IN - ~(2,2)~ : w(i,j) > max {i, jj .

We put

~ i , j e l N " e . * e . = e i J ~(i,j)'

Domiaty 199

and define a mapping * : H ~ H -- H by

"*" is billnear and satisfies the norm-product-rule

II a*bll = II a II II b II

for every a, b e H. Finally, we define a multiplication

�9 - H X H -- H, a �9 b = a*b - b*a.

LEMMA. i) The multiplication "." is continuous, bilinear and

skew (i.e., b �9 a = - a . b) in H.

2) For a, b e H, a. b = 0 iff a and b are linearly dependent.

3) For a, b s H and ~, ~ E JR, if a �9 b + ~a + ~b = O, then

a and b are linearly dependent.

The proof consists in straightforward explicit calculation of

the components (with respect to { en} ) of the respective ex-

pression, applying (5). The details are completely elementary,

hence they are omitted.

Now we put for every a, b a H

u = V i l a l l 2 l ib 11 2 - I (a ,b) l 2,

( a . b , b ) ( a , b ) - (a �9 b , a ) lib i[ 2 [Y(a'b)] 2 ' if a �9 b / O

~ (a,b)

O, if a0b = O

(a, b,a) (a,b) - (a �9 II 2 [y(a,b) ]2 ' if a. b / O

~(a,b)

[ O, if a.b = O

and

f : H X H - H; f(a,b) = y(a,b)[a, b + =(a,b)a + 8(a,b)b].

Then ( y(a,b) f(a,b), if f(a,b) ~ 0

O, if f(a,b) = O

is a strict 2-vector-product in H.

To show this, we put fl(a'b) = y(a,b) a �9 b,

200 Domiaty

f2(a,b) = y(a,b)~(a~b)a, and f3(a,b) = y(a,b)8(a,b)b. Trivially,

y(a,b) (_> O) and fl(a,b) are continuous in H. f2(a,b) is con-

tinuous, if y(a,b) ~ 0. On the one hand,

(6) y(a,b) = 0 iffa and b are linearly dependent.

On the other hand,

llf2(a.b)[l <_ lla'bil lJajj!Ibll Because of 17 and 2) in the lemma, f2(a,b) is therefore also

continuous if y(a,b) = O, i.e., f2(a,b) is continuous in H.

An analogous argument shows that f3(a,b) is continuous in H.

Hence

(7) f is continuous in H.

As a consequence of 3) in the lemma and (6) we have

(8) f(a,b) = 0 iff a and b are linearly dependent.

(6), (7) and (8) show immediately that V(a,b) is continuous~

if a and b are linearly independent. In this case,

llv(a.b) li = u This property together with the continuity of y(a,b) and with

(6) shows, that V(a,b) is also continuous, if a and b are

linearly dependent. Therefore V satisfies (i) and (3). An

elementary calculation gives

(f(a,b),a)= (f(a,b),b) = O,

and hence

(V (a, b) ~ a) = (V(a,b),b) = O.

Therefore V satisfies (2). A straightforward calculation shows

that for a, b e H and ~/),~" E IR we ha~e

=(~a + ~b, 2a + ~b) = 6"=(a~b) - 9 8(a,b)

~(#a +wb, f a +~b) = -);=(a,b) + /~(a,b)

y(~a +~b, 9a +~b) = !~-wfI u

Hence, V satisfies (4).

If we put for every E s H(2) with [a,b] = E,

n(E) = V(a,b)llV(a,b)IF 1

then n is a strict normal in H(2).

PROBLEM. Does there exist a bilinear continuous function

f ~ HXH-- H

such that

Domiaty 201

i y (a,b) f (a,b), IIf(a b> II V(a,b) =

O, if f(a,b) = O

is a vector-product in H %

if f(a,b) ~ O

REFERENCES

[i] J.F. ADAMS, Bull.Amer.Math. Soc., 64(1958) 279-282,

Ann.Math., 72 (1960) 20-104

[2] N. BOURBAKI, Varletes. Fascicule de resultats, ~ 1 -9 7.

Hermann, Paris 1971

[3] B. ECKMANN, Commentarii Math.Helv., 15(1942/43)318-339

[4] B. ECKMANN, Battelle Recontres 1967 (Ed.: C.M. DeWitt-

J.A. Wheeler, Benjamin Inc., New York-Amsterdam 1968),

p. 516-526

[5] I. KAPLANSKY, Proc.Amer.Math. Soc., 4(1953)956-960

t I [6] N.H. KUIPER, Varietes Hilbertiennes. L'Universite de

Montreal, 1971

[7] S. LANG, Differential Manifolds. Addison Wesley,

Reading, Mass., 1972

[8] W.S. MASSEY, Amer.Math.Month., 90(1983)697-701

[9] O. VALDIVIA, Contemporary Mathematics (Amer.Math.Soc.),

12 (1982) 331-338

[i0] B. WALSH, Amer.Math.Month., 74(1967)188-194

[ii] G.W. WHITEHEAD, Commentarii Math.Helv., 37(1962/63)

239-240

[12] P. ZVENGROWSKI, Commentarii Math.Helv., 40(1965/66)

149-152

Rudolf Z. Domiaty Institut fuer Mathematib (c) Technische Universitaet Graz Kopernikusgasse 24 A-8010 Graz, Austria

(Eingegangen am 30. Juli ~1986)