Chapter 14 Notation and Multilinear Mappings

CHAPTER 14 NOTATION AND MULTILINEAR MAPPINGS

Unless stated otherwise, U

locally convex spaces, and

IN, IR

E.

subset of

G

and

E

and

F

will denote complex

will denote a non-empty open denote respectively the sets of

natural numbers, of real numbers and of complex numbers. IN*

{1,2,3, ...).

denotes the set and

SC(E)

SC(F)

denote respectively the sets of conand

tinuous seminorms on

E

DEFINITION 14.1

m E N*.

Let

F.

ra(%;F)

E~ = EXE

all m-linear mappings o f

denotes the set of

x...~ E

times) into F,

(m

the operations of addition and scalar multiplication being defined pointwise. Ca(%;F)

Ca,(%;F)

denotes the subspace of

of all symmetric m-linear mappings. (%;F)

'as

for every

means that

xl,...,x

m E E

and every

set of all permutations of

If A E Sa(%;F), element

As

Thus

of

u E Sm, Sm

{1,2,...,m].

the symmetrization of

.Cas(%;F)

being the

defined by

155

A

is the

14

CHAPTER

156 for

x1,x2

,...,

xm E E .

We d e n o t e by

C(%;F)

v e c t o r subspaces of

and

ga(%;F)

Xs(%;F)

respectively the

gas (%;F)

and

c o n s i s t i n g of

c o n t i n u o u s mappings.

m = 0,

For

we d e f i n e

La(OE;F)

:= L(OE;F) := Ss(OE;F) := F

= A

A

for

E

:= SaS(OE;F) :=

a s v e c t o r s p a c e s , and we s e t

~,(OE;F).

Ak+As

i s a pro-

which maps

S(%;F)

I t i s e a s y t o s e e t h a t t h e mapping j e c t i o n of onto

ea(%;F)

Ss(%;F)

onto

= S(%)

write

Sa(’E;F)

t h e spaces

m

for e v e r y

E

= ga(E;F)

gas

ga(”’E),

= Cs(%).

gs(%;C)

and

and

X(’E;F)

,...,1,

E

(c,

s o t h e mapping

then

A(X1

and

f i n e d as follows:

m = 0,

if

Ax

0

we

If E = C,

Ls(mC;F) a r e

A(1,

...,I),

and

i s an isomorphism.

A E Sa(%;F)

Let

= Sas(%),

For if

= X1...l,

.,.,l) E F

A+-A(l,

DEFINITION 1 4 . 2

,..,X m )

we

m = 1,

= L(E;F).

a l l n a t u r a l l y isomorphic w i t h one a n o t h e r .

hl

(%el

For

gas (mC;F), S(%;F)

Ca(mG;F),

F = C

I n the case

[N.

s ~ ( ? E ; c )=

w r i t e for s i m p l i c i t y ,

L(%;C)

Sas(%;F)

and

x E E.

= A E F.

Axm

m E

If

i s deIN”,

m times

...,x ) . Sa(%;F), x1 ,...,xk E 7----L_\

Axm More g e n e r a l l y , l e t

E N,

m,nl,n2,,..’nk n Axl’.

If

.

n .xkk

A

E

and

= A(x,x,

n = n +n2 1

i s d e f i n e d as f o l l o w s :

m = n > 0,

+...+ If

n

E,

s m.

m = 0,

k E IN”,

Then

nl Axl

.,

n .xkk = A.

157

NOTATION AND MULTILINEAR MAPPINGS

where

each

xi

is repeated

by

5

n n 1 k (AX1 * * * X k)(Y1,***tYm-,) ~

1

n l Axl

Then

, Y ~ ,E ~E

9

...

nk

xk

is symmetric if

times

and

0,

is defined

nk times

,. - J

'(X~Y...YX~Y***,~,*~. ,X~,Y~Y***YY,-,)

nl in each casey and Ax1

E Xa(m-%;F) A

...x2

,-A

=

ni >

times if n m > n, Axl

ni = 0. And if

omitted if

where

ni

is symmetic, and continuous if

...xn

k

k

is

A

continuous. LEMMA 14.1 (Newtonfs Formula). x1

,...,xk E

E,

k E N*,

m,n E IN

A(x 1+X 2+...+x~)~ =

PROOF: A(xl+

+...+nk

The case

...+xk)

n

then f o r each

n = 0

A E Las(%;F),

and

n c m.

' ...

the sum being taken over all n = n,

Let

nl!

n

n!

nk!

nl,...,nk E

is trivial.

Then

If

...+x~,...~x1+...+x (y, ,...,ymen) E Em-n,

= A(xl+

ax^ l [N

nk k '

f o r which

m = n > 0, then

k).

If m > n >

0,

In each case, the given expression may be expanded, using the fact that

A

is multilinear and symmetric, and it is easy to

see that the number of occurrences of.:xA the number of permutations of

xl,.. .,xk,

.

.xF

where

is equal to x1

is re-

158 peated

CHAPTER

nl

times,

this number is

...,xk

... n!

nl!

!nk!

14

i s repeated

nk

times.

t h e lemma is proved.

Since

Q.E.D.