Chapter Twelve Holomorphically Significant Properties of Locally Convex Spaces

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CHAPTER TWELVE HOLOMORPHICALLY SIGNIFICANT PROPERTIES OF LOCALLY CO?IVEX SPACES

12.1 P r e l i m i n a r i e s . E and F s t a n d f o r complex l o c a l l y convex spaces. U i s a non-void open subi s t h e l i n e a r space o f a l l holomorphic f u n c t i o n s f : U 4 F.

s e t o f E and W(U,F) N o t a t i o n 12.1.1: +F.

Fc(U,F)

T G ( U , F ) i s t h e space o f a l l G-holomorphic mappings f:U-

i s t h e space o f a l l f k Z G ( U , F ) such t h a t f i s bounded on com-

(U,F) i s t h e space o f a l l f & g G ( U , F ) such t h a t f i s hY continuous on compact subsets o f U. q D ( U , F ) i s t h e space o f a l l f6%?c(U,F) p a c t subsets o f U.

such t h a t $ m f ( x ) E P ( m E , F ) P r o p o s i t i o n 12.1.2:

f o r a l l rn and x i n U.

~ ~ ( u , F ) c ~ P ~ ( u , F ) c ~ P ~ ~ (CuX, ~F () U , F )c%',(u,F).

Proof: Only t h e i n c l u s i o n %,(U,F)C$( (U,F) hY p a c t subset o f U, x a p o i n t o f K and f c g D ( U , F ) . such t h a t V : = ( z t E :

r(z-x)&a-l)

I f q C c s ( F ) , TAYLOR'S remainder f o r m u l a

shows t h a t , f o r a l l z C V A K and m=1,2,.., M/am(a-l),

Sni:e

t h i s i s true f o r

f i s continuous a t x as d e s i r e d . / /

I f E i s a k-space (E,3.3.21),

(E,3.3.25)

m

q(f(z)-f(x))lq( z(j!)-%'f(x)(z-x))

where M : = s u p ( q ( f ( ( l - b ) x + b z ) ) : z € K ; l b l = a ) .

arbitrary a ) l ,

there i s r c c s ( E )

i s c o n t a i n e d i n U and (1-b)x+bzCU f o r a l l

b C C w i t h l b l & a and f o r a l l z i n V .

+

needs p r o o f . L e t K be a comGiven a ) l ,

e v e r y open s u b s e t U o f

E i s a g a i n k-space

and hence t h e f o l l o w i n g r e s u l t i s immediate.

P r o p o s i t i o n 12.1.3:

I f E i s k-space,

P r o p o s i t i o n 12.1.4:

Every (DFM)-space i s k-space (see 8.9).

then g(U,F)

=

ZhY(U,F).

I n particular,

(LS)-spaces and quasi b a r r e l l e d (DcF)-spaces a r e k-spaces. P r o o f : By 8.3.40(a),

(DFM)-spaces a r e u l t r a b o r n o l o g i c a l and, c l e a r l y , (DF)-

spaces. According t o t h e theorem o f BANACH-DIEUDONNE (see K l , $ Z l . l O . ( l ) ) ,

a

f u n c t i o n d e f i n e d on a (DFM)-space w i t h v a l u e s i n a Banach space i s c o n t i n u o u s

450

BARRELLED LOCALLY CONVEX SPACES

i f and only i f i t i s continuous on compact subsets of E and t h e conclusion follows. 8.5.26 shows t h a t (LS)-spaces a r e strong duals of FS-spaces and hence (DFM)-spaces. On the o t h e r hand, i f a (DcF)-space i s q u a s i b a r r e l l e d , i t i s a (DF)-space since i t i s a (gDF)-space. I t i s enouah t o apply 8.3.49 and 8 . 3 . 1 0 ( i i ) t o ensure t h a t i t i s a (DFM)-space.

I/

Definition 1 2 . 1 . 5 : A l o c a l l y convex space E i s holomorphically bornolog i c a l ( s h o r t l y h-bornological) i f a ( ( U , F ) = XC(U,F)f o r every F and U .

Tfc(U,F)

i s t h e space of a l l ftaPG(U,F) such t h a t f i s bounded on f a s t

compact subsets of U. A l o c a l l y convex space E i s holomorphically ultraborf o r every F nological ( s h o r t l y h - u l t r a b o r n o l o g i c e ) i f y(U,F) = Zfc(U,F) and U. By 6.1.16 ( r e s p . , 6.1.22) h-bornological ( r e s p . , h-ultrabornolog cal ) spaces a r e bornological ( r e s p . , ul trabornological ) . Theorem 12.1.6: ( i ) I f E i s metrizable, then E i s h-bornoloqical ( i i ) I f E i s a (DFM)-space, then E i s h-bornological. Proof: ( i ) Let U be a non-void open subset of E and f C y c ( U , F ) . By 0.5.9, i t i s enough t o show t h a t f i s amply bounded. I f t h i s i s not t h e case, t h e r e i s p G c s ( F ) and a vector x 6 U such t h a t pof i s not bounded on any x-nghb

contained in U. Since E i s metrizable, s e l e c t a countable basis ( V n : n = 1 , 2 , . ) of x-nghbs w i t h VnCU and vectors x ( n ) C V n w i t h p ( f ( x ( n ) ) ) & n f o r each n . Since ( x ( n ) : n = 1 , 2 , . . ) converges t o x in E , f i s not bounded on the compact s e t ( x ) U ( x ( n ) : n = 1 , 2 , .. ) , a contradiction. ( i i ) Let U and f be as above. By 12.1.3 and 1 2 . 1 . 4 , Y ( U , F ) = X h Y ( U , F ) f o r each F which may be assumed normed ( F : = ( F , l l . l l ) ) . By 1 2 . 1 . 2 , i t i s enough t o show t h a t f C y D ( U , F ) . Given a compact subset K of E there i s a > O such t h a t x+byGU f o r each y c K and b C C w i t h ( b l d a , x beinq a fixed vector of K . CAUCHY's integral formula (0.5.10) shows t h a t S m f ( x ) ( y )E ( m ! ) a - m . E x ( f ( x + b y ) : y c K ; / b l = a ) and hence 5 " f ( x ) i s bounded on K f o r each m. The followinq claim f i n i s h e s t h e proof: clajm: Let E be a b a r r e l l e d bornoloqical (DF)-space and l e t F be a space. I f ~ s 6 7 ( ~ E , Fi )s bounded on compact subsets of E , then D E P ( " E , F ) . Indeed, t h e p o l a r i z a t i o n formula ( 0 . 5 . 1 ) shows the existence of a unique A in 'ajs("E,F) such t h a t 3 = p and A i s bounded on compact subsets 3f Em. I t i s enough to prove t h a t A is continuous. Since E i s a b a r r e l l e d (DF)-space,

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45 1

i t s u f f i c e s t o show t h a t A is s e p a r a t e l y continuous (K2,$40.2.(11)). Moreover, s i n c e A i s symmetric, i t i s enough t o check t h a t , i f we f i x a 2 , . . , a m i n E, the l i n e a r mapping f : E + F defined by f(x):=A(x,a2,..,am) i s continuous. B u t this is immediate s i n c e f is bounded on compact subsets of E and E i s bornological (6.1.16)

-//

Observation: The notion of h-ultrabornological space appears i n GALINDO, GARCIA,MAESTRE ,( 1) where they prove t h a t Fr6chet spaces a r e h-ul trabornol ogical and so a r e (LS)-spaces. I t i s not c l e a r i f (DFM)-spaces a r e h - u l t r a bornological. Proposition 12.1.7: I f F i s complete and E i s h-bornological, t h e space (%(U,F),to) i s complete. Proof: The space M(U,F) of a l l functions f:U- F bounded on compact s u b s e t s of U i s complete i f endowed w i t h the compact-open topoloqy to s i n c e F is complete. Recall t h a t holomorphic functions a r e continuous, hence bounded on compact sets and t h e r e f o r e V ( U , F ) C M(U,F). Let us check t h a t g ( U , F ) i s closed i n ( M ( U , F ) , t o ) : I f L i s a net i n ';14(U,F) convergina t o a c e r t a i n f i n (M(U,F),t,), WEIERSTRASS theorem shows t h a t f i s G-holomorphic since F is complete. Since E i s h-bornological , f t y ( U , F ) a s d e s i r e d . / / Observation 12.1.8: ( a ) compact subsets cannot be replaced by bounded subsets i n the d e f i n i t i o n of h-bornological space: Although holomorphic functions a r e bounded on compact s e t s , they can be unbounded on bounded s e t s as t h e following example shows S e t E:=co endowed w i t h i t s usual sup-norm topology and d e f i n e f : E - + C , m f ( x ) : = z ( x ( m ) ) m f o r each x:=(x(m):m=1,2,. . ) i n E . Clearly, f E y ( E , C ) . I f y, i s the vector of E having m of i t s coordinates equal t o 1 and 0 t h e remain i n g ones, t h e n lly,,,ll=l, b u t f(ym)=mand hence f i s unbounded on t h e closed u n i t ball of E . Due t o a deep r e s u l t i n Banach Space Theory (DI,4.9), i n every i n f i n i t e dimensional complex nonned space there i s an entire complex-valued function on i t which i s unbounded on some bounded subset of the space. ( b ) F cannot be replaced by C i n t h e d e f i n i t i o n of h-bornological space. In 2.5.2 we gave a topology s on t h e space co(A) f o r which t h e space was transseparable b u t not separable. Set t f o r t h e usual sup-norm topology on co(A). Clearly, s i s s t r i c t l y c o a r s e r than t and i t can be checked t h a t

BARRELLED LOCAL L Y CON VEX SPACES

452

both topologies share the same bounded sets, s i n c e they share the same compact s e t s . Moreover, ( c o ( A ) , t ) i s h-bornological. F i r s t , we check t h a t ( c o ( A ) , s ) i s not bornological (and hence not h-bornological): Indeed, the 1 inear canonical i n j e c t i o n (co(A) ,s)--t (co(A) ,t) maps bounded s e t s in bounded s e t s b u t i t i s not continuous. JOSEFSON,(l) proves t h a t , i f U i s a non-void open subset of ( c o ( A ) , s ) (and hence open in ( c o ( A ) , t ) ) , then g(U)i s t h e same regardless of t h e f a c t t h a t C o ( A ) i s endowed w i t h t h e topology s o r t . Now, i f f : U - + C belonqs t o y c ( ( U , s ) , C ) ( = z C ( ( U , s ) )), then f c g ( ( U , t ) ) , since ( c o ( A ) , t ) i s h-bornological and t h e r e f o r e f C g ( (U,s)). ( c ) f a s t compact s e t s i n t h e d e f i n i t i o n of h-ultrabornological spaces can be replaced by f a s t convergent sequences ( s e e 6 . 1 . 2 2 ) . Now we turn our a t t e n t i o n t o t h e holomorphic analogon of b a r r e l l e d and

quasi barrel led spaces. The fol lowing concepts will play an important r z l e : Definition 12.1.9: Let E and F be spaces and 3; a family of mappinqs f : W F. i s s a i d t o be amply bounded a t x , x being a vector of U, i f f o r every q C c s ( F ) , t h e r e i s a x-nghb V contained in U such t h a t s u p ( q ( f ( y ) ) : y r V;f&) i s f i n i t e . F i s s a i d t o be amply bounded i f i t i s amply bounded a t every vector of U .

5

If the topology of F is described by a s i n g l e seminorm, we w r i t e l o c a l l y bounded --

a t x o r l o c a l l y bounded f o r the family F i n s t e a d of amply bounded a t

x o r amply bounded r e s p e c t i v e l y . Observe t h a t , i f F i s amply bounded i n

X(U,F),then % i s

bounded i n

(X(U,F), t o ) . Proposition 12.1.10: Let F b e a subset of % ( U , F ) . The followinq condit i o n s a r e equivalent: i s amply bounded (i) ( i i ) $ i s bounded in ( ~ ( U , F ) , t o ) a n d i t i s equicontinuous i s pointwise bounded and equicontinuous. (iii) Proof: I f ( i ) holds, q i s c l e a r l y bounded in ( g ( U , F ) , t o ) . Given x in U and q in c s ( F ) , there i s p ( c s ( E ) and a p o s i t i v e constant M such t h a t , i f B:=(y t E : p ( y ) 4 b ) with bCl,@then V:=x+B i s contained in U; s u p ( q ( f ( z ) ) : z L lAk V;fcp)_LMand lim ( q ( f ( z ) d f ( x ) ( z - x ) ) = 0 uniformly f o r z &V and

F

z(k')0

CHAPTER 12 f in

453

F. According

11 ( m ! ) - ' P f ( x )

11 p,q:=

ce p(z-x)_C b c 1, on:

I(m!

)-'Gmf(x)

\\

t o t h e CAUCHY i n e q u a l i t i e s (0.5.11), sup(q((m!)-l~mf(x)(y):

one has

yCB)LFul. Then, i f z CV and hen00

g

has t h a t q ( f ( z ) - f ( x ) ) 6 qq((m!)-l$mf(x)(z-x))

p y . , ~ ( p ( z - x ) ) mC M $bm

(bM)/(l-b)

=

and t h u s F i s equicon-

t i n u o u s . T h e r e f o r e (ii) i s satisfied. h o l d s . There i s a poC l e a r l y , ( i i ) i m p l i e s ( i i i ) . Now suppose t h a t (iii) s i t i v e c o n s t a n t M such t h a t , i f x C U and q G c s ( F ) , t h e n q ( f ( x ) ) h M f o r each f in

3;. Moreover,

t h e r e i s a x-nghb V c o n t a i n e d i n U such t h a t q ( f ( z ) - f ( x ) ) q ( f ( z ) ) Ll+M and (i) holds.

h l f o r f i n y a n d z i n V. A c c o r d i n g l y ,

//

L e t U be a non-void open subset o f a space E and l e t

P r o p o s i t i o n 12.1.11:

F be a seminormed space. I f 3; i s a p o i n t w i s e bounded subset o f % ( U , F ) i f g:U+

and

lm( 3 , F ) i s d e f i n e d b y g ( x ) ( f ) : = f ( x ) f o r each x i n U and f i n

the following conditions are equivalent:

(i) i s l o c a l l y bounded ( i i ) g i s l o c a l l y bounded

(iii) g i s holomorphic P r o o f : C l e a r l y , (iii) i m p l i e s (ii). I f ( i i ) holds, f i x x i n U. There i s a x-nghb V and a p o s i t i v e c o n s t a n t M such t h a t s u p ( q ( g ( y ) ) : y r V)d M, q b e i n g t h e seminorm which d e s c r i b e s t h e t o p o l o q y o f F. Set q* f o r t h e seminorm which d e s c r i b e s t h e t o p o l o g y o f 1Oo(S,F). Then, s u p ( q ( f ( y ) ) : y c V , f c S ) S M a n d y i s l o c a l l y bounded a t x . Thus, (i) i s satisfied. Suppose t h a t ( i ) h o l d s and l e t us c h e c k ( i i i ) , t h a t i s l e t us show t h a t q i s l o c a l l y bounded and G-holomorphic.

By assimption, o u r f i r s t c l a i m i s t r u e

s i n c e , i f x i s a f i x e d v e c t o r o f U , t h e r e i s a x-nghb \$CU and a p o s i t i v e c o n s t a n t M such t h a t s u p ( q ( f ( y ) ) : y t W,fCs) GM and hence sup(q*(q(y)):ycW),( M.

NOW, l e t S be a f i n i t e - d i m e n s i o n a l subspace o f E i n t e r s e c t i n g U and t a k e

x(SnU.

Determine a 0-nghb V i n E such t h a t x+VCW. By t h e CAUCHY i n e q u a l i -

t i e s (0.5.11),

f o r a c e r t a i n p o s i t i v e constant

sup(q((mI)-'Gmf(x)(y):ycV)LN

N. Then, l i m q ( f ( x + y ) - ? ( m ! ) - ' a m f ( x ) ( y ) )

=

0 u n i f o r m l y on V. L e t K be an

a b s o l u t e l y convex compact 0-nqhb i n S such t h a t x + K C ( x + V ) A S . $',F) s i t i v e i n t e g e r rn, s e t Pm':E -lm( y gV, q * ( P m ' ( y ) )

< N and

For each po-

f o r Pm'(y)(f):=(mI)-l~mf(x)(y).

hence Pm' € P("E,F)

by 0.5.2.

For

Hence i t s r e s t r i c t i o n

Pm t o S belongs t o P(mS,F). Our c o n c l u s i o n f o l l o w s i f we show t h a t t h e l i m i t m

( f o r n ) o f q * ( g ( x + y ) - z P m ( y ) ) equals 0 u n i f o r m l y on ( Z - ' ) K , m

v a l e n t t o show t h a t l i m s u p ( q ( f ( x + y ) - t(mi)-l$mf(x)(y):

fe3)

which i s equi=

0 uniformly

BARRELLED LOCALLY CONVEXSPACES

454

Definition 1 2 . 1 . 1 2 : A space E is said to be holomorphically barrelled (shortly h-barrelled) if, for every non-void open subset U of E and every space F, each subset of v ( U , F ) , which is bounded on finite-dimensional compact subsets of U, is amply bounded. A space E is said to be holomorphically quasibarrelled (shortly h-quasibarrelled) if the same condition holds replacing "finite-dimensional compact subsets" by "compact subsets" of U. Observation 12.1.13: For a given family 3;'in X(U,F) the followinq conditions are equivalent: (i) y is bounded on finite-dimensional compact subis bounded on every fast convergent sequence in U: sets o f U and (ii) Indeed, clearly, (ii) implies (i). If (i) holds, let (x(n):n=1,2,..) be a fast convergent sequence in U converging to some x in U. There is a Banach disc A in E such that the sequence converges to x in EA ' Since F/(UAEA) = (f/(UnEA) : f € z ) C y(UnE,,F) is bounded on finite-dimensional compact subsets of U A E A and EA is h-barrelled (see our next proposition 12.1.18), f /(UAEA) is amply bounded and hence is bounded on the compact subset (x)U(x(n):n=1,2,..). By 4.1.3, 4.1.4 and 12.1.10 one has Proposition 12.1.14: Every h-barrelled space is barrelled and every hquasibarrelled space is quasibarrelled. As in the linear setting, we may replace F by the field C in 12.1.12 in sharp contrast with the h-bornological case, see 12.1.8(b). Proposition 12.1.15: A space E is (h-barrelled) h-quasibarrelled if and which is bounded on (finite-dimensional) only if every subset o f y(U), compact subsets of U, is locally bounded. Proof: Only for the h-barrelled case. Take any space F and let %be a subset of X ( U , F ) which is bounded on finite-dimensional compact subsets of U. s e t % : = ( v E F ' : l < y , v > \ < q ( y ) for each y in F). Accordinq to Given q(cs(F), HAHN-BANACH theorem, q(y) = sup( Idy,v>l :vcfl) and set y':=(vof:vtR , f c F )

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which i s contained i n %(U) by 0.5.6. Clearly, 5 ' i s bounded on f i n i t e - d i mensional compact subsets of U and hence !Xi i s l o c a l l y bounded by assumpt i o n . T h u s , 'f: is amply bounded.// Proposition 12.1.16: A space E i s h-bornological i f and only i f E i s hquasi barrel 1 ed and qC( U)= 'de ( U ) . Proof: First assume t h a t E i s h-bornological and l e t

x(U,F)

'3: be

a subset of

which i s bounded on compact subsets of U. We may assume F normed. Consider g:U-l@(%,F) defined by g ( x ) ( f ) : = f ( x ) . g i s well-defined and i t i s bounded on compact subsets of U . I f S is a finite-dimensional subspace of E i n t e r s e c t i n g U, g:UAS-+l@( F,F) i s l o c a l l y bounded and hence holomorphic by 12.1.11. T h u s , g i s G-holomorphic and hence holomorphic, since E i s hbornological. 12.1.11 again implies t h a t !X i s l o c a l l y bounded. a member of '&(U,F). Given q c c s ( F ) , s e t % : = ( v c F ' : Now take f : U - F i s G-holomorphic and bounded on compact I(y,v>JSq(y);yEF). Each v.f,v&, subsets of U and hence vof E%(U) f o r v i n & % . Set T : = ( v o f : w g ) cy(U). $ i s bounded on compact subsets of U and hence l o c a l l y bounded, since E i s

:v&) , qaf i s l o c a l l y bounded. T h u s , h-quasibarrelled. Since q(y)=sup(l4yYv)) f i s amply bounded (and G-holornorphic), i . e . f i s holomorphic by 0.5.9. T h u s , E i s h-bornological a s desired. //

Let ( E , t ) be a space and s e t E ' : = ( E , t ) ' . Recall t h a t ( E , t ) i s a Mackey space if one of the following conditions i s s a t i s f i e d : ( i ) t i s the f i n e s t l o c a l l y convex topology of the dual p a i r ( E , E ' ) ; ( i i ) f o r every space F, the spaces L((E,t),F) and L ( ( E , t ) , ( F , s ( F , F ' ) ) coincide and ( i i i ) f o r every space F and f o r every f < z ( E , F ) , f belongs t o L ( E , F ) i f vofEE' f o r each v E F ' . Definition 12.1.17: A space E i s said t o be a Mackey-Nachbin space (shortl y a MN-space) i f , f o r every non-void open subset U of E and every space F, H(U,F) coincides w i t h H(U,F,), where Fs stands f o r ( F , s ( F , F ' ) ) . By 0 . 5 . 1 4 ( i i ) , every finite-dimensional space i s a MN-space and our comments above show t h a t every MN-space i s a Mackey space. Proposition 12.1.18: I f E is h-quasibarrelled, then E s a MN-space. Proof: Let U be a non-void open subset of E , F a space and f : U 4 F a function such t h a t v o f E'#!(U) f o r each v i n F ' . For every f i n te-dimensiona

BARRELLED LOCALLY CONVEXSPACES

456

subspace S o f E i n t e r s e c t i n g

U one has t h a t

f

E

H(UAS,F).

It suffices t o

show t h a t f i s amply bounded. L e t K be a compact subset o f U. C l e a r l y , f ( K ) i s bounded i n Fs

and hence bounded i n F. Thus, f i s bounded on compact

subsets o f U . Given any s t r o n g l y bounded bounded subset A o f F ' , t h e f a m i l y

"s : = ( v o f : v f A

) C Y ( U ) i s bounded on compact subsets o f U and hence 3: i s

amply bounded, s i n c e E i s h - q u a s i b a r r e l l e d . Takinq q c c s ( F ) and A : = ( v C F ' : ((y,v>[Cq(y) , y c F ) , one has t h a t q o f i s l o c a l l y bounded, hence f i s amply bounded as d e s i r e d .

/I

P r o p o s i t i o n 12.1.19:

A space E i s h - b o r n o l o g i c a l

y(U).

MN-space and T c ( U ) =

P r o o f : N e c e s s i t y f o l l o w s f r o m 12.1.18.

i f and o n l y i f E i s a

Now assume t h a t f € g C ( U , F ) f o r g i -

ven F and U. F o r each V E F ' , v e f i s G-holomorphic and bounded on compact subs e t s o f U. According t o h y p o t h e s i s , v o f E % ( U ) f o r each v t F ' . Since E i s a MN-space,

f EH(U,F)

which i n t u r n i m p l i e s t h a t f i s amply bounded and t h u s

f € % (U,F) ./,

P r o p o s i t i o n 12.1.20:

L e t 9:E-c H be a continuous open s u r j e c t i v e l i n e a r

mapping. I f E i s r e s p e c t i v e l y h - u l t r a b o r n o l o g i c a l

, h-barrelled,

h-bornologi-

c a l , MN-space, t h e n so i s H. P r o o f : We c o n s i d e r o n l y t h e h - b o r n o l o g i c a l case. L e t U be a non-void open subset o f H and f:U--.F

a G-holomorphic f u n c t i o n bounded on compact subsets

of U. I f K i s a compact subset o f E , g(K) i s compact i n H and hence f o g ( K ) i s bounded i n F. Thus, t h e G-holomorphic mapping fog:g-l(U)-, F i s bounded 1 on compact subsets o f t h e open subset g- (U) o f E and hence f o g i s amply 1 and x c g - ( y ) , y b e i n g a p o i n t o f U, t h e r e i s a x1 nghb V c o n t a i n e d i n g- (U) such t h a t q o ( f - g ) i s bounded on V and hence q o f

bounded. Given q ( c s ( F )

i s bounded on t h e y-nghb g(V). Thus, f i s amply bounded.

C o r o l l a r y 12.1.21:

h-ultrabornological

/I

, h - b o r n o l o g i c a l , h - b a r r e l l e d and

MN-spaces a r e s t a b l e by H a u s d o r f f q u o t i e n t s and hence b y complemented subspaces. P r o p o s i t i o n 12.1.22:

L e t G be a dense subspace o f a space

E.

I f G i s h-

( q u a s i ) b a r r e l l e d , t h e n so i s E. P r o o f : L e t U be a non-void open subset o f E and

3: a

subset o f % ( U , F )

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457

bounded on f i n i t e - d i m e n s i o n a l compact subsets o f and d e f i n e f*(z):=f(z+x) Set y * : = ( f * : f € $ ) ,

U. Take q r c s ( F ) and x c U

f o r a l l z E U - x and f C 2 ( U , F ) .

C l e a r l y , f*@U-x,F).

V:=(U-x)A G and d :=(f*/V w i t h f*t: F*). C l e a r l y , & i s

contained i n X(V,F)

and i t i s bounded on f i n i t e - d i m e n s i o n a l

o f V. Since G i s h - b a r r e l l e d , #

compact subsets

i s amply bounded. Since OEV, t h e r e i s a po-

M and an open 0-nghb LM f o r z i n W A G and f i n s . Since WnG, q.f(z+x) = q o f * ( z ) S V f o r z i n s i t i v e constant

W i n E such t h a t

WAGCV

and (q.f*/V)(z)

W i s contained i n the closure i n E o f W and f i n

3;. Thus, 9.y i s bounded on

xtW and t h e c o n c l u s i o n f o l l o w s . The h - q u a s i b a r r e l l e d case i s s e t t l e d analogouslY * / /

12. 2 Examples

.

We have a l r e a d y shown t h a t m e t r i z a b l e and (DFM)-spaces a r e h - b o r n o l o q i c a l N (12.1.6). I n p a r t i c u l a r , C and C") a r e h - b o r n o l o g i c a l . h - b o r n o l o g i c a l spaces have poor s t a b i l i t y p r o p e r t i e s as t h e f o l l o w i n g r e s u l t shows

1

P r o p o s i t i o n 12.2.1:

;

E be

m e t r i z a b l e . Then ExEIb i s h - b o r n o l o g i c a l

i f and o n l y i f E i s norn j ( E l b i s t h e s t r o n g dual o f E ) .

P r o o f : I f E i s normec i f ExEtb i s h - b o r n o l o g i c a l

so i s E l b and ExEIb i s h - b o r n o l o g i c a l . Conversely,

, e v e r y ?-homogeneous p o l y n o m i a l i s c o n t i n u o u s if

i t i s bounded on compact subsets o f ExEtb. To r e a c h o u r c o n c l u s i o n , we p r o -

ceed as i n DIY1.23,p.17 d e f i n i n g p(x,u):=

Lx,u>

which i s bounded on compact

s e t s s i n c e each bounded s e t o f E l b i s E-equicontinuous. Thus, p i s c o n t i n u o u s and hence t h e r e a r e 0-nghbs V and W i n E and E l b r e s p e c t i v e l y such t h a t lp(Y,W)I 5 1 from where i t f o l l o w s t h a t V C W " . C o r o l l a r y 12.2.2:

Thus, E i s normed.

//

The space CNxC(N) i s n o t h - b o r n o l o g i c a l and so i s e v e r y

space which c o n t a i n s CNxC(N) complemented. P r o o f : See 1 2 . 2 . 1 and 12.1.21.// P r o p o s i t i o n 12.2.3:

(Assuming t h e Continuum H y p o t h e s i s ) C(')

i s h-borno-

l o g i c a l i f and o n l y i f I i s c o u n t a b l e . Proof: I f

I

i s countable, C(')

i s a (LS)-space and hence a (DFM)-space.

I f I i s uncountable, t h e Continuum Hypothesis shows t h a t c a r d ( 1 ) a c . Our d e s i r e d c o n c l u s i o n f o l l o w s i f we c o n s t r u c t p ~ @ ( ~ c ( ' ) \ ) P( 2 C ( 1 ) ) : Indeed,

BARRELLED LOCAL L Y CON VEX SPACES

458

i t i s bounded on compact subsets

such a polynomial i s c l e a r l y G-holomorphic, o f C(')

(compact subsets a r e f i n i t e - d i m e n s i o n a l ) and i t i s non-continuous.

S e l e c t an i n f i n i t e c o u n t a b l e subset J1 o f

C l e a r l y , F,G and FxG a r e complemen-

c a r d ( J 2 ) = c . D e f i n e F:=C(J1) I T \ and G:=C(J2). t e d subspaces o f E:=Cll'.

and a subset J2 o f I \J1 w i t h

I

Set q(l):FxG-+F,

q(2):FxG-G

t h e canonical p r o j e c t i o n s . The mappings q , q ( l )

and q:E-FxG

for

ar?d q ( 2 ) a r e continuous. The

a l g e b r a i c d u a l o f F can be i d e n t i f i e d a l g e b r a i c a l l y w i t h G. D e f i n e p:E+

2

C

belongs t o @ ( E). Since E induces on

by p ( x ) : = .p

FxG i t s own s t r o n g e s t l o c a l l y convex topology, which i n t u r n i s t h e t o p o l o g i c a l p r o d u c t o f t h e s t r o n g e s t l o c a l l y convex t o p o l o g i e s o f F and G, p i s n o t continuous i f we check t h a t p*:FxG * C ,

p*(x,u):=

(x,u)

i s n o t continuous,

Suppose t h a t p* i s c o n t i n u o u s . Then t h e r e a r e seminorms r on F and t on 6 such t h a t II < r ( x ) t ( u ) f o r x i n F and u i n G. Since F i s i n f i n i t e - d i mensional, t h e r e i s a l i n e a r f o r m v on F which i s n o t continuous on ( F , r ) . On t h e o t h e r hand, I ( x , v ) l C r ( x ) t ( v ) t i n u o u s on (F,r),

f o r a l l x i n F. Thus, v has t o be con-

a c o n t r a d i c t i o n . The p r o o f i s complete.

//

Our n e x t aim i s t o prove t h a t , if E i s t h e s t r i c t i n d u c t i v e l i m i t o f a sequence o f Frechet-Monte1 spaces and i f E has a continuous norm, t h e n E l b i s h-bornological.

I n p a r t i c u l a r , i f E:=D(X)

,

l l f II :=sup( I f ( x ) l :x EX) i s a

continuous norm on i t s n a t u r a l ( L F ) - s t r u c t u r e and hence D ' ( X ) g i c a l . I f E=s-ind(En:n=1,2,..), and ( E n ' ,b(Enl ,En))

i s h-bornolo-

we w r i t e E ' and En' t o denote (E',b(E',E))

r e s p e c t i v e l y . We s h a l l s t a r t w i t h s e v e r a l remarks

Observation 12.2.4:

( a ) i n 9.1.46 we proved a s t r o n q v e r s i o n o f DE WILDE

c l o s e d graph theorem. Observe t h a t 9.1.46

implies that, i f E i s ultraborno-

l o g i c a l , F has an a b s o l u t e l y convex C-web and f : E +

F i s a l i n e a r mapping

w i t h c l o s e d graph i n ExF, t h e n F i s c o n t i n u o u s . From here i t i s immediate t o deduce t h e f o l l o w i n g open-mapping theorem: "If E i s u l t r a b o r n o l o g i c a l , F a space w i t h an a b s o l u t e l y convex C-web and f:E-F

a continuous s u r j e c t i v e

l i n e a r mapping, then f i s open". be a p r o p e r s t r i c t i n d u c t i v e l i m i t o f FrPchetL e t E=s-ind(E :n=1,2,..) n Montel spaces E ,n=1,2,... Then, n ( b ) E and E ' a r e b o t h complete Montel spaces. I f Jn:En+

E i s t h e canonical i n j e c t i o n (which i s a t o p o l o g i c a l homomor-

phism), t h e n i t s transposed mapping Qn:E' --En' mapping) i s c o n t i n u o u s and s u r j e c t i v e . Moreover,

(which i s the r e s t r i c t i o n

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459

E i s r e g u l a r (8.5.14(a))

.) where t h e

and hence E'=proj(En':n=1,2,.

l i n k i n g mappings (Qn:n=1,2,..)

a r e even open: Indeed, E ' has an a b s o l u t e l y

convex C-web (K2,$35.4.(13)),

En' i s u l t r a b o r n o l o g i c a l ( s i n c e i t i s a (DFM)-

space) and o u r o b s e r v a t i o n ( a ) a p p l i e s . Thus, ( c ) E ' i s t h e s t r i c t p r o j e c t i v e l i m i t o f a sequence o f (DFM)-spaces and moreover, i t i s even a d i r e c t e d p r o j e c t i v e l i m i t (0.3.3), i . e . i f U i s an -1 a b s o l u t e l y convex open subset o f E' t h e r e i s m w i t h U = Qm (Qm(U)). ( d ) Given a compact subset Km o f En',

t h e r e i s a compact subset K o f F'

such t h a t Qn(K)=Kn: Indeed, t h e r e i s an a b s o l u t e l y convex 0-nghb V i n En such t h a t i t s p o l a r V" i n En' c o n t a i n s Kn. Since 0-nghb W i n

E' i s s t r i c t , t h e r e i s a

E w i t h V 3 W n E n . Since E ' i s Montel, W" i s compact i n E ' and

Q(W")=V" by HAHN-BANACH theorem. Now t h e c o n c l u s i o n f o l l o w s . D e f i n i t i o n 12.2.5: y c ( U ) . A subset K o f

L e t U be a non-void open subset o f a space H and

H i s s a i d t o be d e t e r m i n i n g

for f

i f KT\U #

4 and

fc

f/(KAU) = 0 i m p l y t h a t f vanishes i d e n t i c a l l y . Lemma 12.2.6:

L e t G be a b o r n o l o g i c a l space and

K a t o t a l a b s o l u t e l y con-

vex subset o f G. Then, K i s d e t e r m i n i n g f o r each f which belongs t o

xc(V).

f o r e v e r y open a b s o l u t e l y convex s u b s e t W o f G. f/(K(IW)

# $.

L e t f be a v e c t o r of Wc(W) and suppose t h a t -ifin = 0. Since f = Z ( n 1 ) 2 f ( O ) , we have t h a t (n')-l$nf(0) /(KAW) = 0

Proof: Clearly, K f l W

cm

f o r each n and hence i t s u f f i c e s t o show t h a t K i s d e t e r m i n i n g f o r n-homogeneous p o l y n o m i a l s bounded on compact subsets o f G, n=1,2,..

. We

proceed by

i n d u c t i o n . F o r n=1, such a p o l y n o m i a l p i s continuous on G, s i n c e G i s b o r n o l o g i c a l and hence P E G ' . Moreover,

b p E K " f o r each b r C and, s i n c e K i s

t o t a l i n G, p=O as d e s i r e d . Now suppose t h a t K i s d e t e r m i n i n g f o r each k-homogeneous polynomial bounded on compact subsets o f G f o r each k C n and l e t p be a n-homogeneous p o l y nomial bounded on compact subsets o f G such t h a t p/K = 0 and l e t L be t h e symmetric n - l i n e a r form correspondinq t o p. According t o t h e P o l a r i z a t i o n Formula (0.5.11, Kx..xK.

L is. bounded on compact subsets o f Gx..xG

F i x x i n K and d e f i n e Lx:G-C,

Lx(z):=L(x,z,..,z).

and vanishes on Lx i s a (n-1)-

homogeneous polynomial bounded on compact subsets o f G v a n i s h i n o on K and hence Lx=O by i n d u c t i o n h y p o t h e s i s . L e t y be an a r b i t r a r y v e c t o r o f G. Then Ly:G-C,

LY(~):=L(z,y,..,y)

i s a l i n e a r form on G bounded on compact sub-

s e t s o f G and vanishes on K and hence on G by i n d u c t i o n h y p o t h e s i s . I n p a r -

BARRELLED LOCALLY CONVEXSPACES

460

t i c u l a r , Ly(y)=p(y)=O f o r y i n G. Thus p=O on G as d e s i r e d .

/I

Lemma 1 2 . 2 . 7 : L e t U be a non-void open subset o f a b o r n o l o q i c a l space E-sd-proj(En:n=l,2,..) of spaces En w i t h continuous p r o j e c t i o n s On,n=1,2,. L e t K be a t o t a l compact a b s o l u t e l y convex subset o f E.

...

( i ) i f fc%c(U), t h e n f f a c t o r i z e s through some En, i . e . s i n c e t h e r e i s rn -1(Qm(U)), t h e r e e x i s t s n and a mappinq f * : Q n ( U ) + C y f * ( x ) : = such t h a t U=Q, f ( x ' ) f o r x ' E U w i t h Q n ( x ' ) = x , such t h a t f = f * o Q n . (ii) i f T i s a bounded subset o f ( v ( U ) , t o ) ,

then ? f a c t o r i z e s

uniformly

t h r o u g h some En. P r o o f : Suppose n>m. n=1,2,..)

Suppose t h a t ( i ) does n o t h o l d . F i n d sequences (z(n):

y(n))#f(z(n)).

The f u n c t i o n A ( z ) : = f ( z + y ( n ) ) - f ( z )

W i n E.

a b s o l u t e l y convex 0-nghb

there e x i s t x(n)E Kfl(Z-lU) gn(b):=f(x(n)+by(n)) 0.5.12(ii),

i n U w i t h On(y(n))=O;

and (z(n)+y(n):n=l,Z,..)

By 12.2.6,

y(n)#O and f ( z ( n ) +

belongs t o r)Cc(W) f o r some

K i s d e t e r m i n i n q f o r A and hence

such t h a t f ( x ( n ) + y ( n ) ) # f ( x ( n ) ) .

i s a non-constant (qn(0)#gn(l)) e n t i r e f u n c t i o n . By

choose b ( n ) C C such t h a t l q n ( b ( n ) ) / > n + l f ( x ( n ) ) l and hence

If(x(n)+b(n)y(n))\) (b(n)y(n):n=1,2,..)

n. Since K i s compact and s i n c e Qr(y(n))=O f o r r S n , i s a n u l l sequence and (x(n)+b(n)y(n):n=l,Z,..)

is a

r e l a t i v e l y compact subset o f U whose c l o s u r e i s c o n t a i n e d i n U by c o n s t r u c t i o n . That i s a c o n t r a d i c t i o n s i n c e f i s unbounded on t h i s sequence. ( i i ) IfF d o e s n o t f a c t o r u n i f o r m l y through some En, proceed as above t o f i n d a sequence (x(n)+b(n)y(n):n=l,Z,. and a sequence ( f n :n=1,2,..)

in

contradicts the f a c t t h a t ? i s

5 such

.) in 2-'U

which i s r e l a t i v e l y compact

t h a t l f n ( x ( n ) + b ( n ) y ( n ) ) ] > n and t h i s

bounded in ( % ( U ) , t o ) . / /

Proposition 12.2.8: L e t E be t h e s - d - p r o j e c t i v e l i m i t o f a sequence o f (DFM)-spaces and suppose t h a t E has a t o t a l compact a b s o l u t e l y convex subset. Then, E i s h - b o r n o l o g i c a l . P r o o f : Set E = s - p r o j ( E :n=1,2,..)

n

convex subset o f E. By 12.1.16, r e l l e d and t h a t y c ( U ) = $ ( U ) such t h a t U=Qm-l(Qm(U))

x(U):

and l e t K be a t o t a l compact a b s o l u t e l y

i t i s enough t o check t h a t E i s h-quasibar-

f o r U a non-void open subset o f E . There i s m

Qm b e i n g t h e m-th continuous p r o j e c t i o n .

Indeed, suppose t h a t ftyc(U).F i r s t observe t h a t E i s bor%,(U)= n o l o g i c a l : l e t T:E*F be a l i n e a r mapping bounded on compact subsets o f E, F being any space. According t o t h e f a c t o r i z a t i o n lemma ( D I y l . 1 6 ) , t h e r e i s a l i n e a r mapping

Tn*:En-+F

such t h a t

T=Tn*oQn f o r some n.

By 12.2.4(d),

Tn*

is

CHAPTER I2

461

bounded on compact subsets o f En and c l e a r l y i t i s G-holomorphic. Since En i s a (0FM)-space,

i t i s h - b o r n o l o g i c a l (12.1.6)

and hence Tn* i s continuous.

Thus, T i s c o n t i n u o u s as d e s i r e d . By 1 2 . 2 . 7 ( i ) ,

f f a c t o r i z e s through some En w i t h f a c t o r i z a t i o n mapping f*.

As above, f * i s bounded on compact subsets o f Q n ( U ) . S i n c e En i s a (DFM)-

space, f* i s continuous and so i s f . Thus, f C % ( U ) . E i s h - q u a s i b a r r e l l e d : Indeed, l e t y b e a subset bounded i n ( d ( U ) , t o ) . By 12.1.15,

i t s u f f i c e s t o show t h a t F i s l o c a l l y bounded. By 1 2 . 2 . 7 ( i i ) ,

. The

% f a c t o r s u n i f o r m l y t h r o u g h some En

t i o n mappings i s bounded i n ( g ( Q n ( U ) , t 0 ) , i s h - q u a s i b a r r e l l e d . Thus,

C o r o l l a r y 12.2.9:

3; i s

sequence ( f * : f E x )

o f factoriza-

hence l o c a l l y bounded, s i n c e E

l o c a l l y bounded on U . / /

n

The s t r o n g dual o f a p r o p e r s t r i c t i n d u c t i v e l i m i t o f a

sequence o f F r k h e t - M o n t e 1 spaces i s h - b o r n o l o g i c a l i f i t has a c o n t i n u o u s norm. I n p a r t i c u l a r , D ' ( X ) i s h - b o r n o l o g i c a l . Observation 12.2.10:

FLORET,(13)

has shown t h e e x i s t e n c e o f a n u c l e a r Fr6-

c h e t space (G,t) w i t h an i n c r e a s i n g sequence o f c l o s e d subspaces Fn each hav i n g a continuous norm b u t t h e s t r i c t i n d u c t i v e 1 i m i t E:=ind( (Fn,t):n=l,2,.) does n o t admit any continuous norm. I t would be n i c e t o know i f t h e s t r o n g dual of a p r o p e r s t r i c t i n d u c t i v e l i m i t o f Frechet-Monte1 spaces En, n=1,2,

..,

each h a v i n g a continuous norm, i s h - b o r n o l o g i c a l . The answer t o t h i s q u e s t i o n i s c l e a r l y a f f i r m a t i v e i f such an i n d u c t i v e l i m i t has u n c o n d i t i o n a l b a s i s , because i n t h i s case i t i s isomorphic t o a c o u n t a b l e d i r e c t sum and hence has a c o n t i n u o u s norm and we can a p p l y 12.2.9. P r o p o s i t i o n 12.2.11:

L e t E=s-ind(En:n=1,2,..)

of FrGchet-Monte1 spaces En, n=la2,..,

be a s t r i c t i n d u c t i v e l i m i t

such t h a t E l b i s h - b o r n o l o q i c a l . Then,

each En has a c o n t i n u o u s norm. P r o o f : W i t h o u t loss o f g e n e r a l i t y , suppose t h a t E does n o t have a c o n t i 1 N nuous norm. By 2.6.13, El c o n t a i n s C complemented. Set y(n) f o r t h e n - t h N c a n o n i c a l u n i t v e c t o r i n C C E=E" ( E i s r e f l e x i v e ! ) . S e t x ( n ) t : En\En-l m

and d e f i n e t h e 2-homogeneous p o l y n o m i a l on E ' p ( .):= F y ( n ) ( . ) x ( n ) ( . ) . p i s continuous on compact subsets o f E l : Indeed, l e t K be a compact subset

of E ' which i s c o n t a i n e d i n t h e p o l a r o f some 0-nghb V o f E b y r e f l e x i v i t y . Since VAEl

i s a 0-nghb i n El,

where W i s a 0-nghb i n Cq.

VOCN

W

c o n t a i n s a 0-nghb o f t h e f o r m W x n C , 9+' Thus, t h e r e i s k such t h a t b y ( n ) C V f o r each n>k

BARRELLED LOCALLY CONVEXSPACES

462

\ b y ( n ) ( x ) [ _ L l f o r a l l bGC, n)/k

and a l l b c C . Hence

t h e r e s t r i c t i o n of p t o K equals ,$y(n)(.)x(n)(.)

and x M " .

Since

KCV",

and t h e c o n c l u s i o n f o l l o w s .

p i s n o t continuous on E ' : Indeed, suppose p c o n t i n u o u s and a p p l y t h e F a c t o r i z a t i o n Lemma ( D I , l . l 6 ) t o o b t a i n t h e e x i s t e n c e o f m such t h a t , i f Qm-l(y)=O,

t h e n p(x+y)=p(x) f o r a l l x ( E ' . Take x as t h e m-th canonical u n i t N v e c t o r , which belongs t o ( C ) ' = C ( N ) and extend i t t o E. Thus, x C E ' and Go

y ( m ) ( x ) = l and y ( n ) ( x ) = O f o r n#m. For a l l y w i t h Q m - l ( y ) = 0 , F y ( n ) ( . ) x ( n ) ( x ) = i?y(n)(

. ) x ( n ) ( x + y ) and hence x(m)(x)+x(m)(y)=x(m)(x).

T h e r e f o r e x(m)(y)=O

-

and t h e n x(m) E(E, 1 1 ) ' = Em-1 and t h a t i s a c o n t r a d i c t i o n . By 12.2.10,

/I

a c o u n t a b l e p r o d u c t o f (0FM)-spaces each h a v i n g a t o t a l com-

p a c t subset i s h - b o r n o l o g i c a l . We s h a l l extend t h i s r e s u l t t o a r b i t r a r y p r o d u c t s o f such spaces. F i r s t a t e c h n i c a l lemma i n s p i r e d i n t h e technique o f 12.2.7: Lemma 12.2.12:

Let E=n(Ei:i

61) be a p r o d u c t o f b o r n o l o g i c a l spaces Ei

each h a v i n g a t o t a l a b s o l u t e l y convex subset Ki and suppose t h a t E i s borno-

C I)and U:=TT(Ui:i

l o g i c a l . Set D:= @ ( K i : i vex subset o f Ei and Ui=Ei

for all

L e t F be a space and f L y c ( U , F ) .

G I ) w i t h Ui open a b s o l u t e l y con-

i c I \ J , J b e i n g a f i n i t e subset o f I .

I f f does n o t f a c t o r t h r o u g h any f i n i t e

t h e r e e x i s t sequences unC UA(n-'D)

p r o d u c t o f spaces Ei,

and znC D such

t h a t f ( u n + z n ) # f ( u n ) f o r each n . Moreover, J and N n : = ( i G I : z n ( i ) # O )

.

wise d i s j o i n t , n=1,2,.

are p a i r -

P r o o f : Set L1 : = J . Since f does n o t f a c t o r t h r o u g h T ( E i :i&Ll), t h e r e a r e 1 1 1 such t h a t f ( x +y ) # f ( x ) . D e f i n e gl(y):= f ( x 1+ y ) - f ( x 1) , gl: TT(Ei:i&Ll)--+F, which i s G-holomorphic and bounded on

x

1-EU and y 1& E w i t h y 1(i)=O,iELly

compact subsets. D1:=

@(Ki:iLI\L

(with 1 ) i s d e t e r m i n i n g f o r g1 b y 12.2.6 1

t h e obvious m o d i f i c a t i o n s i n t h e n o n - s c a l a r c a s e ) . So, t h e r e i s z E D w i t h 1 gl(z 1) = 0 and hence f ( x 1+z 1) # f ( x 1) . An easy i n d u c t i o n argument shows t h e e x i s n tence o f sequences (xn:n=1,2,..) i n U and ( z :n=1,2,..) i n D with f(xn) # f ( x n + z n ) f o r each n w i t h t h e r e q u i r e d p r o p e r t i e s on t h e index s e t s . F o r each x t U , x + z n L U and, f o r f i x e d n, t h e mapping A ( z ) : = f ( z + z n ) - f ( z ) i s well-defined.

I t i s G-holomorphic and i t i s bounded on compact subsets of

U. Since E i s b o r n o l o g i c a l ,

i s un C (n-l)D/'IU

( n - 1D) i s d e t e r m i n i n g f o r A and t h e r e f o r e t h e r e

w i t h f ( un+zn)#f ( un) .//

P r o p o s i t i o n 12.2.13:

Let E=TT(Ei:i&I)

be a b o r n o l o g i c a l p r o d u c t o f a

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463

f a m i l y o f (DFM)ispaces Ei.

Then, E i s h - b o r n o l o g i c a l i f and o n l y

f each Ei

has a t o t a l compact subset Ki. Proof: Since each Ei

i s complete, we may suppose Ki a b s o l u t e l y convex.

We keep t h e n o t a t i o n i n 12.2.12 and we suppose F normed (F:=(F, 11.11). be a member o f T c ( U , F ) .

Our p r e v i o u s arguments i n 12.2.8

Let f

show t h a t i t

s u f f i c e s t o prove t h a t f f a c t o r s through some f i n i t e p r o d u c t o f t h e spaces ( a (DFM)-space!). I f t h i s i s n o t t h e case, a p p l y 12.2.12 t o f i n d sequen1 n i n D w i t h f(un+zn)#f(un) ces ( u :n=1,2,..) i n U A ( n - D) and (zn:n=1,2,..) Ei

f o r each n. D e f i n e hn(b):=f(un+bzn)),

b c C , which a r e non-constant e n t i r e

>

t o o b t a i n b ( n ) LC w i t h \ \ f ( u n + b ( n ) z n \ \ n f i s unbounded on t h e r e l a t i v e l y compact subset

f u n c t i o n s and a p p l y 0 . 5 . 1 2 ( i ) f o r each n. As i n 12.2.7, (un+b(n)zn:n=1,2,.

.) whose c l o s u r e i s c o n t a i n e d i n U, a c o n t r a d i c t i o n .

I f some Ek does n o t have a t o t a l compact subset, 9.1.12

and 9 . 1 . 1 4 ( i i )

show t h a t Ek c o n t a i n s C ( N ) complemented and hence E c o n t a i n s CNxC(N) complernented. By 12.2.2,

E i s n o t h-bornological

We keep t h e n o t a t i o n o f 12.2.13. @(Ei:i

*//

I n 6.2.9

and 6.2.10 we showed t h a t

6 I)endowed w i t h t h e t o p o l o g y induced b y E and Eo were b o r n o l o g i c a l

spaces. Moreover, we s h a l l p r o v e ( 1 2 . 2 3 8 ) t h a t CNxC(N) i s n o t even h-quasib a r r e l l e d . I f t stands f o r t h e t o p o l o g y on C ( N ) &l(C(N)yt)

induced b y CN,

t h e space

i s n o t h - b o r n o l o g i c a l , f o r i f i t were C(N)xCN would be h-quasi-

b a r r e l l e d (12.1.22),

a c o n t r a d i c t i o n . W i t h t h e hypotheses o f 12.2.13,@(Ei:

i G I ) c o n t a i n s C(N)x(C(N),t)

complemented i f some Ei does n o t have some t o t a l

compact subset and hence @(Ei:i

L I ) cannot be h - b o r n o l o g i c a l .

I f some Ei

does n o t have any t o t a l compact subset, Eo c o n t a i n s CNxC(N) Complemented and hence i t i s n o t h - b o r n o l o g i c a l . Making t h e obvious changes, we have P r o p o s i t i o n 12.2.14:

Let E=lT(Ei:i

f a m i l y o f (DFM)-spaces Ei.

Then, @(Ei:i

& I ) be a b o r n o l o g i c a l p r o d u c t o f a

G I ) and Eo a r e h - b o r n o l o g i c a l i f d3 (Ei:i t I ) and Eo

and o n l y i f each Ei has a t o t a l compact subset, where a r e endowed w i t h t h e t o p o l o g y induced b y E. C o r o l l a r y 12.2.15:

If C

I i s bornological

, then

(C('),t)

i s h-bornological.

Our n e x t r e s u l t improves 12.2.15 P r o p o s i t i o n 12.2.16:

L e t E = T ( E i : i E I)be a p r o d u c t o f m e t r i z a b l e spaces

BARRELLED LOCALLY CONVEX SPACES

464

E i . Then, ( i ) i f C I i s bornological, E i s h-bornological and ( i i ) f o r any index s e t I , Eo and @ ( E i : i C I ) a r e h-bornological. Proof: ( i ) We keep t h e notation in 1 2 . 2 . 1 2 . Let f be an element of Yc(U,F) and suppose F:=(F, J/,/]) normed. Since f i n i t e products of metrizable spaces a r e again metrizable and hence h-bornological, i t s u f f i c e s t o show t h a t f f a c t o r s through some f i n i t e product. If t h i s i s not the case, taking K i : = E i n n t h e r e a r e sequences ( u : n = 1 , 2 , . . ) i n UnD and ( z : n = 1 , 2 , . . ) i n D=$(Ei:i&I) w i t h f ( u n + z n ) # f ( u n )f o r each n with J and N n , n = 1 , 2 , . . , pairwise d i s j o i n t , by 12.2.12. S e t P : = ( i & I : u n ( i ) # O ; n = 1 , 2 , . . ) . P i s countable and hence t h e spac e n ( E i : i L P ) i s metrizable. Let ( V n : n = 1 , 2 , . . ) be a b a s i s of absolutely convex 0-nghbs i n this space w i t h V l c u n l T ( E i : i t P ) . For each n , define An: UAIT(Ei:iL P ) - - t F , A n ( z ) : = f ( z + z n ) - f ( z ) . Each An i s G-holomorphic and does not vanish i d e n t i c a l l y . Since VnA@(Ei:i C P ) i s absolutely convex and dense i n t h e bornological space V ( E i : i L P ) , 12.2.6 shows the existence of wnC V n A @ ( E i : i t P ) w i t h f(wn+zn)#f(wn)f o r each n . Take e n t i r e functions hn:C-+ F defined by hn(b):=f(wn+bzn). Since each hn i s non-constant, 0 . 5 . 1 2 ( i ) shows t h a t llf(wn+b(n)zn))ll> n f o r each n and some sequence of complex numbers ( b ( n ) : n = 1 , 2 , . . ) . The sequence ( w n + b ( n ) z n : n = 1 , 2 , . .) converges t o zero in D and hence f i s unbounded on some compact s e t , a c o n t r a d i c t i o n . ( i i ) Recall t h a t @(Ei:i G I ) and Eo a r e bornological (6.2.9 arid 6.2.10). The obvious modifications above and in 1 2 . 2 . 1 2 give t h e desired conclusion

I/

Corollary 12.2.17: ( C ( ' ) , t )

i s h-bornological.

Now we t u r n o u r a t t e n t i o n t o find l a r g e c l a s s e s o f spaces which a r e h barrel l e d . Lema 12.2.18: Let E be a space, F a normed space and p & P ( " E , F ) . I f a , b r E , then lIp(b)ll A s u p ( I\p(a+sb)l\ : s E C with I s l l l ) . Proof: By t h e maximum p r i n c i p l e , we may replace " 1 S I C 1" by " j s / = l " and then the mapping s -l / s shows t h e e q u a l i t y sup(jlp(sa+b)ll : I s l b l ) sup(l\p(a+sb)l\ : isl'l) from where the conclusion follows.

=

//

Theorem 12.2.19: I f E i s a Baire space, then E i s h-barrelled. Proof: Without loss of g e n e r a l i t y , we suppose F : = ( F , q ) normed. Let U be a non-void open subset o f E a n d y c w U , F ) bounded on every compact subset of U of t h e type (a+sb: \ s l L l ) , a and b being vectors of E . Fix x in U and take

CHAPTER 12

465

an open absolutely convex 0-nghb V such t h a t x+VCU. By 0.5.11, the set'll:= ( (m!)-l$mf(x): f t ~ m = 1 , 2 ..) , i s pointwise bounded on V , since 3;'is bounded on ( x + s y : l s l ~ l ) y, b e i n g a vector of V . By 1.1.5, V i s of second category i n E. Since 2' 4 i s pointwise bounded on V , there i s a vector a g V where 'LLis l o c a l l y bounded, i . e . there i s an absolutel y convex 0-nghb W such t h a t a+WcV and ' M i s bounded on a+W: Indeed, s e t V n : =(x(V: q ( f ( x ) ) L n f o r every f t U ) . Since uis pointwise bounded, V= n = 1 , 2 , . . ) . Since each Vn i s closed i n V and V i s o f second category i n E, there i s a positive integer p such t h a t V has non-void i n t e r i o r . I f a 4 P i n t ( V p ) , u i s l o c a l l y bounded a t a . By 12.2.18, Gis bounded on W. Take M)O such t h a t q ( g ( t ) ) / M f o r every tLW and g e u .Given any f e q there i s a 0-nghb ZC2-lW such t h a t lim q ( f ( x + t ) - $(m!)-'Gmf(x)(t)) = 0

u(Vn:

uniformly f o r t i n Z . Observing t h a t s u p ( q((m!)-l$mf(x)(t)): t t 2 - l W ) = 2-msup( q((m!)-'Gmf(x)(t): t tW), we have t h a t s u p ( q ( Z ( m ! ) - ' i m f ( x ) ( t ) :

t C2-'W)C z s u p ( q ( ( m ! ) - l i m f ( x ) ( t ) ) : t 62-lW) C M and therefore M % sup( q ( f ( x + t ) ) : t L Z ) f o r every f i n 3;. T h u s , f i s l o c a l l y bounded a t x and since x was a r b i t r a r y , % i s 1ocal l y bounded.

//

T h u s , we have Frichet

-

Metr i zabl e

Baire

h-barrel led

\P h-bornological

h-quasibarrel1edjMN/

'

Let us show t h a t (LS)-spaces a r e h-barrelled. Unfortunately, we do not know i f (DFM)-spaces a r e h-barrelled. An inspection of the proof of 12.2.19 shows t h a t a subset F o f Z ( U , F ) i s bounded on each finite-dimensional compact subset of U i f and only i f i t i s bounded on every compact subset of U of the type ( a + s b : l s l $ l ) f o r each a,btU. Indeed, t o prove this, we may r e s t r i c t a t t e n t i o n t o the case when E i s f i n i t e dimensional and hence a Baire space. So we have Theorem 12.2.20: (BANACH-STEINHAUS-NACHBIN THEOREM) If E i s a Frechet space, each subset FC%(U,F) i s equicontinuous i f bounded on every compact subset of U of the type (a+sb: Islrl).

BARREL LED LOCAL L Y CON VEX SPACES

466

Lemma 12.2.21: Let (E,t)=ind((En,tn):n=l,2,.

.) be a (LS)-space and U a

non-void open subset of ( E , t ) . Then ( U , t ) c a r r i e s t h e f i n e s t topology such t h a t the inclusions ( U / \ E n , t n ) - + ( U , t ) a r e continuous f o r each n . Proof: S e t s f o r t h e f i n e s t topology on U f o r which t h e canonical inclusions a r e continuous. Clearly, s i s f i n e r than t on U . I f V i s an open sub-

set of (U,s), t h e n VAEn i s open i n ( U / \ E n , t n ) and, s i n c e UOEn i s open i n ( E n , t n ) , we have t h a t VOEn i s open i n (En,tn) f o r each n . T h u s , V i s open i n (E,t**)=(E,t) by 8 . 5 . 2 8 ( i i ) . T h u s , V i s open i n ( U , t ) . / / Lemna 12.2.22: Let ( E , t ) and U be a s above. Suppose t h a t U A E l # 4 . Let f:U-.F, F being any space, be a mapping and s e t f ( n ) f o r i t s r e s t r i c t i o n t o UI\En f o r each n . Then, f&%(U,F) i f and only i f f ( n ) C . ' d e ( ( U n E n , t n ) , F ) f o r each n . , n e c e s s i t y is c l e a r . To prove s u f f i c i e n c y , Proof: According t o 0.5.4 we s h a l l see t h a t f i s G-holomorphic and continuous ( 0 . 5 . 9 ) . Let S be a finite-dimensional subspace of E i n t e r s e c t i n g U . There i s k w i t h S C E k and s i n c e f ( k ) Lg((UnEk,tk),F),we have t h a t f L % ( ( U n S , t ) , F ) which proves our f i r s t a s s e r t i o n . According t o 0 . 5 . 4 ( a ) , each f ( n ) : ( U ( \ E n , t n ) - + F i s continuous. By 1 2 . 2 . 2 1 , f i s continuous.

//

Proposition 12.2.23 : Let ( E, t ) = ind( ( En, t n ) :n = l , 2 , . .) be a ( LS) -space. Then ( E , t ) i s h-barrelled. Proof: Let U be a non-void subset of ( E , t ) and ?-a subset of X(U,F) bounded on finite-dimensional compact subsets of U . Without l o s s o f g e n e r a l i t y , and set Fnf o r t h e s e t of a l l r e s t r i c t i o n s t o we may assume t h a t UOE1# UAE, of t h e members of f o r each n . i s amply bounded i f and only i f each FnC g ( ( U n E n , t n ) , F )i s Claim: ----amply bounded. e may assume F normed. Since each ?,,i s Only s u f f i c i e n c y needs proof. W pointwise bounded (12.1.10), i t follows t h a t F i s pointwise bounded and hence t h e mapping g:U- 1@(? , F ) defined in 12.1.11 i s well-defined. I f gn stands f o r t h e r e s t r i c t i o n of g t o U / I E n , 12.1.11 shows t h a t each gn belongs t o % ( ( U n E , t ) , l @ (% , F ) ) , since each Fn i s l o c a l l y bounded. By 1 2 . 2 . 2 2 , g & w 5 n i s l o c a l l y bounded. g ( U , l ( 3, F ) ) . Again by 1 2 . 1 . 1 1 , Clearly, each Fn i s bounded on finite-dimensional compact subsets of U . Moreover, each ( E n , t n ) can be taken as a Banach space (hence h-barrelled by 12.2.19) and hence each i s amply bounded. Now apply t h e claim.,,

4

yn

CHAPTER 12

467 L e t E=T(Ei:i 6 I ) be an a r b i t r a r y product o f (LS)E i s h-barrelled, i f each Ei has a t o t a l compact subset Ki.

P r o p o s i t i o n 12.2.24: spaces. Then

Proof: We

keep t h e n o t a t i o n i n 12.2.12.

t h a t Eo i s h-barrelled.

L e t y b e a subset o f X(U,F)

subsets o f U (see 12.1.13) t e d i n 12.1.11.

By 12.1.22,

and s e t g:U--lm($:,F)

i t i s enough t o show

bounded on f a s t compact

f o r t h e f u n c t i o n construc-

The conclusion f o l l o w s i f we show t h a t g f a c t o r i z e s throuqh

some f i n i t e product o f f a c t o r spaces. Since D:= @(Ki:i6?I)

i s determining

f o r g, proceed as i n t h e p r o o f o f 12.2.12 t o f i n d sequences (un:n=1,2,..) and (zn:n=1,2,. .), w i t h p a i r w i s e d i s j o i n t supports, such t h a t (nun:n=1,2,..) n n n contained i n U A D and w i t h g ( u +z )#g(un) f o r each n, i f and ( z :n=1,2,..)

g does n o t f a c t o r through any f i n i t e product o f f a c t o r spaces. Proceeding as i n 12.2.13,

take a sequence (b(n):n=1,2,..)

i n C withIlg(un+b(n)zn))l't>n f o r

K i s a Banach d i s c and (un:n=1,2,..) convereach n. I f K:=ZE?(nun:n=1,2,..), N n ges t o zero i n EK. Now sp(z :n=1,2,..) i s c l e a r l y isomorphic t o C and hence the n u l l sequence (b(n)zn:n=1,2,..)

i s f a s t convergent t o 0. Thus, g i s un-

bounded on t h e f a s t convergent sequence (un+b(n)zn:n=1,2,.

.)CU, a contradic-

t i o n . Thus, g i s holomorphic and hence F i s l o c a l l y bounded by 12.1.11. C o r o l l a r y 12.2.25:

/I

D'(X) i s h - b a r r e l l e d .

Proof: The conclusion follows from 12.2.24 r e c a l l i n g t h a t D'(X) i s isomorp h i c t o ( s ' ) N by 7.6.10.//

Our next purpose i s t o prove t h a t t h e a r b i t r a r y product o f B a i r e metrizab l e spaces i s t?-barrelled,

f o r what we need some preparation.

L e t U be a non-void open subset o f t h e product o f spa-

D e f i n i t i o n 12.2.26:

ces ExG and suppose t h a t f i s a mapping from U i n t o any space F. f i s s a i d t o be separately holomorphic i f t h e f u n c t i o n s fa(b):=f(a,b)

and fb(a):=f(a,b)

are holomorphic on ( ( a ) x G ) A U and on ( E x ( b ) ) n U r e s p e c t i v e l y f o r a l l a t E and b EG. Observation 12.2.27:

HARTOGS proved t h a t every separately holomorphic

f u n t i o n defined on an open subset of Cn i s holomorphic. For a separately holomorphic f u n c t i o n as i n 12.2.26,

t h e f u n c t i o n ( z , z ' ) w f(a+zc,b+z'd)

, with

z,z' f C and a , c t E and c,dCG i s separately holomorphic, hence holomorphic by HARTOGS' theorem.

In

Thus, f i s G-holomorphic.

p a r t i c u l a r , i t i s holomorphic on the diagonal z = z ' .

BARRELLED LOCAL L Y CONVEX SPACES

468

P r o p o s i t i o n 12.2.28:

L e t E be a B a i r e space, G a B a i r e m e t r i z a b l e space

such t h a t ExG i s B a i r e and U a non-void connected open subset o f ExG. I f f : U-F

i s s e p a r a t e l y holomorphic, t h e n f i s holomorphic.

P r o o f : By DI,2.28 (ZORN's theorem), i t s u f f i c e s t o show t h a t f i s bounded on some open subset c o n t a i n e d i n U. L e t V:=VlxV2

be an open subset c o n t a i n e d

i n U w i t h Yl and V 2 open subsets o f E and G r e s p e c t i v e l y . F i x y o E V 1 and l e t (Wn:n=1,2,..)

be a b a s i s o f yo-nghbs c o n t a i n e d i n V2.

Set A(n,r):=(x€V1:

llf(x,y)ll Lr f o r a l l yGWn), where 11.1 stands f o r t h e norm o f F ( F may be assumed normed ) . Since A ( n , r ) = n ( l l f lomorphic, each A(n,r) Since

lt-'([O,r]): y € W n ) and each f i s hoY Y n,r=l,Z,..). i s c l o s e d i n V and a l s o V,=U(A(n,r):

E i s Baire, there e x i s t

n and r such t h a t i n t ( A ( n , r ) ) #

t h e r e i s a p o i n t x i n V1 and an open x-nghb Z w i t h

c ZXV,.

16

and hence

I l f ( x , y ) \ l S r f o r a l l (x,y)

//

Theorem 12.2.29:

L e t E ba a B a i r e space, G a m e t r i z a b l e space, (F,II.II)

notmed space and U a non-void open s u b s e t o f ExG. L e t f : U +

a

F be a s e p a r a t e l y

holomorphic mapping which i s bounded on t h e subsets o f U o f t h e form KxL, K being a finite-dimensional

compact subset o f E and L a compact subset o f G .

Then f i s holomorphic. P r o o f : We may suppose U=UlxU2. function g(j,x):=f(

5

7 +2x,y)

Since CxG i s m e t r i z a b l e and 1 2 . 1 . 6 ( i ) ,

the

d e f i n e d on U * : = ( k C : $ + 2 x CU1)xU2 i s holomor-

U1 and x i n E. F i x (a,b)CU and l e t W1xW2 be an open absol u t e l y convex subset o f ExG such t h a t (a,b)+WlxW2 CU. Set Bn:=(x&W1:

phic f o r a l l

in

Il(m!)-l~mf(a,b)(x,y)ll,cn,

f o r a l l y C V n and m=0,1,2,..), where (Vn:n=1,2,.)

i s a d e c r e s i n g b a s i s o f 0-nghbs i n G w i t h V1CGJ2. Given y.CW2, d e f i n e f*: W1x(%y: ' X t C and Ay(W2)-F,

by f * ( x , % y ) : = f ( ( a , b ) + ( x , % y ) ) .

Since Ex(hy: r mf(0,O) IXEC) i s B a i r e ( 1 . 4 . 1 ) , f* i s holomorphic by 1.2.28 and hence ( m ! ) - l9

i s continuous f o r e v e r y m=0,1,2,. . Since (m!)-l$mf*(O,O)(x,y)=(m!)-%mf(a,b) (0,O)

f o r every x 6Wly

t h e mapping x t t ( m ! ) - 1 6 m f ( a , b ) ( x , y )

i s continuous on

. ) : Indeed, i f f o r some xCW1 and f o r each m y t h e r e e x i s t l m E C w i t h \ > m l C 1 and ym(Vm such t h a t I l f ( ( a , b ) + ( ; t m x , y m ) ) \ I m y then g(a,x) i s unbounded on t h e compact

W1 and hence Bn i s c l o s e d i n W1.

Moreover, W1=u(Bn:n=l,2,.

>

) , a c o n t r a d i c t i o n . Thus, g i v e n x&W1, s e t (2.x: I X k l ) x ( (b+y :m=l,Z,..)L'(b) m t h e r e i s n such t h a t I l f ( ( a , b ) + ( fix,y))ll 5 n f o r every1AcC w i t h I ? \ S l and A m By DIy1.l3, Il(m!)-l'$mf(a,b)(x,y)l) d sup( 11 (m!) -19 f(a,b)(lx,z)! e v e r y y EV,.

: l ~ l $ lz, €Vn),(sup(

IIf((a,b)+(fix,z))\\

: 12\L1,z€Vn) L n f o r every m=O,l,

...

and hence x E B n . Now a c a t e g o r y argument shows t h e e x i s t e n c e o f a p o s i t i v e

CHAPTER 12

469

i n t e g e r n such t h a t i n t ( B n ) # d and hence t h e r e i s a p o i n t x E B n and an absol u t e l y convex 0-nghb V w i t h x+VCBn. By 0 . 5 . 1 2 ( i i ) , ~ ~ ( m ! )- l . a’ m’ f ( a , b ) ( x , y ) l l ,C n f o r each (x,y)EVxVn and each m=0,1, (a,b)

...

The TAYLOR s e r i e s expansion a t

shows t h a t f i s bounded on (a,b)+2-lVxVn

P r o p o s i t i o n 12.2.30:

which concludes t h e p r o o f .

//

.) be a c o u n t a b l e p r o d u c t o f Bai-

L e t E=T(En:n=l,2,.

r e m e t r i z a b l e spaces. Then E i s h - b a r r e l l e d . P r o o f : F o r each m y s e t Gm:=TT(En:n=m+l,m+2,..). space and

7a

subset o f x ( E , F )

s e t s o f E. L e t us check t h a t

be a normed

bounded on f i n i t e - d i m e n s i o n a l compact sub-

3; i s

l o c a l l y bounded, o r e q u i v a l e n t l y t h a t t h e i s l o c a l l y bounded. I f t h i s i s

d e f i n e d i n 12.1.11

function g : E - t l m ( ~ , F )

L e t (F,II.lI)

which i s holomorand c o n s i d e r g * E +l””(?,F) Y‘ 1 i s h - b a r r e l l e d (12.2.19), y t G 2 . I f , f o r e v e r y x L E 1 and v C E ,

n o t t h e case, w r i t e E=ElxG2 p h i c s i n c e El

t h e mapping gV:sp(x)xG2-+lp(

5,F)

, gv(sx,y):=g(v+(sx,y))

i s holomorphic i n

we a p p l y 12.2.29 t o o b t a i n t h a t g i s l o c a l l y

a nghb o f ( s x : s C C , ( s l & l ) x ( O ) ,

bounded. Thus, t h e r e i s v C E and x GE1 w i t h g (sxlyy) i s n o t bounded on a 1 V GE1xE2 and d e f i n e f S y t : G 2 4 F nghb of (sxl:s €C, I s l L l ) x ( O ) . W r i t e v=(vl,zl) by fsyt(y):=f(sxl+tvl,y).

The f a m i l y 5 1 : = ( f S y t : f 4 F

i n G2 and, consequently,

i s unbounded on any zl-nghb

, s,t CC, I s l L 2 , It1 -L 2)

Fl

i s not locally

bounded on G2 but i s c l e a r l y bounded on f i n i t e - d i m e n s i o n a l compact subsets o f G2. By i n d u c t i o n , determine sequences (vn:n=1,2,..) E such t h a t t h e f a m i l y F k : = ( f s j=l,..

L C Y \ S . \ C 2 , \ t . \ ~ 2, J J on f i n i t e - d i m e n s i o n a l compact

( y ) : = f ( SlXl+tlV1,. - - , s Ytk skxk+tkvkyy) w i t h Y C G ~ + ~ Then, . gj;eh’k, there exists f(k)CF,

subsets o f Gk+ly

EC, j = l , . . , k ,

w i t h fs

and u(k)rZGk+l

sk(k)xk+tk(k)vk,u(k))

w i t h l\f(k)(sl(k)x

sp((xn,vn)~(u(k)(n):k=1,2,.

and u ( k ) ( j ) = O f o r j=l,Z,..,k.

.I).

o f En and hence H:=mHn:n=l,2,..) family o f a l l

f/H,fCT

on f i n i t e - d i m e n s i o n a l i n H (12.2.19).

.di s

fl(snxn+tnvn:

s.(k),t.(k) J J

t o E with We s e t Hn:=

i s a Fre‘chet space. NOW, s e t

fi f o r

the

a f a m i l y o f holomorphic f u n c t i o n s on H bounded

bounded on t h e compact subsets o f H. L e t us

sn,tn CC, I s n \ 6 2 ,

?is

y

................. ,

compact subsets o f H and hence& i s l o c a l l y bounded

Thus,&is

and we a r e done. Thus,

.... . ....

Each Hn i s a f i n i t e - d i m e n s i o n a l subspace

i s n o t bounded: s e t K:=

c o n s t r u c t a compact subset o f H i n w h i c h & 00

+t (k)vly

1 1 Il>k. We may c o n s i d e r each u ( k ) b&nging

c o o r d i n a t e s (u(k)(n):n=1,2,..)

in

,t : f c 3 , s j y t j

,k) i s n o t l o c a l l y boundeh’bh;*b&kdeb k=1,2,..,

and (xn:n=1,2,..)

ItnlL2) +

( (u(k):k=l,Z,..)U(O)

)

l o c a l l y bounded.

Now l e t U be a non-void open subset o f E and Y&(U,F)

a f a m i l y bounded

BARRELLED LOCALLY CONVEXSPACES

470

on finite-dimensional compact subsets of U . We consider f i r s t U=UlxG2, where U1 i s an open subset of E l . I f we f i x v:=(v(n):n=1,2,..) i n U , take W1 a s an open absolutely convex subset of El w i t h xl+WICU1 and s e t W:=WlxG2. -1"m , f ( 5 ) i s bounded on finite-dimensioClearly, :=( ( m ! ) d f ( v ) : m = O , l , . . nal compact subsets of W . For each xcW1, set y x : = ( P x : P & ) w i t h Px(y):= P(x,y) f o r each y(G2. Clearly, ~ x C ~ ( G 2 , F i s ) bounded on finite-dimensio-

9

nal compact subsets of G2. Let (Zk:k=1,2,..) be a basis of 0-nqhbs i n G2. According t o the f i r s t p a r t of t h i s proof, there exist r,k such t h a t IIPx(y)II

_Cr f o r each y(Zk and P i n p, hence W 1 = u ( A ( k , r ) : k , r = 1 , 2 , . . ) ,A(k,r):= (xhW1: ~ ~ P x ( y ) ~y~hLZ kr , P c ~ ) Each . A(k,r) i s closed i n the Baire space W1 and hence 11P!x7y)111r f o r each (x,y)(VxZk f o r some k,r and a 0-nghb V contained i n W1 (see 0 . 5 . 1 2 ( i i ) ) . T h u s , 3; i s bounded on v+2-lVxZk. A repeated use of the former argument shows the conclusion t r u e . //

Theorem 12.2.31: The a r b i t r a r y product E=-TTfEi:i GI) of Baire metr zabl e spaces Ei i s h-barrelled. Proof: Let 3; be a subset of %(U,f) w i t h U=TT(Ui:iCI) bounded on f i n i t e -dimensional compact subsets of U and set g:U---lP($,F) f o r the G-ho omorphic mapping defined i n 12.1.11. The s e t @ ( E i : i & I ) i s determining f o r g . Proceeding a s we d i d i n 12.2.13, i f g does not f a c t o r through a f i n i t e product of E i ' s , take sequences ( w n : n = 1 , 2 , . . ) and ( v n : n = 1 , 2 , . . ) i n Un&(Ei:iQ) n n w i t h IIg(w +v )\i> n f o r each n . The set K:=(wn+vn:n=1,2,..)U(0) i s compact. Q : = ( i EI:wn(i)#O o r v n ( i ) # O ) i s countable and G:=n(E CQ) i:i i s h-barrelled (12.2.30) and hence g / ( U n G ) i s holomorphic b u t unbounded on K . That i s a contradiction, hence g f a c t o r i z e s through a f i n i t e product of f a c t o r spaces from where the conclusion fol lows.

//

Our next aim i s t o find a c l a s s of spaces f o r which the concepts of hb a r r e l l e d , h-bornological and h-quasibarrelled coincide. The core of the proof of 1 2 . 1 . 6 ( i i ) was the claim in i t and the f a c t t h a t , i f f C q c ( U 7 F ) , t h e n 6 m f ( x ) i s bounded on compact subsets of U f o r each x t U and m=0,1,2.. We study two related r e s u l t s : Proposition 12.2.32: IfyC%(U,F) i s bounded on (finite-dimensional) i s bounded on (finite-dimensional) compact subsets of U , then (^dmf(x):ftF) compact subsets of E f o r each x CU and m = = 1 , 2 , . . Proof: Let K be a compact subset of E . There i s a > O such t h a t x+bzCU

CHAPTER 12

471

f o r a l l bCC such t h a t Bf :==(

bl
. Clearly,

(2"f ( x ) (K) :f

($1

$mf(x)(K)Ca-m(mI)Bf, C a - m ( m ! ) u ( Bf

i s bounded on the compact s e t (x+bz: I bl=a,xCK),

and, since _.

acx(f(x+bz):

we have t h a t

I bl=a,fc'5;)

E

9)=

($f ( x ) ( z ) :z 6 K,f

( 3 )C a-"(m!

:fLF)

the s e t A:=

C A,

i s bounded i n F. Therefore, since U ( B f : f E $ )

u(i m(fx ) (K) :f

P r o p o s i t i o n 12.2.33:

P(mE,F)

I

f (x+bz) : I b l =a,z E K)

)A.

//

L e t E be a b a r r e l l e d (DF)-space and F a space.

i s bounded on f i n i t e - d i m e n s i o n a l compact subsets o f

E,

then

Ifxc

3; i s

amply bounded. Proof: Set y*:=(AELs(mE,F):

F* i s

%C$).

bounded on f i n i t e - d i m e n s i o -

nal compact subsets o f Em. By t h e P o l a r i z a t i o n Formula,

F* is

bounded as a

subset o f L(mE,F) f o r t h e topology o f the simple convergence. By K2,540.2.

( l l ) ,F * i s equicontinuous and hence F i s equicontinuous. By 12.1.10,the

conclusion f o l l o w s .

//

D e f i n i t i o n 12.2.34: $mf(x)€P(mE,F)

%,*(U,F)

i s t h e space o f a l l f6%G(U,F)

fo a l l x i n U and m=1,2,

P r o p o s i t i o n 12.2.35:

X(U,F)= %',(u,F)

E

Proof: Suppose

...

Clearly, %,(U,F)L

I f E i s h-quasibarrelled (resp.

l y ) . L e t x E U and fs%D(U,F).

m

, h - b a r r e l l e d ) , then

I t s u f f i c e s t o show the existence o f an open

p(z-x).La-')

There e x i s t s p(cs(E)

i s contained i n U and (1-b)x+bzGU f o r a l l b 4

C w i t h I b K a and z C V and f ( z ) - T ( j ! ) - % j f ( x ) ( z - x ) f((1-b)x+bz)

T,*(U,F).

(resp., X ( U , F ) = X,,(U,F)), f o r any u i n E. h-quasi b a r r e l l e d ( t h e h-barrel 1ed case f o l lows analogous-

x-nghb VCU such t h a t f / V i s continuous. F i x a ) l . such t h a t V:=(z:

such t h a t

= (2~i)-'J,~,=~

... .Set fm:V-+F :=( fm:m=l ,2,. - ) . 9i s bounded i n

/ bm+l(b-l) db f o r every z C V and m=1,2,

( x ) (z-x) and fm( z) := &(j ! (%(V,F) ,to) and hence equicontinuous,

since E i s h-quasibarrelled.

converges t o f over t h e p o i n t s o f V, the s e t ~ ( z ) : = ( f m ( z ) : m = 1 , 2 , . . ) l a t i v e l y compact i n F f o r each z i n compact i n (C(V,F),to)

v.

by

Since i s re-

By ASCOLI's theorem, 's; i s r e l a t i v e l y

and hence i t has an adherent p o i n t g LC(V,F).

For

every q G c s ( F ) and Z G V , we have q ( f ( z ) - g ( z ) ) = O . Since F may be assumed Hausdorff, f / V coincides w i t h g and t h e conclusion f o l l o w s . Theorem 12.2.36:

//

L e t E be a bornological b a r r e l l e d (DF)-space such t h a t

E and F. The f o l l o w i n g c o n d i t i o n s are equivalent: ( i ) E i s h-bornological; ( i i ) E i s h - b a r r e l l e d and ( i i i ) E i s V,(U,F)=

gD*(U,F)

f o r each U i n

472

BARRELLED LOCALLY CONVEXSPACES

h-quasibarrelled. Proof: Clearly, (if implies (iii). If (iii) holds, 12.2.35 shows that %(U,F)= PD(U,F) and by assumption, gD*(U,F). Fix an open subset U in E. In order to prove (ii), it suffices to show that, ifycp(U) is bounded on finite-dimensional compact subsets of U, then ?is locally bounded by 12.1.15. Define g:U+lm(~,F) as in 12.1.11. By 12.2.32 and 12.2.33, gC&’D,(U,F) and hence gEx(U,F). Thus 3; is locally bounded by 12.1.11. Now suppose that (ii) holds. By 12.2.35, %(U,F)= gD*(U,F) and hence Z(U,F)= XD(U,F) by assumption. By the claim in the proof o f 12.1.6(ii), Yc(U,F)= YD(U,F) and hence %(U,F)= Pc(U,F) as desired./,

x(U,F)=

Theorem 12.2.37: In sequentially retractive (s.r.1 (LB)-spaces, the notions of h-bornological, h-barrelled and h-quasibarrelled coincide. Proof: By 8.5.48, we may suppose that the (LB)-space (E,t)=ind((En,tn): n=1,2,..) is compactly regular. By 12.2.36, it suffices to prove that ZD(U,F) and K a compact subset of (E,t). =vD*(U,F) for any U and F. Let f&’J4D*(U,F) Since (E,t) is compactly regular, there exists a positive integer p such that K is contained and it is compact in ( U O E ,t ) . Set f :=f/(Uf\E ) . Clearly, P P P P Gnfp(x) = ^anf(x)oJ for each xt U n E and every n. Thus, f Cp?,,,((UnEP,tp) P P P , F ) . By 12.2.19, (E ,t ) is h-barrelled and hence 12.2.35 shows that f 6 P P P %((UnEp,tp),F). Thus, f(K)=fp(K) is bounded in F as required. //

In what follows we shall provide examples which show that the holomorphically significant properties introduced in this chapter are different. First observe that any metrizable non-barrelled space is an example o f a hbornological space which is not h-barrelled. If x(C R ( C R ) o , the space E:= sp( (CR ),U(x) ) endowed with the topology induced by CR is a Baire space which is not bornological (6.2.16). Thus, E is h-barrelled (12.2.19) but not h-bornological. Unfortunately, the topoloqical product of two h-barrelled or h-bornological spaces need not be even h-quasibarrelled as the following examples shows: Example 12.2.38: Let Xo be an infinite-dimensional complex Banach space. Ek:=Xo@ @(Xm:m=l,.,k) Set X,:=C for m=1,2,.. and E:=Xo @@(Xm:m=1,2,..), for each k. Xo is h-barrelled (12.2.19) and h-bornological (12.1.6(i)) and @ (Xm:m=1,2,..) is h-barrelled (12.2.23) and h-bornological (12.1.6(ii)). We shall see that E is not even h-quasibarrelled. If this is true, E is an

CHAPTER 12

473

example of a s t r i c t (LB)-space E = s - i n d ( E k : k = 1 , 2 , . .) which i s not h-quasibarr e l l e d and a l s o of a non-h-barrelled countable inductive l i m i t of h-barrelled spaces. By 12.1.8(a), f o r every positive integer m y there i s a function g m t % ( X 0 ) unbounded on Bm:=(xCXo: I l ~ l l S m - ~ )I .f x : = ( x k : k = O , l , . . ) belongs t o E , s e t 01 .& f(x):= ~ g m ( x o ) x m and f k ( x ) : = c g m ( x o ) x m f o r k = 1 , 2 , . . Since f/Ek i s holomorphic and E i s compactly regular, f i s bounded on compact subsets of E and ? : = ( f k : k = 1 , 2 , . . ) i s a l s o bounded on compact subsets of E . Moreover,xconverges pointwise t o f . Should 3;' be amply bounded, then f would be l o c a l l y e shall see t h a t f i s not l o c a l l y bounded a t 0 from where our debounded. W s i r e d conclusion follows. Suppose the existence of a 0-nqhb V i n E and a positive constant M such t h a t s u p ( I f ( y ) l :yCV) CM. Clearly, there i s k and such t h a t , i f Cm:=(a&C: l a \ & a ( m ) ) , t h e n positive s c a l a r s a(m), m=1,2,.., Bk O$CmCV and hence ff(y,O, O,a(k),O, . . . ) \ GM f o r every y & B k . T h u s , I g k ( y ) ( -L Ma(k)-' f o r each yEBk and t h a t i s a contradiction since gk i s unbounded on B k . Observe t h a t , since each f k belongs t o g(E)and since %converges t o f uniformly on compact subsets, we have a l s o seen t h a t ( X ( E ) , t o ) i s not sequentially complete.

..,

Now replace Xo i n the construction by CN. By DI,4.25, t h e r e e x i s t s a sequence (g : m = l Y 2 , . . ) , g , & Y ( C N ) such t h a t , given any 0-nqhb V i n C N , then m some g, i s unbounded on V and the above construction can be repeated t o obtain t h a t C N x C ( N ) ( which i s an inductive l i m i t of copies of CN o r a s t r i c t directed projective l i m i t of copies of C ( N ) ) i s not h-quasibarrelled. Proposition 12.2.39: Let E=s-d-proj(E :n=1,2,. .) be the s t r i c t directed n projective l i m i t of a sequence of spaces E n in which weakly holomorphic mappings on open s e t s a r e holomorphic. T h e n , y ( U , F s ) = F ( U , F ) f o r every open s e t U of E. Proof: Let Q n : E 4 En the n - t h continuous projection and suppose t h a t U= Q, -1(Ql(U)) and Qn+l-l(0)CQn-l(O) f o r each n . I t s u f f i c e s t o show t h a t f f a c t o r i z e s through some E n . I f not, s e l e c t sequences ( x n : n = 1 , 2 , . . ) in U and (yn:n=1,2,..) i n E such t h a t Qn(yn)=Oand f ( x n + y n ) # f ( x n )f o r each n . Define b h f(xn+byn)-f(xn) f o r b CC. This mapping i s non-constant and e n t i r e and, by 0 . 5 . 1 2 ( i ) , choose (b(n):n=1,2,..) i n C such t h a t ~ ~ f ( x n + b ( n ) y n ) - f ( x >n )nl,I 0 . I I being the norm on the Banach space F. Given v C F ' , vof i s holomorphic and hence vof f a c t o r s through some Em. Then, vof(xn+b(n)yn)-v*f(xn)= 0 f o r a l l n 7 / m , a contradiction since (f(xn+b(n)yn)-f(xn):n=l,2. . ) converges

BARRELLED LOCALLY CONVEXSPACES

474

weakly t o zero.

/I

C o r o l l a r y 12.2.40: Example 12.2.41: nor h-bornological.

CNxC(N) i s a MN-space,

but n o t h-quasibarrelled.

A h - q u a s i b a r r e l l e d space which i s n e i t h e r h - b a r r e l l e d L e t G be t h e t o p o l o g i c a l p r o d u c t T ( E i : i

t I ) o f metri-

z a b l e spaces Ei such t h a t c a r d ( I ) = c and w i t h a t l e a s t one o f then which i s n o t barrelled. I f xCG\Go, barrelled nor bornological

s e t E:=sp(GoU(x)).

. On

w i t h respect t o the i n j e c t i o n s T ( E i : i t a b l e p a r t s o f I (see 0.1.3

By 6.2.17,

E i s neither

t h e o t h e r hand, Go c a r r i e s t h e f i n a l topology 6P)-

5,

P r u n n i n g through a l l coun-

) . Proceedina as i n t h e p r o o f o f 12.2.21,

if U

i s a non-void open subset o f Go and i f t stands f o r t h e p r o d u c t topology, (U,t)

has t h e f i n a l topology w i t h r e s p e c t t o t h e canonical i n c l u s i o n s

UA'rr(Ei:icP)-+ hence (E,t)

(U,t).

lde s h a l l see t h a t (Go,t)

w i l l be h - q u a s i b a r r e l l e d by 12.1.22.

L e t ? be a subset o f X(U,F) -la(F,F)

i s h - q u a s i b a r r e l l e d and

bounded on compact subsets o f U and s e t g:U

f o r t h e G-holomorphic f u n c t i o n d e f i n e d i n 12.1.11.

t o show t h a t g i s continuous. Since each F ( E i : i t P ) hence h - q u a s i b a r r e l l e d ) , g/(UATT(Ei:i

CP))

ments above, g i s continuous as d e s i r e d .

I t i s enough

i s m e t r i z a b l e ( and

i s continuous and, by o u r com-

/I

12.3 Notes and Remarks.

A p r e c i s e understanding o f t h e c o n t e n t o f t h i s chapter r e q u i r e s a knowledge o f t h e b a s i c t h e o r y o f I n f i n i t e Holomorphy and those r e s u l t s which a r e used droughout o u r e x p o s i t i o n a r e l i s t e d i n Chapter 0. The i n t e r e s t e d reader should l o o k a t t h e p i o n e e r i n g work o f NACHBIN,(l) and t h e more r e c e n t books BARROS0,(5), CHAE,(l), DINEEN,(DI) and MUJICA,(5). B o r n o l o g i c a l , b a r r e l l e d , quasi b a r r e l l e d and Mackey spaces appear when l o c a l l y convex spaces a r e c l a s s i f i e d according t o t h e i r behaviour w i t h r e s p e c t t o b a s i c p r i n c i p l e s and p r o p e r t i e s o f L i n e a r F u n c t i o n a l A n a l y s i s . Replacing l i n e a r mappings by holomorphic mappings, h - b o r n o l o p i c a l (12.1.5) h - b a r r e l 1ed and h-quasi b a r r e l 1ed (12.1.12) and h-Mac key (here c a l l ed MN-) spaces can be d e f i n e d (see NACHBIN,(E) ,(3) and BARROSO,MATOS,NACHBIN,(4)). T h i s l a s t r e f e r e n c e i s a readable account o f t h e b a s i c t h e o r y o f holomorphic a l l y s i g n i f i c a n t p r o p e r t i e s w i t h f u l l p r o o f s and t h i s has been o u r p r i n c i p a l source o f i n s p i r a t i o n f o r p r e s e n t i n g t h i s m a t e r i a l . The f o l l o w i n g r e s u l t s can be seen t h e r e : 1 2 . 1 . 6 ( i ) ; 12.1.7; 12.1.11; 12.1.14; 12.1.15; 12.2.18; 12.2.19; 12.2.20; 12.2.23; 12.1.18 and 12.1.19. There i s another source which has i n f l u e n c e d o u r p r e s e n t a t i o n , namely ARAGONA,(l) where one can f i n d 12.1.2; 12.2.32; 12.2.33; 12.2.34; 12.2.35; 12.2.36 and 12.2.37. I n BARROSO,MATOS,NACHBIN,(4) one can f i n d t h e p r o o f o f every (LS)-space

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b e i n g h - b o r n o l o g i c a l T h i s r e s u l t was extended by DINEEN,(l) t o (DFM)-spaces ( 1 2 . 2 . 6 ( i i ) ) and o u r p r o o f uses i d e a s t o be found i n ARAGONA,(l). 12.1.8 f o l l o w s NACHBIN,(3). The f a c t t h a t i n e v e r y i n f i n i t e - d i m e n s i o n a l complex normed space t h e r e e x i s t s an e n t i r e complex-valued f u n c t i o n unbounded on some bounded s e t f o l l o w s from a deep r e s u l t o f JOSEFSON,(1),(2) and NISSENZ V E I G , ( l ) , namely 12.3.1: I f E i s an i n f i n i t e - d i m e n s i o n a l normed space, t h e r e e x i s t s a sequeni s a n u l l sequence ce(f(m):m=l,Z,..) i n E ' such t h a t ( :m=1,2,..) f o r any x i n E and y e t llf(m)ll = 1 f o r a l l m. which s o l v e d a l o n g - s t a n d i n g open q u e s t i o n i n Banach Space Theory. The i d e a behind 12.2.5 and 12.2.6 appears i n BOLAND,DINEEN,(l) and those r e s u l t s t o e t h e r w i t h 12.2.4; 12.2.7; 12.2.8; 12.2.9 and 12.2.11 can be seen i n MORAES,~l),(P), where she asks i f s t r i c t i n d u c t i v e l i m i t s o f F r e c h e t Monte1 spaces such t h a t t h e i r s t r o n g d u a l s a r e h - b o r n o l o g i c a l have a c o n t i nuous norm. 12.2.38 i s based on i d e a s o f DINEEN,(4) and appears i n NACHBIN,(3) (see a l s o BARROSO,MATOS,NACHBIN,(4) which i n c l u d e s a l s o 12.2.3). The Continuum Hypothesis i n 12($)3 can be e l i m i n a t e d as showed by JECH and a l s o DINEEN,(3), Prop. 2 . Then, C f o r uncountable A serves as an example o f an u l t r a b o r n o l o g i c a l space h i c h i s n o t a MN-space, s i n c e a non-continuou -homogeneous polynomial p:C?A) ~ ~ ( A X can A ) be c o n s t r u c t e d such t h a t p:CTAf-+l2(AxA), i s continuous (see DINEEN,(3)). 12.1.20; 12.1.21 and 12.1.22 can be seen i n GALINDO,GARCIA,MAESTRE,(l). 12.2.12; 12.2.13; 12.2.14 and 12.2.16 appear i n MAESTRE,(l) where t h e t e c h n i q u e s o f MORAES,(1),(2) a r e used t o s t u d y p r o d u c t s . 12.2.7 i s due t o BARROSO,NACHBIN,(P). 12.2.28 i s a HARTOGS' t y p e theorem (see NOVERRAZ,(l),p.31) and 12.2.29 extends r e s u l t s o f BOCHNAK,SICIAK,(l) ( w i t h t h e e x t r a assumption o f ExG b e i n g B a i r e ) and MATOS,(l) ( w i t h E a F r 6 c h e t space), and can be seen i n BONET,GALINDO,GARCIA,MAESTRE,(G). 12.2.30 and 12.2.31 can be found i n MAESTRE (1). 12.2.39 and 12.2.40 a r e s u b s t a n t i a l l y due t o DINEEN,(3) and o u r formuwhere t h e f o l l o w i n g r e s u l t can t i o n f o l l o w s BONET,GALINDO,GARCIA,MAESTRE,(6) a l s o be seen 12.3.2: I n ( C ( X ) , t o ) , t h e n o t i o n s o f b a r r e l l e d and h - b a r r e l l e d space c o i n c i d e a n d e same i s t r u e f o r t h e q u a s i b a r r e l l e d case. T h i s l a s t r e s u l t i s o b t a i n e d v i a techniques due t o SORAGGI (see DI,Ch.6). The n o t i o n s o f p o l y n o m i a l l y b o r n o l o g i c a l , b a r r e l l e d , e t c . . can a l s o be t r e a t e d (see ARAGONA,(l)) b u t we s h a l l n o t e n t e r t h i s t o p i c here.