A locally c o n v e x space w h i c h is n o t an ~ - s p a c e
ERIK GRUSELL
T h e p u r p o s e of this p a p e r is to e x h i b i t an e x a m p l e of a locally c o n v e x topological v e c t o r space w h i c h is n o t an og-space.
T h e class o f locally c o n v e x spaces for which u n i f o r m h o l o m o r p h y holds was f i r s t i n t r o d u c e d b y N a c h b i n [3]. I t is e x t e n d e d b y t h e class o f (o-spaces whose d e f i n i t i o n was f i r s t p u b l i s h e d b y D i n e e n [1]. D i n e e n has since t h e n g i v e n a new definition, w h i c h covers m o r e general s i t u a t i o n s [2]. I n this p a p e r we use t h e t e r m
"m-spaces" for spaces which in D i n e e n ' s n e w t e r m i n o l o g y are called "C-m-spaces".
T h e n o t i o n o f an o J-space is of a c e r t a i n i m p o r t a n c e in infinite dimensional h o l o m o r p h y , as D i n e e n shows in [1] a n d [2].
W e h a v e r e c e n t l y b e c o m e a w a r e t h a t M. S c h o t t e n l o h e r has also c o n s t r u c t e d a space which is n o t a n ~o space. His e x a m p l e is a s u b s p a e e o f t h e space o f holo- m o r p h i e m a p p i n g s f r o m a c e r t a i n B a n a c h space into itself.
Definition.
A locally c o n v e x space E is ano~-space
if for e v e r y f u n c t i o n f: U--* C, w h e r e U ~ E is o p e n a n d c o n n e c t e d , a n d f is G g t e a u x h o l o m o r p h i e a n d continuous, t h e r e exists a s e q u e n c e {pl}~~ o f c o n t i n u o u s s e m i n o r m s such t h a t f is also c o n t i n u o u s in t h e t o p o l o g y g e n e r a t e d b y t h e s e q u e n c e {p~}~~ 1.L e t E ~ @ten+ C algebraically, i.e.
E = {(Xr)ren+; X~ ~ 0 o n l y for a f i n i t e n u m b e r o f coordinates}.
I f A c R+ we let (Xr)reA d e n o t e t h e p o i n t (Yr)~ea+ w h e r e y, = x~ if r E A a n d y ~ = 0 otherwise.
D e f i n e a t o p o l o g y on E b y t h e set o f s e m i n o r m s
{p}
u{p~k; s c l t + \ Z + , ~ e Z+},
w h e r e2 1 4 E 1~ii( GRUSEL:5
a n d
p ( x ) = ~ [Xrl if X~--(X~)~eR +,
rCR+
psi(X) = ~ Ixjl " k ~
j = l
Here {(n,j)j~l; s E R + ~ Z + } is t h e set of all strictly increasing sequences of n u m b e r s belonging to Z+, where t h a t set has been p u t in a one-to-one
correspondence w i t h R + ~ Z + .
PROPOSITIOn. Define a function f: E --~ C by
f(x) = f((xr)reR+) ~-- ~ x~ " f,(x),
s E R + ~ Z +
where
L(x) = L((x~)) = ~ xj- x~<
j = l
Then f is Gdteaux-holomorphic and continuous, but
f
is not continuous in any semi- metrizable locally convex topology on E, weaker than the given one.Proof. a) f is continuous. The functions x----(Xr)reR+ ~-~ X~o are continuous for e v e r y r o C R+, for t h e y are linear a n d p(x) < s implies IX~ol < e.
F o r s C R + ' ~ Z + t h e seminorms p~k, k EZ+, are chosen so t h a t f~ is continuous:
z 0
L e t x ~ (x ~ be given a n d choose m > m a x { r ; x r ~ 0}. T h e n
j = l j = l j = m + l
for a n y x E E, a n d f,1 is clearly continuous in t h e topology d e f i n e d b y t h e semi- n o r m p, i.e. Ef, l(x ~ § y) --fsl(X~ < e if p(y) is small enough. N o w
If,2( x~ + Y) --f,2(x~ ---- If,2( x~ + Y)I --~ ] ~ Yj( x~ § Ys)nSJl <Psk(Y)
j=m+ l
if p(y) < 1 a n d k >
lx~
+ 1.Thus f~l a n d fs2 are b o t h continuous in x ~ which implies t h a t fs ~ f,1 § f,2 is continuous there. B u t x ~ was a r b i t r a r y , a n d fs is t h e n continuous everywhere,
L e t x ~ = (x~ still denote a n a r b i t r a r y clement of E. T h e n we h a v e x O = = (xO)r A +
where A c R + ~ Z + a n d B C Z+ a n d b o t h are finite.
A LOCALLY CON-VEX SPACE W I I I C I t IS :NOT A N (D-S:PACE 215
f(x) = E x~L(x) = E x~ .L(x) + E x~L(x) = L ( x ) +A(x)
r C R + ~ Z + r ~ A r E M
for a n y x E E , if M = ( R + ~ Z + ) ~ A .
Now, f t is a f i n i t e s u m o f f i n i t e p r o d u c t s o f c o n t i n u o u s f u n c t i o n s a n d t h u s c o n t i n u o u s . F o r t E M a n d y E U, w h e r e
u = {y; p(y) < [1 + ~ ix~ -~} n {y; p(y) < 1},
=1
we h a v e
rfXx ~ + y)l = i E (x 2 + yj)yr'Jl _< ~ i(x] + yj). y,I. ly, I ~ <
j = l j = l
< ly,["q -1 [yt[ j = 2.
- - j = l "= "=
This gives
If2( x~ -~ Y) --f2(x~ = If2( x~ -~ Y)[ = I Z y , f ~ ( x~ § y)[ -< 2 . Z lytl ~< 2 . p ( y ) .
t E M t ~ M
T h u s , f2 is c o n t i n u o u s in x ~ T h e n f = f i - k f2 is c o n t i n u o u s in x ~ b u t x ~ was a r b i t r a r y a n d t h e r e f o r e f is c o n t i n u o u s in all o f E .
b) f is not continuous i n a n y semi-metrizable locally convex topology weaker than the given one.
L e t P be a d e n u m e r a b l e subset o f t h e set of s e m i n o r m s d e f i n i n g t h e t o p o l o g y on E . T h e n t h e r e is a n u m b e r t E R + ~ Z + such t h a t
n , j - - j
- - - + ~ w h e n j - ~ oo nsj
for e v e r y s c o r r e s p o n d i n g t o a s e m i n o r m in P .
E v e r y z e r o - n e i g h b o u r h o o d in t h e t o p o l o g y d e f i n e d b y P contains' a set of t h e
f o r m :
v = {x; p,k(x) < ~, p.k e ~1} n {x; p(x) <
~},
where P1 is a f i n i t e s u b s e t o f P .
I f q(x) < 1 a n d x = (Xs)sez+ t h e n x E U, p r o v i d e d 1
q ( x ) = - 9 } ]xjl . h ' J
e j=l where
a n d
h = m a x { j ; p s i E P 1 for some s E R + ~ Z + } n i = m a x {nsi; p~k C P1 for some k E Z+}.
216 E~IK G~VSELL
L e t
x ' ~ s . 2 ) ~ 9 , m = 1 , 2 . . . .
= 1
a n d x 0__ (x~),el~+,0 x~~ = 0 i f s =#t, x ~ 2, w h e r e t is t h e n u m b e r m e n t i o n e d a b o v e ( t h u s ( n , j - - j ) / n j - - > ~ w h e n j - - > ~ ) . T h e n q(x m) < 1, m = 1 , 2 , . . . a n d
S 2ntj m 2'~tj J
f ( x ~ 4- x ~) ---- x~ 9 f , ( x ~ + x ' ) : 2 . 2F+~ h,~j - - e - ~ h"J
j = t j = l
I f m is c h o s e n l a r g e e n o u g h , t h e l a s t t e r m a b o v e is a r b i t r a r i l y l a r g e , i.e. f(:c ~ 4- U) is n o t b o u n d e d .
References
1. I)INEE~ *, S., t I o l o m o r p h i c a l l y complete locally convex vector spaces, Sdm. P. Lelong, ] 971/72, Lecture notes in mathematics 332 (1973), 77--111.
2. --~>-- Applications of surjeetive limits to infinite dimensional holomorphy. To appear in Bull. Soc. Math. France.
3. •ACHBIN, L., Uniformitd d'holomorphie et t y p e exponentiel. Sdm. P. Lelong, 1969/70, Leetu,re notes in mathematics 205 (1971), 216--224.
Received October 1'[, 1973;
in revised f o r m A p r i l 30, 197~
Erik Grusell
D e p a r t m e n t of M a t h e m a t i c s S y s s l o m a n s g a t a n 8 S-752 23 Uppsala Sweden
