PACIFIC
JOURNAL OF MATHEMATICS
Vol. 120, No. 2, 1985
GROTHENDIECK
LOCALLY CONVEX SPACES
OF CONTINUOUS VECTOR VALUED FUNCTIONS
FRANCISCO
J. FRENICHE
Le ^{X, E) be he space o
con inuous
unc ions
om
he com-
ple ely
egula
Hausdo
space X
in o
he
Hausdo
locally con ex
space E,
endowed
wi h
he
compac -open
opology.
Ou aim is o
cha ac e ize
he ^(X, E) spaces
which
ha e
he
ollowing
p ope y:
weak-s a
and
weak
sequen ial
con e gences coincide in he
equicon inu-
ous subse s o ^(X, E)'. These spaces a e
he e
called
G o hendieck
spaces. I is
shown
ha
in he
equicon inuous
subse s o E' he σ(E', E)-
and β(E',
^-sequen ial
con e gences coincide, i ^(X, E) is a
G o hendieck
space and X con ains an
in ini e
compac
subse .
Con-
e sely,
i X is a
G-space
and E is a s ic
induc i e
limi
o
F eche -Mon el spaces ^(X, E) is a
G o hendieck
space.
The e o e,
i
is
p o ed
ha
i £ is a
sepa able
F eche space,
hen
E is a Mon el space
i and
only
i
he e
is an
in ini e
compac
Hausdo
X
such
ha
,
E) is a
G o hendieck
space.
1. In oduc ion. In his pape X
will
always deno e a comple ely
egula Hausdo opological space, E a Hausdo locally con ex space,
and
&( X, E) he space o con inuous unc ions om Xin o E, endowed
wi h he compac -open opology. When E is he scala ield o eals o
complex numbe s, we w i e ^(X) ins ead ^( X, E).
I
is well known ha Ή{X, E) is a Mon el space whene e ^(X) and
E
so a e, hence, i and only i X is disc e e and E is a Mon el space (see
[5],
[16]).
We s udy wha happens when X has he ollowing weake p ope y:
he
compac subse s o X a e G-spaces (see below o de ini ions).
We ob ain in Theo em 4.4 ha i £ is a F eche -Mon el space and X
has
ha p ope y, hen ^( X, E) is a G o hendieck locally con ex space.
The
key in he p oo is he ollowing ac : e e y coun able equicon inuous
subse o ^(X, E)' lies, ia a Radon-Nikodym heo em, in a sui able
L τ9
Eβ). As a consequence o a heo em o Mύjica [10], he same esul
is ue when E is a s ic induc i e limi o F eche -Mon el spaces.
In
§3 we s udy he con e se o 4.4. In Co olla y 3.3 i is p o ed ha i
X con ains an in ini e compac subse , E is a F eche sepa able space and
#(
X, E) is a G o hendieck space, hen E is a Mon el space. This p ope y
cha ac e izes
he Mon el spaces among he F eche sepa able spaces.
345
346
FRANCISCO
J.
FRENICHE
Finally,
in §5 we s udy he G o hendieck p ope y in &(Σ, E), he
space o Σ- o ally measu able unc ions, by using he esul s o
%?{X,
E).
2.
Gene ali ies.
A compac Hausdo opological space K is called
a
G-space whene e ^(K) is a G o hendieck Banach space, i.e. he
weak-s a and
weak
sequen ial con e gences coincide in %>(Ky [6].
We ex end he e his concep o comple ely egula spaces.
2.1.
DEFINITION.
X is a G-space i
e e y
compac subse K o X is a
G-space.
I XΊs compac , bo h de ini ions coincide [6]. Le us ema k ha he e
exis
non-compac non-disc e e (/-spaces. Indeed, he opological subspace
o he S one-Cech compac i ica ion o a coun able disc e e se ob ained
emo ing a clus e poin , is such a space.
We in oduce a new de ini ion o G o hendieck locally con ex space,
so ha
<&{
X) is a G o hendieck space i and only i X is a G-space.
2.2.
DEFINITION.
E is a G o hendieck space whene e he σ(E', E)-
and
σ(E E'^-sequen ial con e gences coincide in he equicon inuous
subse s o E'.
In [17] he ΓG-spaces a e de ined as hose spaces E in which he
σ(E', E)- and σ(E Zs'^-sequen ial con e gences coincide. When one
deals wi h ^( X) spaces, ou de ini ion seems o be mo e easonable
han
ha
o [17] (see 2.4 and 2.5).
The
ollowing
pe manence p ope ies o he
class
o G o hendieck
locally con ex spaces a e
easy
o see, hus we s a e hem wi hou
p oo .
2.3.
PROPOSITION,
(a) E is a G o hendieck space i and only i e e y, o
some, dense subspace o E so is.
(b)
Le T: E
—>
F be a linea con inuous ope a o such ha o e e y
bounded subse B o F he e is a bounded subse C o E so ha B is con ained
in
he
closu e
o T(C).
Then
F is a G o hendieck space i E so is.
(c)
// E is he induc i e limi o he sequence (En) o G o hendieck
spaces, and i e e y bounded subse o E is con ained in some En, hen E is a
G o hendieck space.
2.4.
THEOREM.
<g(X, E) is a G o hendieck space i and only i
< (K,
E)
so is o e e y compac subse K o X. In pa icula , X is a
G-space
i and
only
i %?(X) is a G o hendieck space.
GROTHENDIECK
LOCALLY
CONVEX
SPACES
347
P oo .
Le us
ecall ha ,
i K is a
compac
subse
o X, he
es ic ion
map
Γis a
con inuous linea ope a o om
^(X, E)
in o
^(K, E).
I
B c %{K, E) is
bounded, hen
he
bounded
subse
C o
<V(X9
E
whose
elemen s
g
can
be
w i en
g =
Σπ<w
/„(-)en wi h/M
€ ^(
X),
0 < /„
<
1,
Σn<m n
^ 1? and
<?„
e
U{A(*:):
Λ e 5},
sa is ies
Γ(C)
z>
5 (see
[14,1.5.3]).
I
«χ X, £) is a
G o hendieck space, #(
JBΓ,
E) so is by
2.3(b).
Con e sely,
le (g'n) be an
equicon inuous
and
σ(V(X,
E) V(X9
£))-null sequence.
By
[14,
III.3
and
III.4],
he e
exis
a
compac
subse
K o X
and
an
equicon inuous sequence
(h'n) in ^(K, E)'
such ha g'n=h'noT
o all /ι e N.
Since
(A'J is
σ(V(K9
E)'9
T( (K, £)))-null
and
equicon inuous,
i is
also
σ(<g(K9
E)
<V(K9
E)")-nυΆ
i V(K, E) is a
G o hendieck space.
I
ollows
ha
(g'n)
is σ(ί (Z, E) V(X,
E)")-mύl.
2.5.
REMARK. We use an
example
o
Haydon [4]
o
show
ha ,
while
in
he
class
o
ba elled spaces
he
ΓG-spaces
and he
G o hendieck spaces
do coincide, his
is no
ue
in
gene al.
Choose,
o
each in ini e sequence
in N, a
clus e poin
in he
S one-Cech compac i ica ion
o
N,
and le X be he
opological subspace
o ha compac i ica ion, o med
by N and
hese clus e poin s. Then
e e y
compac
subse
o X is
ini e,
V(X) is
in aba elled
and
e e y
/e
V(X) is
bounded.
By
Theo em
2.4, X is a
G-space.
Le
/„'(/)
=
n~ι (n)
o all/ e
<g(X)
and n e
N. Then
(/;) is a
σ(^(X) , V(X))-nuΆ
sequence
in &(X)
ha
is no
o((S(X) <^{Xyynu because
i is no
equicon inuous.
3.
Necessa y condi ions
o
Ή(X,
E) o be a
G o hendieck space.
I
is
well
known,
and
easy
o see,
ha
Φ(X) and E a e
opologically
isomo phic
o
complemen ed subspaces
o V(X9 E). By
2.3(b),
V(X) and
E mus
be
G o hendieck spaces
i ΦζX, E) is
such
a
space.
Howe e , unless
X is
pseudo ini e,
i.e.
hei compac
subse s
a e
ini e
(hence
ί (-Y,
E) is a
G o hendieck space
i and
only
i E so is, by
Theo em
2.4), E has a
s onge p ope y
i Φ(X9 E) is a
G o hendieck
space,
as we
p o e
in
he nex heo em.
To
p o e
i we
ecall
he
ollowing
esul
o [2]:
THEOREM
A. Le E and F be
Hausdo locally con ex spaces,
and
suppose
ha
F
con ains
a
subspace opologically isomo phic
o he
subspace
o c0 whose elemen s ha e only ini ely
many
non-ze o coo dina es.
I
he in ec i e enso p oduc
F
<8>εE
is a
G o hendieck space, hen
he
σ(E
E)- and β(E
E)-sequen ial con e gences coincide
in he
equicon inu-
ous subse s
o E .
348
FRANCISCO
J.
FRENICHE
As was no ed in [2], i X is no pseudo ini e, hen ^( X) con ains a
subspace opologically isomo phic o he abo e men ioned subspace o cQm
Mo eo e , he injec i e enso p oduc <g(X)®εE can be linea and
opologically iden i ied wi h a dense subspace o ^{X, E), namely, he
subspace o all ini e dimensional alued elemen s o ^{X, E). Thus we
ob ain
om Theo em A and P oposi ion 2.3 (a):
3.1.
THEOREM. // ^(X, E) is a
G o hendieck
space
and X
con ains
an
in ini e
compac
subse ,
hen
he σ(E', E)- and β(E
E)-sequen ial
con e -
gences
coincide
in he
equicon inuous
subse s
o E'.
3.2. REMARK. By Theo em 2.4, i X is pseudo ini e and £ is a
G o hendieck
Banach space, ^(X, E) is a G o hendieck space. Howe e ,
i E is in ini e dimensional, he conclusion o Theo em 3.1 does no hold
[11].
Using Theo em 3.1 and [7, 11.6.2], we ob ain he
ollowing
co olla y,
con e se o Theo em 4.4:
3.3.
COROLLARY.
I E is a
F eche
sepa able
space,
X is no
pseudo ini e
and ^(X, E) is a
G o hendieck
space,
hen
E is a Mon el
space.
3.4. REMARK. I is unknown o us i Co olla y 3.3 is ue wi hou he
sepa abili y assump ion on E. This is ela ed wi h he
ollowing
ques ion
aised in [7, pg. 247]: is a F eche space E al eady a Mon el space i
e e y
σ(E', ^-con e gen sequence in E' con e ges o β(E', E)Ί
4.
Su icien
condi ions
o V(X9 E) o be a
G o hendieck
space.
We shall need some ac s abou ec o in eg a ion, many o hose can be
ound in [1] and [15].
Le (X, Σ, T) be a comple e measu e space wi h τ(X) < 1. We deno e
by S?(Σ,E) ( esp. @(Σ,E), L τ, E),
L°°(τ,
E)) he ec o space o
Σ-simple ( esp. Σ- o ally measu able, τ-in eg able, τ-essen ially bounded)
E- alued (classes o ) unc ions. Recall ha S {Σ,E) and @(Σ, E) a e
endowed wi h he uni o m con e gence opology, and ha he opology o
Lι{τ, E) is de ined by he semino ms u -> /p(u(x))
dτ(x),
whe e/? uns
o e he se o all con inuous semino ms in E (unless con a y speci ica-
ion,
all in eg als
will
be ex ended o X).
The
ollowing
Radon-Nikodym heo em is p o ed in [1]:
THEOREM B. // E is a
quasi-comple e
(CM)-space,
μ: Σ -> E is a
coun
ably
addi i e
ec o
measu e,
o
bounded
a ia ion
and
τ-absolu ely
con inuous,
hen
he e
exis s
u e Lι(τ, E)
such
ha μ{A) = jA u(x) dτ(x)
o
e e y
A ^ Σ.
GROTHENDIECK
LOCALLY CONVEX SPACES
349
Le
us
ecall ha
E is a
quasi-comple e (CM)-space,
i , o
ins ance,
i
is ei he
a
F eche -Mon el space
o a
(DF)-Mon el space [1].
Fi s ly
we
ex end
he
classical duali y heo em
Lι - L00 o L τ9
Eβ),
whe e
E is a
F eche -Mon el space.
The
ollowing
lemma
can be
easily
p o ed.
As
usual,
pL
will
deno e
he
gauge
o
he absolu ely con ex
se L in i s
linea span.
4.1.
LEMMA.
// u
e^(Σ,
E'
namely,
u =
Σi^mχAe/
i
wi h
(Ai)i^m
disjoin
in Σ,
hen
ί
pBo(u(x))dτ{x)
< τl U
AήsuppBo(e;)
o
e e y
bounded
subse
B o E.
4.2. THEOREM. Le
E be a
F eche -Mon el
space.
The
ela ion
(1) u'{u)=
u{x){υ{x))dτ{x)
o
all
u €=
Z/(τ,
E'β)
de ined
o u e
L τ> Eβ)' and
e
L°°(τ,
E), is an
algeb aic
isomo phism
be ween
L τ,
E'β)'
andU°(τ9
E
P oo .
Le υ e
L°°(τ,
E).
The map
x -»
u{x)(υ(x))
is
measu able
o
e e y
u e L 9
Eβ), because
is
s ongly measu able and
he
asse ion
is
clea ly ue when
e
^(Σ,
E).
Fu he mo e,
i Z
G
Σ is a
τ-null
se
such ha
B =
(S Z)
is
bounded, hen
we
ha e
(2)
u(x)( (x)) <pB0(u(x))
o
e e y
x ^ X Z.
Hence
x ->
w(jc)(ί;(x))
is
τ-in eg able,
and we can
de ine
a
linea
o m
«' on Lι(τ,
Eβ)
by (1).
Mo eo e ,
i
ollows
om
(2)
ha
u' is
con inuous.
Con e sely,
ix u'
G
LX(T,
ί^)'.
The e
exis s
a
bounded subse
B o E
such ha
(3)
ί
PB°(U(X))
dτ(x) < 1
implies
(w^w))
< 1
o
e e y
w
e L τ,
Eβ).
We de ine
a
map
μ: Σ -*
£"'
by
(4)
μMXeO
= «'(χ^0
o
e e y
A e Σ and e' e £' (i
ollows
easily
om Lemma
4.1 and (3)
ha
μ(A) e E").
Since
£ is
e lexi e
we
can suppose ha
μ(A) e £.

350
FRANCISCO
J.
FRENICHE
Clea ly,
μ: Σ
->
E is a
ini ely addi i e ec o measu e.
We
shall show
ha
μ is
coun ably addi i e:
le A be
he union
o
he disjoin sequence
(An)
in Σ.
Gi en
an
absolu ely con ex ze o-neighbo hood
U in E and
ε
> 0, we
chodse
λ
wi h
0 < λ < oo
such ha
B c
λU, and
a0 e N
such
ha
λτ(U >w J( )
< ε o
e e y
m > m0.
Since
e'{μ{A))
- £
e'(μ{An))
=
u'(xun>mAn
e')
n<m
i
ollows
om Lemma 4.1 and (3) ha
e'(μ(A))
- Σ
e'(μ(An))
< e
n<m
o e e y
m > m0
and
e' e
C/°,
as
desi ed.
Fu he mo e,
iΐ A
=Un<mAn whe e {An)n<m
is
disjoin
in
Σ, and
i
ε
> 0,
he e
exis s
(e'n)n<min
U°
such ha
Σ
P (μ(An))
< Σ
e'MAH)) + ε =
«'( Σ XΛΔ
+ e.
n<m
n<m n<m
Hence
hep^ a ia ion
o μ
sa is ies
he inequali y
Vpuμ(A)
<
λτ(A),
om Lemma 4.1 and (3) again.
Thus
μ is
τ-absolu ely con inuous
and has
bounded a ia ion.
By
Theo em
B, he e
exis s
e
Lx(τ,
E)
such ha
(5)
μ(A) = ί
υ{x)dτ{x)
o
e e y
A
<Ξ
Σ.
JA
We claim ha
υ is
τ-essen ially bounded and
sa is ies
(1). Indeed,
le
{UJ)J
be a
coun able
basis
in E o
absolu ely con ex ze o-neighbo hoods.
Choose,
o
eachj
e
N,
λy
such ha
0 < λy <
oo and
B c
λ^I/,.
By Lemma 4.1, (3), (4) and (5), we ha e
(6)
( e'( (x)) dτ(x) <λ τ(A)
JA
o
all e'
<Ξ
U/,
A e Σ
andy
<Ξ
N.
Le (e'j
k)k be a
sequence
in U
such ha
pυ(e) =
supk e'jk(e)
o
e e y
e & E.
By (6), he e
exis s
Z e Σ
wi h τ(Z)
= 0
such ha e
jΛ{ (x))
< λj
o sΛ x
^ X Z
and
ally,
c
G N. Hence
υ(X
Z)
is
bounded
in
£
Finally,
i
ollows
om (4) ha (1)
is
ue
o all
MGy(Σ, £"), and,
by densi y,
o
e e y
u ^ Lι(τ,
Eβ). This concludes he
p oo .
Assume ha
X is
compac Hausdo
and Σ
con ains
he
Bo el
subse s
o X.
Fo each
u e L τ,
Eβ), deno e
by u
he ec o measu e
o
densi y
u
wi h espec o T.
I p is a
con inuous semino m
in E,
he subse
GROTHENDIECK
LOCALLY CONVEX SPACES
351
F
o L τ,
Eβ) de ined
by he
condi ion
Vp u{X)
<
oo,
is a
linea
sub-
space.
I
M
G JF hen
u has
bounded semi a ia ion, hus
i
de ines
a
con inuous
linea o m on <^(Σ, £), which ex ends
by
con inui y
o he
whole space
&(Σ, E) [15]. Le
ΓwG <g(X,
E)' be he
es ic ion
o
V(
X, E) o
his linea o m,
i.e.
(7) (Tu)(g) =
g(x)
d u(x)
o e e y
g
€Ξ
<g(X9
E).
4.3.
LEMMA.
ΓΛe map T: F
-> #(*, J?)' έ e/ineJ
y (7) is
<2
linea
con inuous
ope a o ,
when
^{Xy E)' is
endowed
wi h
he
s ong
opology
wi h
espec
o
<g(X9
E).
P oo .
We
ha e,
o
each
u e F,
(8)
(Tu)(g) = u(x)(g(x)) d (x)
o e e y
gE
^{X9E). Indeed, he domina ed con e gence heo em and
a
s anda d densi y a gumen show ha
i
su ices
o see (8)
when
g
belongs o5^(Σ,
E),
ha
is
i ially ue.
Le
H be a
bounded subse
o
ί?(
JST,
£). Then
B =
U{g(X):
g e i }
is
a
bounded subse
o E.
Hence,
by
(8), (Tu)(g)
<
jpBo(u(x))
dτ(x)
and
he lemma
ollows.
We a e now eady o p o e he su icien condi ion:
4.4.
THEOREM.
Le X be a
comple ely
egula
Hausdo
G-space
and
E
a
F eche -Mon el
space.
Then
^(X, E) is a
G o hendieck
space.
P oo .
By 2.4 we can
suppose, wi hou
loss
o
gene ali y, ha
X is
compac .
Le (g
n)n
be an
equicon inuous sequence
in ^(X, E)'. By
[14, IΠ.4.5]
he e
exis s
a
con inuous semino m
p in E
such ha
Vpμn(X)
< 1, o
e e y
n e
N, whe e
μn is
he ep esen ing measu e
o g
n
[14, III].
Le
T =
Σn2~nVpμn.
T is a
coun ably addi i e [0,l]- alued Bo el
measu e,
by
[14,
ΠI.2.5].
Le
Σ be
he comple ed
σ- ield
o
he Bo el
ield
o
X
wi h espec
o T. We
shall deno e also
by T and μn he
na u al
ex ensions
o
he ea lie measu es
o Σ.
352
FRANCISCO
J.
FRENICHE
Since E is a Mon el space, he measu e μn: Σ -> E£ is coun ably
addi i e. Clea ly
Vpμn
< 2", hus μn has bounded a ia ion and is τ-abso-
lu ely con inuous (when i is conside ed as an £^- alued measu e).
We apply Theo em B, ob aining, o each n G N, a unc ion un G
L^T,
2?^) such ha μw is he ec o measu e o densi y un wi h espec o
T.
Clea ly un G i7 and
7M
n = g^, o
e e y
n G N.
Fix g" G <g(X, E)". By Lemma 4.3 and Theo em 4.2, he e
exis s
υ
e
L°°(τ,
£) such ha
g /(g^)
= / WΠ(JC)(I;(JC)) dτ(x) o
e e y
/i G N.
Le Z be a se in Σ wi h τ(Z) = 0 and (X Z) bounded. The
unc ion
λ =
χX Z
is o ally measu able, because E is Mon el and
me izable.
Gi en ε > 0, we can choose 2 G y(Σ, E) such ha /?(ί;3(jc)) < ε/2,
o
e e y
x G X9 i u3 = λ — 2. Hence,
(9)
un(x)( 3(x))
dτ(x) ϋ3(x) dμn{x) <ε/2
o
e e y
n G N, because F^μ ( X) < 1.
On
he o he hand, i (g'n) is σ(^(X, £) , ί (Z,
£))-nuU,
hen
(μn(A)(e)) is a null sequence, o
e e y
e G E and ^EI Indeed, since X
is a G-space, o each e G 2S, he weak-s a null sequence (μn( )(^)) in
#(Λy, is also weak null, hence (μn(A)(e)) is null o
e e y
Bo el subse A
o X, and so o
e e y
A ^ Σ.
Since u2 is simple, i
ollows
ha
(10) lim [
un(x)( 2(x))
dτ(x) = 0.
By (9) and (10), (g/ (g^)) is a nuH sequence, and we ha e shown ha (g'n)
is
σ(V(X9
E) V(X9 E)")-n M.
4.5. COROLLARY. Le X be a
comple ely
egula
Hausdo
G-space
and
E
he
induc i e
limi
o he
sequence
(En) o
F eche -Mon el
spaces,
such
ha
e e y
bounded
subse
o E is
localized
in
some
En. Then ^(X, E) is a
G o hendieck
space.
P oo .
We can again suppose X compac . By [10], he induc i e limi
o he sequence (#(Z, En)) is a dense opological subspace o <#(X, E).
By P oposi ion 2.3 (a) and (c), and Theo em 4.4, i
ollows
ha <g(X, E)
is a G o hendieck space.
GROTHENDIECK
LOCALLY CONVEX SPACES
353
4.6. COROLLARY.
Le E be a
F eche
sepa able
space.
The
ollowing
condi ions
a e
equi alen :
(a)
E is a
Mon el
space.
(b)
The e
exis s
a
non-pseudo ini e
comple ely
egula
Hausdo
space
X such ha
Ή(X, E) is a
G o hendieck
space.
(c)
Fo
e e y
comple ely
egula
Hausdo
G-space
X,
<&(X9
E) is a
G o hendieck
space.
P oo .
Use
4.4 and 3.3.
5.
Applica ion
o
spaces
o
o ally
measu able
unc ions.
Le X be a
nonemp y
se and Σ a
ield
o
subse s
o X. We
will
say
ha
a
subse
B o
X
is
open
i o
e e y
x e B
he e
is A e Σ
wi h
x ^ A and A c B.
Endowed
X
wi h his opology,
le X* be he
Hausdo space associa ed
o
X,
<π
X -> X*
he quo ien map,
and Σ* =
{π(A):
A e Σ}.
The
ollowing lemma
is
easily es ablished:
5.1.
LEMMA
(a) X* is a
comple ely
egula
Hausdo ze o-dimensional
opological
space.
(b)
The map A ^ Σ
->
π(A) ^ Σ* is a
Boolean
isomo phism.
(c)
The map g e
^(Σ*, £)->g°τ G ^(Σ,
£) w α
opological
iso-
mo phism,
and i s
es ic ion
oS (Σ*,
E) so is
on oS (Σ,
E).
(d)
The map x* e X* -> {5* e Σ*:
x*eΰ*}e
^(Σ*)
is
By using
5.1,
when
one
s udies
he
linea opological p ope ies
o
,
/?),
i can be
supposed ha
X is a
dense subspace
o a
Hausdo
compac
ze o-dimensional opological space
K
(namely,
he
S one space
o
he
Boolean algeb a Σ),
and Σ is he
ace
in X o he
Boolean algeb a
o
open
and
closed subse s
o K. In
his con ex
we
ha e
he
ollowing
heo em:
5.2.
THEOREM.
The e
exis s
a
subspace
o SS(Σ,
E),
con aining &'(Σ,
E),
ha
is
opologically
isomo phic
o ^(K, E).
P oo .
I is
easy
o
check ha
he se o
es ic ions
o X o all
elemen s
o ^(K, E) is
such
a
subspace.
By P oposi ion
2.3 (a), i
ollows ha SS(Σ,
E) is a
G o hendieck
space
i and
only
i ^(K, E) so is.
Hence
we can
apply
o 38(Σ, E) he
esul s
o
§§3
and 4.