Rev~~ta Colomb~ana de Matema~~a~
Vol. XI X (198 5),
YJa9~.
31 3 - 32 2
BEST APPROXIMATION
VALUED
IN VECTOR
FUNCTION
SPACES
by
ROSHD I KHALI L
AB 5T RAe T. Let T be the unit circle, and m be the
normalized Lebesgue measure on T. If H is a separable
Hilbert space, we let LOO(T,H) be the space of essentially bounded functions on T with values in H. Continuous
functions with values in H are denoted by C(T,H), and
Hoo(T,H) is the space of bounded holomorphic functions
in the unit disk with values in H. The object of this
paper is to prove that (Hoo+C)(T,H) is proximinal in
Loo(T,H). This generalizes the scalar valued case done
by Axler, S. et al. We also prove that
(Hoo+C)(T ,'1,00) HOO(T,'1,00) is an M-ideal of Loo(T,'1,00) ~(T,Q,OO),
and V(T,Q,OO)is an M-ideal of LOO(T,Q,OO)
whenever V is an
M-ideal of Loo, where V(T,Q,oo)= {g E LOO(T,Q,OO):
<g(t),o > E V for all n}.
I
I
n
§o.
Introduction.
normalized
Lebesgue
tegrable
functions
denoted
by LP(T,X),
Let T be the unit circle and m be the
measure
In-
on T with values in a Banach space X
1 ~ p <
the space of essentially
in X. LP(T,X)
on T. The space of p-Bochner
00.
bounded
If P =
00,
functions
is
then Loo(T,X) is
on T with values
are Banach spaces with the usual norms:
313
UII f(-r)IIPdm)lip,
=
T
IIfll
=
00
es s .su
t
[3],
We refer to,
pll f( t )II, p =
00
for the basic structure
The problem
of best approximation
of much interest.
The existence
C(Q) to functions
in Lco(Q), was proved in
paracompact,
and later,
of these spaces.
1
co
In Land
L is
of best approximation
[13,16],
[9],
in
where Q is
was generalized
to the
vector valued case, where X is assumed to be uniformly
con-
vex.
00
The existence
of best approximation in H +C to funcco
tions in Lco(T) was proved in [2], where H is the space of
bounded
analytic
functions
the space of continuous
in the unit disk and C is C(T),
functions
on T.
The object of this paper is to try to generalize
result in
[2]
to the vector valued
to be a Hilbert
space. Further,
case, where
X
the
is assumed
if V is a closed subspace
of
00
L (T)
and
V(T,9, ) = g
00
{
e: L
co(T,9, ): <g(-r),o
00
n
>e:
V for all n
}
co
in L (T,9, ) whenever V
co
in L (T), noting that 9, is not uniformly con00
00
we prove that V(T,9, ) is proximinal
00
is proximinal
vex.
In Section
co
some results on (H+C){T,H),
1, we prove
similar to the scalar valued case. A representation of the
co
dual of L (T,H) is included. In Section 2, we prove that
co
(H +C)(T,H) is proximinal
in L (T,H). In Section 3 we give
co
an example, 9, , where C(T,X) is proximinal in L (T,X),
00
00
without
314
X being uniformly
convex.
Throughout
the paper, if X and Yare
then X0 Y, X~ Y denote the projective
of X and Y, respectively.
sor products
of bounded
Banach spaces,
and the injective tenL(X,Y) is the space
from X into Y. The dual of
linear operators
a
Banach space X is X*. The complex numbers are denoted by ~.
§1.
Vector
val ued
funct
ion
spaces.
We let HP(T) be
00
Hardy spaces, 1 ~ P <
the classical
space of bounded
00,
and H (T) is the
analytic functions in the unit disk. One
can consider HP(T) as closed subs paces of LP(T), 1 ~ p ~
for which
J
= 0
f(t)eintdm(t)
for all n <
0;
f
E:
LP(T),
00,
[8].
T
If X is a Banach space, we define
,;';
X
This definition
E:
x"~}.
saves us the trouble of proving the existence
of the radial limit if we were to consider HP(T,X) as functions on D =
{z
E:~:
Izi
< 1}.
As in the scalar valued case one can prove:
1. 1.
THEOREM
LP(T,X),
1 ~
P ~
HP(T ,X) J./.:, a uo.oe..d
.oub.opac.e..
on
00
Now, we take our Banach space to be a separable
bert space H. Consequently
every element f
E:
Hil-
HP(T,H) has a
representation:
00
f =
I fe,
n n
n=l
(e ) is some orthonormal
n
basis of H. For p = 2, one has
315
00
IIfl/
=
(I/IfnI12)\
[7J.
Since H is reflexive,
ive Banach spaces,
Loo(T,H),
[3J.
LP(T,H),
and LP(T,H)
= [LP'(T,H)]*,
From the definition
that the HP(T),
1 ~ P ~
are reflex-
00
1
L (T,H) * =
of HP(T ,H) and the fact
are w*-closed
00
1 < p <
in LP(T,H)
the fol-
lowing result follows:
.-.
LEMMA 1.2.
HP(T,H)
iJ.:,
w"-c1o.oed ~n LP(T,H),
1< P
Set A(T ,H) = H (T ,H) n C(T ,H). If f e::A(T ,H), then
00
00
f =
I
fe,
n=l n n
If Y
f
e::A(T), the disk algebra.
n
of the Banach space X, then for
a subspace
lS
x e::X, d(x,Y) = inf{~x-yll: y e::V}. The proof of the following result is the same as for the scalar valued case and
[6J.
will be omitted,
THEOREM
Let fe:: C(T,H). Then dU,H
1.2.
00
(T,H)) =
d(f,A(T,H)).
00
The subspace
subspace,
[6].
00
H +C
c
This result
L
was proved to be a closed
is true for the vector valued
case:
THEOREM
00
a c.10.0ed .oub-
1.3.
H(T,H)+C(T,H)
1.4.
[Loo(T,H)]'"iJ.:, iJ.:,omebU..~a.U.y
iJ.:,
J.lpa~e 06 Loo(T,H).
THEOREM
p~~
iJ.:,omoJt-
to the .opa~e 06 MMtuy addct.cve: vec.to« mea.ouJteJ.l
w~~h
vaniJ.:,hon m-nutt .oe~, eq~pped
w~h
the total
vaJti~on
noJtm.
PJtoo6. Since one can integrate a vector valued func316
tion against
finitely
proof is similar
Best
[41,
vector measures,
to the scalar valued case,
the
[5], and will
Q.E.D.
be omitted.
§2.
additive
approximation
in
loo(T,H).
space and Y be a closed subspace
Let X be a Banach
of X. For x
E
X, an ele-
b~t apphox1mant 06 x in Y if
ment y E Y is called a
Ilx-yll= d(x,Y) = inf{llx-zll,z EY}.
If every element
lS
called
problem
x E X has a best approximant
p~o~nal
subspace
to determine
whether
nach space is proximinal
that Hoo+C is proximinal
of X. It is an interesting
a given closed subspace of Ba-
[2J,
or not. In
[12],
in Loo(T). Luecking,
We refer to the references
ideals.
gave
a
[2J
In
to prove that (H +C)(T,H)
Hilbert
A closed
lS
subspace
of X if
{ep E X-::;ep(n
=
lS
in L (T,H),
Y of X is called an
and if x = Y + y' E Y G) Y' then
yA.-
proximinal
for
space H.
X if there exists a subspace
an M-ideal
lS
00
00
every separable
for further re-
The object of this section
sults on best approximation.
L-~ummand of
Y' of X such that X = Y
II xii =
G)
Y'
lIyll+h'~. Y is called
an L-summand
of X*, where Y~ =
o}.
THEOREM 2.1.
(Hoo+C)(T,H) IHoo(T,H) ~
an M-ideal
Loo(T,H)!Hoo(T,H).
P~o6.
~
Axler et al. proved
proof for the same result, using the idea of M-
different
06
in Y, then Y
We identify
with Hoo(T,H)1
oo
with (H +C)(T,H)1
(Loo(T,H)IHoo(T,H»*
Loo(T,H)*, and «~+C)(T,H)I~(T,H»l
317
00
sLoo(T,H)"', [6]. Let F
e:: H (T ,H)
J.
. Being
an element
of
00
Loo(T,H)*, it follows
00
(fJ),
fJ
n
n
(e ) of H, where
n
some sequence
basis
e::
L
I
1.8 that F =
from Theorem
fJ e for
n--l n n
orthonormal
.'.
('n",
and a (fixed)
00
=
<F(E) ,x>
for every Lebesgue
I
fJ (E)<e ,x>
n=l n
n
measurable
set E c: T, and x
00
the maximal
space of continuous
space of regular
function
the
s,
on X. Hence L (T)" ~ M(X),
Borel bounded
measures
on X,
[6].
the
Conse-
fJ
every element
considered
H. I f X is
ideal space of L (T), then L (T) ~ C(X),
00
quently,
e::
00
in the representation
of F can be
n
as a countably additive measure on X. Set
N
FN
Clearly
<FN(E),x>
in X and each x
measure
theorem,
[10],
e::
be the Lebesgue
F
a
m is the lifting
X, [6]~ It
follows
Loo(T) ~ C(X) and x
F as a countably
where
H, where
for each m-measurable
additive
vector
decomposition
of the Lebesgue
from Grothendieck's
e::
H. Thus we can consider
measure
on X.
Let F +F
a
of F with respect
is rn-continuous and xF
s
is singular
to m for all x e:: H. One can easily
00
each v
s
= F
[3J,
to m,
with respect
shew that if
r
=
f
e
Inn
is orthogonal
= \w e , then V is m-continuous
and w
s
n
n
!n n
to m for all n.
It follows from Pettis theorem,
and F
set E
that
,
for ecery g
I
u e .
n=l n n
~ <F(E),x>
e::
m on T to
=
a
[3], that
is absolutely continuous with respecto to n. Since
n
(e ) is an orthonormal basis for H, it follows that u = V +w
n
n
n n
318
is the Lebesgue decomposition
.L
00
of ~
-
A
with respect to m.
n
00.1.
Since r £ H (T,H) , it follows that ~ E:H
for all n.
n
Consequent ely by the abstract Riesz theorem, [6], w £H
n
oo.t
and V £ H
Define the following map:
ool..
n
00
P:H (T,H)
.L
00
~ (H +C)(T,H)
.L
,
P( f) = F .
s
It follows from the above argument and theorem 2.4 of
.L
00
II
II
[12] ,
II II,
that rs E:(H +C)(T,H).
rurther, since r ~ = r a 11+ r s
[3 J ,
one has P bounded, where r] is the total variation of r.
II
To complete the proof of the theorem, we have to show that
00
.L
00
P is onto. Let <jl E: (H +C)(T ,H) , and <jl =
00
u
n
lJ
is orthogonal
n
= ¢, and
Hence p(<jl)
\'
L lJ e
for some
n-1 n n
(H +C) , and by Theorem?
4 of
00.L
E: L (T) . Then each lJ
[12J,
m.
~'~
n
to
€:
m.
Consequentely
P is onto.
<jl is singular to
Q.E.D.
00
THEOREM
2.2.
(H
+C) (T
,H)
if.!
pflo~vwl_
,tv[
00
L (T,H).
PflOOn. Since an M-ideal of a Banach space X is prox00
00
iminal in X, [lJ, it follows that (H +C)(T,H)/H
00
proximinal
00
in L (T,H)/H (T,H). Lemma
1.2.
(T,H)
is
together with a
00
compactness
argument imply that H (T,H) is proximinal
00
00
L (T,H1. Thus for f
E:
In
00
L (T,H). there exists g £ (H +C)(T,H)
such that
for some g
o
approximant
E:
Hoo(T,H). Thus g+g
E: (Hoo+C)(T,H) is a best
0
of f.
PROBLEM..
Q.E.D.
Is Th e ore m 2.2 true if H is replaced
by arbitrary Banach space? If not, what are those Banach
spaces for which the result is true?
319
§3.
Proximinal
space
of
bounded
I
suplf(n)
n
ity
sequences,
,£(0).
Let
so that
£00 be the
00
f E £ , then
if
Banach
II f~oo =
,w pJto~na£
in
Loo(T,£oo).
00
00
1
Since
L (T,£
)=(L(T,£))
00
00
(T,£).
Let fEL
00
00
= £ (N,L),
*
1
1
(N,L))
Le.t V be a pJtoxJ..mi.na1. .6u.b.6pac.e On
3.1.
Then V(T,£oo)
PJtOon.
= (£
r
< 00.
THEOREM
Loo(T).
L"r
in
it
follows
1 ~':
that
00
o
IIfll
= IIJ/n nll
= s¥p
s up] f
=
° (j)
Here,
noon
function
00
g
n=l n n
I
°
is
n =
a best
° (j)
IIf
approximant
in
. Pft00n.
C(T)
of
are
from
proximinal
Let
in
16
3.3.
-<'.6 an M-idea1
Loo(T,£oo) with
320
= d(f
n
n
=
,V).
If
the
g =
II f-gl.
Q.E.D.
V(T,£oo).
06
fi
3.1
and
06
be
any
three
open
.
and
l
fact
that
Loo(T),
then
Loo(T,£oo).
I-
radii
ep
balls
in
~
r.
such
l
BUi,r.)
the
Q.E.D.
Loo(T).
V i.6 an M--<.dea1.
B(fi,r.)
centers
V(T,£oo)
f in
Theorem
l
ep and
-in II
Consider
vhe s pac. ell C(T , £(0) and (Hoo+C)( T,r)
3. 2.
Follows
PJto06.
I-
n
= 0 otherwise.
L00(T , £(0) .
THEOREM
V(T,£oo)
II .
= sup] f -f II. ~ sup] f -g II
n
n n
n
n n
aJte pno ~na1
s~plfn(t)1
then
COROLLARY
Hoo+C and
j,
= s~p
n 00
n-
V(T,£oo)
£
n
I1~n °n , where
1=
II f-fll
Hencr~ f
= 1 if
sgplfn(t)1
for
i
= 1,2,3.
that
II B(fi,ri)
i= 1
Let
I
oo
B(fi,r.). If gi =
giO and
l
n- 1 n n
fl : n~lf~On' then supllf~-g~11< rio He~ce ~~ EBU~,ri)
cL (T). By the same argument, we have
B(fl r.) f ~ for
l-l
n> l
all n. It fOll~ws that Vn(i=lB(f~,ri)
f ¢, for all n, [a]:
gl
V(T,t
E
•
) such that gi
00'
E
.,
.n
n
Let g
n
Further
n
i
E V n (. lBU ,r.)). The function g = L1g 0 E V(T,t ).
l=
n l
n- n n
00
oo
Hence g EV(T,t
V(T,t)
It follows,
) n (i01B(fi,ri».
00
00
Using the same argument
3.4.
[1], that
00
is an M-ideal of L (T,t).
THEOREM
00
Q.E.D.
of Theorem 3.3 one can prove:
00
00 00
(H +C)(T,t
1
00
) H (T,t )
-u:, an M-ide.af
oil Loo(T,too)I Hoo(T,too).
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Ve.paume.~
06 M~he.yyuti~.6
The. U~veMily
06 M~~~gaVl
AVlVl A!Lbo!L, ~~~gaVl
48109
U. S. A.
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(Recibido en abril de 1985).
322
Ku.wa..d U~ve.!L.6ily
Ve.pL 06 M~h~~
P.O. Bo X 59 6 9
Kuwail