H. Mohebi et al: On Sum and Quotient of Quasi-Chebyshev Subspaces. 267. Chebyshev subspace of X. For an arbitrary non-empty convex set A in X we shall ...
Analysis in Theory and Applications
19: 3, 2003, 266--272
ON SUM AND QUOTIENT OF QUASI-CHEBYSHEV SUBSPACES IN BANACH SPACES H. Mohebi (Shahid Bahonar University of Kerman, Iran)
Sh. Rezapour (Azarbaidian University of Tartn'at Moallem , Iran )
Received Sep. 28, 2003
Abstract
It will be determined under what conditions types of proximinality are transmitted to and from quotient spaces, In the final section, by many examples we show that types of proximinality of subspaces in Banach spaces can not be preservaed by equivalent norms.
Key Words
proximinality, Chebyshev subspace, pseudo-Chebyshev subspace, quasi-Cheby-
shev subspace, equivalent norms
AMS(2000)
subject classification
46B25, 46A20, 46A25, 41A65
I
Introduction
Let X be a (complex or real) Banach space and let W be a subspace of X. A point y0 E W is said to be a best approixmation for x E X IIx--yo[I =d(x,W)
if
=inf{ IIx--Yll "YEW}.
For x E X , put P w ( x ) ---- {y 6 W. II x - y II =
d(x,W)}.
It is clear that P w ( x ) is a closed, bounded and convex subset of X. If for each x E X there exists at least one best approixmation in W, then W is called a proximinal subspace of X. If for each x E X
there exists a unique best approximation in W, then W is called a
H. Mohebi et al: On Sum and Quotient of Quasi-Chebyshev Subspaces
267
C h e b y s h e v subspace of X. F o r an arbitrary n o n - e m p t y convex set A in X we shall denote by
l ( A ) = {ax + (1 -- a)y: x , y E A; a is scalar}, the linear manifold spanned by A. For every fixed y E A the set
I(A) --y=
{x--y:
x E I(A)}
is a linear subspace of X satisfying
I ( A ) -- y = I(A -- y). T h e dimension of A is defined by d i m A = d i m l ( A ) . T h e n for every y E A dimA = d i m l ( A ) = dim[-l(A) - - y-] = dim[-l(A --
y)]
we have
= dim(A -- y).
(for more details see I-6"]). We say that W is a pseudo-Chebyshev subspace of X if P w ( x ) is a n o n - e m p t y and finite dimensional set in X for each x E X . W is called a quasi-Chebyshev subspace of X if P w ( x ) is a n o n - e m p t y and compact set in X for every x E X
(see [3-]).
E v e r y p s e u d o - C h e b y s h e v subspace of X is a quasi-Chebyshev subspace of X , but the converse is not true (see [-3,4-]). A l s o , Cheney and Wulbert have proved some results for C h e b y s h e v and semi-Chebyshev subpsaces in quotient spaces (see [-1,7-]).
2 On Pseudo-Chebyshev Subspaces
L e m m a 2. 1.
Let M and W be proxirninal subspaces of a Banach space X. Then,
Pw/M(X-k-M)=~r(Pw(x) ) for every x E X. Proof. EX.
If 7r.X--~X/M is the canonical m a p , then r r ( P w ( x ) ) ~ P w / M ( X + M ) for all x
Let y + M E Pw/M(x+ M). T h e n , y + r n E P w ( x ) for some rnEM. T h e r e f o r e , y + M
E 7r(Pw(x)) and PW/M(X+M) =Tr(Pw(x) ). C o r o l l a r y 2. 2.
Let M be a proxirninal subspace of a Banach space X and let W be a
pseudo-Chebyshev subspace of X such that M is a subspace of W. Then W / M is pseudoChebyshev in X / M . T h e o r e m 2.3.
Let M be a finite dimensional subspace of Banach space X and let W be
a proxirninal subspace of X such that M is a subspace of W. I f W / M is pseudo-Chebyshev in X / M , then W is pseudo-Chebyshev in X.
268
Analysis in Theory and Applications
Let z E X
Proof.
19: 3, 2003
be arbitrary and let y o E P w ( z ) . Then, by Lemma 2. 1,
zr(l(Pvc(z) -- yo) ) = l(Pw/M(x + M) -- (Yo q- M) ).
Since W / M is pseudo-Chebyshev in X / M , d i m ~ l ( P w / M ( z + M ) ) - - ( y o + M ) ] ~ o o .
Thus,
d i m z r ( l ( P w ( z ) - - y o ) ) ~ o o . Since M is finite-dimensional, d i m / ( P w ( z ) - - y 0 ) ~ o o .
There-
fore, W is pseudo-Chebyshev in X.
3
3. 1.
Theorem
On Quasi-Chebyshev Subspaces
Let M be a proziminal subspace o f a Banach space X and let W be a
quasi-Chebyshev subspace o f X such that M is a subspace o f W. Then W / M is quasi-Chebyshev in X / M. Proof.
Since the canonical map ~r is continuous, it is obvious by Lemma 2. 1.
Theorem
3. 2.
Let M be a finite dimensional subspace o f a Banach space X and let W
be a closed subspace o f X. Then, the following are equivalent.
(a) M + W is quasi-Chebyshev in X. (b) ( M + W ) / M Proof.
is quasi-Chebyshev in X / M .
(a)=~(b). This is an immediate consequence of Theorem 3.1.
(b)=~(a). For simplicity, put ~ = X / M
and V~= ( M + W ) / M .
nal in ~" and M + W is proximinal in X. Therefore, P u + w ( z ) ~
Then, IF is proximifor all :rEX. Now, we
show that P~+w(z) is compact for all z E X .
Let x E X and g . E P ~ + w ( x ) be arbitrary (n
=1,2,'").
for all n ~ l .
Then,
][x--g.][ for all w E M + W
[[ z - - g . [[ = d ( z , M + W ) ---- [I ( x - - g . ) ^ and all n ~ l .
[I ~
[[x--g.[[
=d(x,M+W)~
[Ix--wl[
Since M + W is a subspace of X, we have
Ild:- .l[ for all w E M + W
Consider
and all m E M ( n = 1 , 2 , ' " ) .
Ilk-w-nil, It follows that
for all w E I ~ and all n>~l. Hence [[ x - - g . [[ ----d(x,W) for all n>~l. Then g . E P ~ . ( x ) for all n ~ l . Since Pg, ( x ) is compact , there exists a subsequenee {g.j},_a ~ }.-i oo • of {g. such that {g.j}~~ I converges to an element ,~0EPg,(x). Then d o + r n E P u + w ( x ) for some r n E M , and hence do E M + W. Now, consider
H. Mohe&"et al: On Sum and Quotient of Quasi-Chebyshev Subspaces II ,t0 -
for all k ~ l .
g., II =
II (do -- g.)^
inf II do -- g.,
mEM
-
m
II = d(do
-
g.,,M)
Since M is proximinal in X , there exists m , ~ E M such that II do
for all k ~ l .
II -
269
-
g., -
m.,
[I -- d(do -- g . , M ) ,
Hence lim II d o - - g . - - m . , II =tim [[ ~0--g., II = 0 . k~oo
k~c~
Since [I g., [I 421[ x [[ for all k ~ l ,
{m.,}~a is a bounded sequence in M. But, g is a
finite dimensional subspace, then there exists a subsequence {m'i}i%l of {m., }~1 such that {mi}~-l converges to an element m o E M . Let go----do--too. Then, g o E M §
II go
-
II =
g,
tl do
-
mo
-
and g,
II
