The Australian Journal of Mathematical Analysis and Applications AJMAA Volume 5, Issue 2, Article 17, pp. 1-7, 2009
ON ε-SIMULTANEOUS APPROXIMATION IN QUOTIENT SPACES H. ALIZADEH, SH. REZAPOUR, S. M. VAEZPOUR
Received 1 December, 2007; accepted 29 September, 2008; published 8 June, 2009. D EPARTMENT OF M ATHEMATICS , A AZAD I SLAMIC U NIVERSITY, S CIENCE AND R ESEARCH B RANCH , T EHRAN , I RAN D EPARTMENT OF M ATHEMATICS , A ZARBAIDJAN U NIVERSITY OF TARBIAT M OALLEM , TABRIZ , I RAN D EPARTMENT OF M ATHEMATICS , A MIRKABIR U NIVERSITY OF T ECHNOLOGY, T EHRAN , I RAN
[email protected] [email protected] [email protected] URL: http://www.azaruniv.edu/~rezapour URL: http://math-cs.aut.ac.ir/vaezpour
A BSTRACT. The purpose of this paper is to develop a theory of best simultaneous approximation to ε-simultaneous approximation. We shall introduce the concept of ε-simultaneous pseudo Chebyshev, ε-simultaneous quasi Chebyshev and ε-simultaneous weakly Chebyshev subspaces of a Banach space. Then, it will be determined under what conditions these subspaces are transmitted to and from quotient spaces.
Key words and phrases: ε-simultaneous approximation, ε-simultaneous pseudo Chebyshev subspace, ε-simultaneous weakly Chebyshev subspace. 2000 Mathematics Subject Classification. 41A65, 46A15.
ISSN (electronic): 1449-5910 c 2009 Austral Internet Publishing. All rights reserved.
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1. I NTRODUCTION The theory of best simultaneous approximation has been studied by many authors (for example, [1]-[8], [10]-[12]). The concept of ε-approximation has been studied by Singer [9]. In this paper, we introduce the concepts of ε-simultaneous pseudo Chebyshev, ε-simultaneous quasi Chebyshev and ε-simultaneous weakly Chebyshev subspaces of a Banach space. Then, it will be determined under what conditions these subspaces are transmitted to quotient spaces. let X be a normed linear space, W a subset of X and S a bounded set in X. we define d(S, W ) = inf sup ks − wk. w∈W s∈S
An element w0 ∈ W is called a best simultaneous approximation to S from W whenever d(S, W ) = sups∈S ks − w0 k. The set of all best simultaneous approximation to S from W will be denoted by SW (S). In the case S = {x} (x ∈ X), SW (S) is the set of all best approximations of x in W , PW (x). Thus, simultaneous approximation is generalization of best approximation in a sense. Definition 1.1. let X be a normed linear space, W a subset of X and S a bounded set in X. An element w0 ∈ W is called ε-simultaneous best approximation to S from W if sup ks − w0 k ≤ d(S, W ) + ε. s∈S
The set of all ε-simultaneous best approximations to S from W will be denoted by SW,ε (S). It is easy to see that SW,ε (S) is a non-empty, convex and bounded subset of W . In fact if s ∈ S, 0 < λ < 1 and w1 , w2 ∈ SW,ε (S), then ks − λw1 − (1 − λ)w2 k ≤ λks − w1 k + (1 − λ)ks − w2 k ≤ d(S, W ) + ε for all s ∈ S. Hence, sup ks − λw1 − (1 − λ)w2 k ≤ d(S, W ) + ε. s∈S
So, λw1 + (1 − λ)w2 ∈ SW,ε (S). Also, SW,ε (S) is closed whenever W so is. We recall that for an arbitrary non-empty convex set A in X the linear manifold spanned by A which is denoted by `(A) is defined as follows `(A) := {αx + (1 − α)y : x, y ∈ A; α is a scalar}. For every fixed y ∈ A the set `(A − y) is a linear subspace of X satisfying `(A − y) = `(A) − y := {x − y : x ∈ `(A)}. It is clear that for an arbitrary non-empty convex set A in X `(π(A)) = π(`(A)), where π is the canonical map in the correspondence quotient space. The dimension of A is defined by dimA := dim`(A). Then, for every y ∈ A we have dimA = dim`(A) = dim[`(A) − y] = dim`(A − y) = dim(A − y). For more details see [9].
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Definition 1.2. let X be a normed linear space, W a subspace of X and S a bounded set in X. Then, W is called (i) ε-simultaneous pseudo Chebyshev subspace if SW,ε (S) is finite dimensional subset of W for all bounded subset S in X. (ii) ε-simultaneous quasi Chebyshev subspace if SW,ε (S) is compact subset of W for all bounded subset S in X. (iii) ε-simultaneous weakly Chebyshev subspace if SW,ε (S) is weakly compact subset of W for all bounded subset S in X. We shall use the following Lemmas throughout this paper. Lemma 1.1. [2] Let X be a normed linear space and M a proximinal subspace of X. Then, for each non-empty bounded set S in X we have d(S, M ) = sup inf ks − mk. s∈S m∈M
Lemma 1.2. [2] Let X be a normed linear space, M a proximinal subspace of X and S an arbitrary subset of X. Then, S is a bounded subset of X if and only if S/M is a bounded subset of X/M . Lemma 1.3. Let W be a proximinal subspaces of normed space X, M a proximinal subspace of X and M ⊆ W . Then, for each non-empty bounded set S with M ⊆ S ⊆ X we have d(S/M, W/M ) = d(S, W ). Proof. It is easy to see that d(S/M, W/M ) ≤ d(S, W ). Fix w ∈ W . Then, sups∈S ks − w + M k ≥ ks − w + M k for all s ∈ S. Since M is proximinal, there exists ms,w ∈ M such that ks − w + M k = ks − w − ms,w k ≥ inf ks − w0 k. 0 w ∈W
Thus, sups∈S ks − w + M k ≥ inf w0 ∈W ks − w0 k for all s ∈ S. Hence by Lemma 1.1, sup ks − w + M k ≥ sup inf ks − w0 k = inf sup ks − w0 k = d(S, W ), 0 0 s∈S w ∈W
s∈S
w ∈W s∈S
for all w ∈ W . Therefore, d(S/M, W/M ) = inf sup ks − w + M k ≥ d(S, W ). w∈W s∈S
Lemma 1.4. Let W be a proximinal subspaces of normed space X, M a proximinal subspace of X, S a bounded set in X, M ⊆ W and ε > 0. Then, π(SW,ε (S)) ⊆ SW/M,ε (S/M ), where π : X → X/M is the canonical map. Proof. If w0 ∈ SW,ε (S), then by Lemma 1.3 sup ks − w0 + M k ≤ sup ks − w0 k ≤ d(S, W ) + ε = d(S/M, W/M ) + ε. s∈S
s∈S
So, w0 + M ∈ SS/M,ε (S/M ). Lemma 1.5. Let W be a proximinal subspaces of normed space X, M a proximinal subspace of X, S a bounded set in X, M ⊆ W and ε > 0. If w0 + M ∈ SW/M,ε (S/M ) and m0 ∈ SM (S − w0 ), then w0 + m0 ∈ SW,ε (S).
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Proof. By lemmas 1.1 and 1.3, we have sup ks − w0 − m0 k = inf sup ks − w0 − mk = sup inf ks − w0 − mk m∈M s∈S
s∈S
s∈S m∈M
= sup ks − w0 + M k ≤ d(S/M, W/M ) + ε = d(S, W ) + ε. s∈S
So, w0 + m0 ∈ SW,ε (S). Corollary 1.6. Let W be a proximinal subspaces of normed space X, M a simultaneous proximinal subspace of X, S a bounded set in X, M ⊆ W and ε > 0. Then, π(SW,ε (S)) = SW/M,ε (S/M ), where π : X → X/M is the canonical map. Proof. By Lemma 1.4, we have π(SW,ε (S)) ⊆ SW/M,ε (S/M ). Now, suppose that w0 + M ∈ SW/M,ε (S/M ). Since M is simultaneous proximinal, there exists m0 ∈ M such that m0 ∈ SM (S − w0 ). Now by Lemma 1.5, w0 + m0 ∈ SW,ε (S). So, w0 + M ∈ π(SW,ε (S)). 2. M AIN R ESULTS Now, we are ready to state and prove our main results. Theorem 2.1. Let ε > 0 be given, M and W subspaces of normed space X such that M is finite dimensional and simultaneous proximinal and W is proximinal. Then the following are equivalent. (i) W/M is ε-simultaneous pseudo Chebyshev subspaces of X/M . (ii) W + M is ε-simultaneous pseudo Chebyshev subspaces of X. Proof. (i) ⇒ (ii) Let S be an arbitrary bounded set in X and k0 be an element of SW +M,ε (S). Then, by using Lemma 1.2 we have π(`(SW +M,ε (S) − k0 )) = `(π(SW +M,ε (S) − k0 )) = `(SW/M,ε (S/M ) − (k0 + M )). Since W/M is a ε-simultaneous pseudo Chebyshev subspaces of X/M , so dim[`(SW/M,ε (S/M ) − (k0 + M ))] < ∞. Hence, dim[π(`(SW +M,ε (S) − k0 )] < ∞. Since M is finite dimensional, so dim[`(SW +M,ε (S) − k0 )] < ∞. Therefore, W + M is ε-simultaneous pseudo Chebyshev subspace of X. (ii) ⇒ (i) Let S be an arbitrary bounded set in X. Since W + M is ε-simultaneous pseudo Chebyshev subspaces of X, SW +M,ε (S) is finite dimensional. But (W + M )/M = W/M , so we have dim[SW/M,ε (S/M )] = dim[`(SW/M,ε (S/M ))] = dim[`(S(W +M )/M,ε (S/M ))] = dim[`(π(SW +M,ε (S))] = dim[π(`(SW +M,ε (S))] < ∞. Thus, W/M is ε-simultaneous pseudo Chebyshev subspace of X/M .
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Corollary 2.2. Let ε > 0 be given, M and W subspaces of normed space X such that M is finite dimensional and simultaneous proximinal, W is proximinal and M ⊆ W . Then the following are equivalent. (i) W/M is ε-simultaneous pseudo Chebyshev subspaces of X/M . (ii) W is ε-simultaneous pseudo Chebyshev subspaces of X. Theorem 2.3. Let ε > 0 be given, M and W subspaces of normed space X such that M is finite dimensional and simultaneous proximinal and W is proximinal. Then the following are equivalent. (i) W/M is ε-simultaneous quasi Chebyshev subspaces of X/M . (ii) W + M is ε-simultaneous quasi Chebyshev subspaces of X. Proof. (i) ⇒(ii) Let S be a bounded set in X and {`n } a sequence in SW +M,ε (S). Then by Lemma 1.3, for each n ≥ 1 we have sup ks − `n + M k ≤ sup ks − `n k s∈S
s∈S
≤ d(S, W + M ) + ε = d(S/M, (W + M )/M ) + ε. Hence `n + M ∈ S(W +M/M ),ε (S/M ) . Since S(W +M )/M,ε (S/M ) is compact, there exist `0 ∈ W +M and a subsequence {`nk +M }k≥1 of {`n +M }n≥1 such that `0 +M ∈ S(W +M )/M,ε (S/M ) and {`nk + M }k≥1 converges to `0 + M . But, for all k ≥ 1 we have k`0 − `nk + M k = inf k`0 − `nk − mk = d(`0 − `nk , M ). m∈M
Since M is proximinal in X, there exists mnk ∈ PM (`0 − `nk ). Hence, k`0 − `nk − mnk k = d(`0 − `nk , M ). Therefore, lim k`0 − `nk − mnk k = 0.
(2.1)
k→∞
Since `n ∈ SW +M,ε (S) for all n ≥ 1, {`nk }k≥1 is a bounded sequence. Hence by (2.1), {mnk } is a bounded sequence in M . Since M is a finite dimensional subspace of X, without loss of 0 generality we can assume that {mnk }∞ k=1 converges to an element m0 ∈ M . Let ` = `0 − m0 . 0 Thus, ` ∈ W + M and we have k`0 − `nk k = k`0 − m0 − `nk k ≤ k`0 − `nk − mnk k + kmnk − m0 k, ∀k ≥ 1. Thus lim k`0 − `nk k = 0.
k→∞
Since `nk ∈ SW +M,ε (S) for all k ≥ 1 and SW +M,ε (S) is closed, `0 is an element of SW +M,ε (S). Therefore, SW +M,ε (S) is compact. (ii) ⇒(i) Let S be an arbitrary bounded set in X. Then, SW +M,ε (S) is compact. But the canonic map is continuous, so π(SW +M,ε (S)) is compact. Thus by Corollary 1.6, π(SW +M,ε (S)) = S(W +M )/M,ε (S/M ) = SW/M,ε (S/M ). Therefore, W/M is ε-simultaneous quasi Chebyshev subspaces of X/M . Corollary 2.4. Let ε > 0 be given, M and W subspaces of normed space X such that M is finite dimensional and simultaneous proximinal, W is proximinal and M ⊆ W . Then the following are equivalent. (i) W/M is ε-simultaneous quasi Chebyshev subspaces of X/M . (ii) W is ε-simultaneous quasi Chebyshev subspaces of X.
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Theorem 2.5. Let ε > 0 be given, M and W subspaces of normed space X such that M is finite dimensional and simultaneous proximinal and W is proximinal. Then the following are equivalent. (i) W/M is ε-simultaneous weakly Chebyshev subspaces of X/M . (ii) W + M is ε-simultaneous weakly Chebyshev subspaces of X. Proof. (i) ⇒(ii) Let S be a bounded set in X and {`n } a sequence in SW +M,ε (S). Then by Lemma 1.3, {`n + M } is a sequence in S(W +M )/M,ε (S/M ) = SW/M,ε (S/M ). Since SW/M,ε (S/M ) is weakly compact, there exists a subsequence ∞ ∞ ∞ {`nk + M }k=1 of {`n + M }n=1 such that {`nk + M }k=1 converges weakly to an element `0 + M ∈ SW/M,ε (S/M ). Then, `0 + m0 ∈ SW +M,ε (S/M ) for some m0 ∈ M . But since M is proximinal, Tf ∈ (X/M )∗ for all f ∈ X ∗ . Therefore, f (`nk ) = Tf (`nk + M ) → Tf (`0 + M ) = Tf (`0 + m0 + M ) = f (`0 + m0 ). Hence,{`nk }k≥1 converges weakly to `0 + m0 ∈ SW +M,ε (S). Thus, SW +M,ε (S) is weakly compact and hence W + M is ε-simultaneous weakly Chebyshev subspaces of X. (i)⇒(ii) Let S be an arbitrary bounded set in X and {wn + M } a sequence in SW/M,ε (S/M ). Since M is proximinal and S − wn is a bounded set in X for all n ≥ 1, there exists mn ∈ SM (S − wn ) for all n ≥ 1. But by Lemma 1.1, we have sup ks − wn − mn k = inf sup ks − wn − mk = sup inf ks − wn − mk m∈M s∈ S
s∈S
s∈ S m∈ M
= sup ks − wn + M k ≤ d(S/M, W/M ) + ε ≤ d(S, W + M ) + ε. s∈ S
Therefore, {wn + mn } is a sequence in SW +M,ε (S). Since SW +M,ε (S) is weakly compact, there of exists a subsequence {wnk + mnk }∞ k=1 ∞ ∞ {wn + mn }n=1 such that {wnk + mnk }k=1 converges weakly to an element `0 ∈ SW +M,ε (S). By Corollary 1.6, `0 + M is an element of S(W +M )/M,ε (S/M ) = SW/M,ε (S/M ). Note that for every f ∈ (X/M )∗ , we have f (wnk + M ) = f (wnk + mnk + M ) = f ◦ π(wnk + mnk ) → f ◦ π(`0 ) = f (`0 + M ). It follows that {wnk + M }∞ k=1 converges weakly to w0 + M . Hence, SW/M,ε (S/M ) is weakly compact subset of X/M . Therefore, W/M is ε-simultaneous weakly Chebyshev subspaces of X/M . Corollary 2.6. Let ε > 0 be given, M and W subspaces of normed space X such that M is finite dimensional and simultaneous proximinal, W is proximinal and M ⊆ W . Then the following are equivalent. (i) W/M is ε-simultaneous weakly Chebyshev subspaces of X/M . (ii) W is ε-simultaneous weakly Chebyshev subspaces of X. R EFERENCES [1] A. S. HOLLAND, B. N. SHANEY, J. TZIMBALAEIO, On best simultaneous approximation, J. Approx. Theory, 17(1976), pp. 187–188. [2] M. IRANMANESH, H. MOHEBI, On best simultaneous approximation in quotient spaces, Analysis in Theory and Applications, Vol. 23, No. 1 (2007), pp. 35–49. [3] C. LI, G. A. WATSON, On best simultaneous approximation, J. Approx. Theory, 91 (1997), pp. 332–348.
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