Rev. Anal. Num´er. Th´eor. Approx., vol. 35 (2006) no. 1, pp. 17–31 ictp.acad.ro/jnaat
BEST APPROXIMATION IN SPACES WITH ASYMMETRIC NORM
S. COBZAS¸∗and C. MUST ˘AT¸ A†
Abstract. In this paper we shall present some results on spaces with asymmet- ric seminorms, with emphasis on best approximation problems in such spaces.
MSC 2000. 41A65.
Keywords. Spaces with asymmetric norm, best approximation, Hahn-Banach theorem, characterization of best approximation.
1. INTRODUCTION
LetX be a real vector space. Anasymmetric seminorm on X is a positive sublinear functional p:X→[0,∞), i.e. p satisfies the conditions:
(AN1) p(x)≥0;
(AN2) p(tx) =tp(x);
(AN3) p(x+y)≤p(x) +p(y), for all x, y∈X and t≥0.
The function ¯p:X → [0,∞) defined by ¯p(x) =p(−x), x∈X, is another positive sublinear functional onX, called the conjugateof p, and
(1.1) ps(x) = max{p(x), p(−x)}, x∈X, is a seminorm on Xand the inequalities
(1.2) |p(x)−p(y)| ≤ps(x−y) and |¯p(x)−p(y)| ≤¯ ps(x−y)
hold for allx, y∈X. If the seminormpsis a norm on X then we say thatp is anasymmetric normonX. This means that, beside (AN1)–(AN3), it satisfies also the condition
(AN4) p(x) = 0 and p(−x) = 0 imply x= 0.
The pair (X, p), whereXis a linear space andpis an asymmetric seminorm on X is called a space with asymmetric seminorm, respectively a space with asymmetric norm, ifp is an asymmetric norm.
∗“Babe¸s-Bolyai” University, Faculty of Mathematics and Computer Science, 400084 Cluj-Napoca, Romania, e-mail: [email protected].
†“T. Popoviciu” Institute of Numerical Analysis, P.O. Box 68-1, Cluj-Napoca, Romania, e-mail: [email protected].
The functionρ :X×X →[0;∞) defined byρ(x, y) = p(y−x), x, y ∈X, is an asymmetric semimetric on X. Denote by
Bp0(x, r) ={x0∈X:p(x0−x)< r} and Bp(x, r) ={x0∈X:p(x0−x)≤r}, the open, respectively closed, ball inX of centerxand radiusr >0.Denoting by
Bp0 =Bp0(0,1) and Bp =Bp(0,1), the corresponding unit balls then
B0p(x, r) =x+rBp0 andBp(x, r) =x+rBp.
The unit ballsBp0 andBp are convex absorbing subsets of the spaceX and pagrees with the Minkowski functional associated to any of them. Recall that for an absorbing subset C of X the Minkowski functional pC :X →[0;∞) is defined by
pC(x) = inf{t >0 :x∈tC}.
IfC is absorbing and convex, thenpC is a positive sublinear functional, and {x∈X :pC(x)<1} ⊂C⊂ {x∈X:pC(x)≤1}.
An asymmetric seminormpgenerates a topologyτp onX, having as basis of neighborhoods of a point x∈X the family{Bp0(x, r) :r >0}of open p-balls.
The family {Bp(x, r) :r >0} of closed p-balls is also a neighborhood basis at x forτp.
The topologyτp is translation invariant, i.e. the addition + :X×X →X is continuous, but the multiplication by scalars · :R×X → X need not be continuous, as it is shown by some examples (see [7]).
The ball Bp0(x, r) is τp-open but the ball Bp(x, r) need not to be τp-closed, as can be seen from the following typical example.
Example1.1. Consider onRthe asymmetric seminormu(α) = max{α,0}, α ∈R, and denote by Ru the space R equipped with the topology τu gener- ated by u. The conjugate seminorm is ¯u(α) = −min{α,0}, and us(α) = max{u(α),u(α)}¯ = |α|. The topology τu, called the upper topology of R, is generated by the intervals of the form (−∞;a), a ∈ R, and the family {(−∞;α+) : >0} is a neighborhood basis of every point α ∈R. The set (−∞; 1) =B0u(0,1) isτu-open, and the ballBu(0,1) = (−∞; 1] is notτu-closed because R\Bu(0,1) = (1;∞) is notτu-open.
The topology τp could not be Hausdorff even if p is an asymmetric norm on X. A necessary and sufficient condition in order thatτp be Hausdorff was given in [22]. Putting
(1.3) p(x) = inf{p(x˜ 0) +p(x0−x) :x0∈X}, x∈X,
it follows that ˜pis the greatest (symmetric) seminorm majorized by pand the topology τp is Hausdorff if and only if ˜p(x) >0 for every x6= 0.Changing x
to−x and takingx0 = 0 it follows that, in this case,p(x)>0 for everyx6= 0, but this condition is not sufficient forτp to be Hausdorff, see [22].
Spaces with asymmetric seminorms were investigated in a series of papers, emphasizing similarities with seminormed spaces as well as differences, see [1, 2, 3, 7, 17, 18, 19, 21, 22], and the references quoted therein. Among the differences we mention the fact that the dual of a space with asymmetric semi- norm is not a linear space but merely a convex cone in the algebraic dual X# of X. This is due to the fact that the continuity of a linear functional ϕ on (X, p) does not imply the continuity of −ϕ.For instance, ϕ(u) =u is contin- uous on (R, u) but ψ(u) = −u is not continuous. For an other example see [7]. The study of spaces with asymmetric norm was motivated and stimulated also by their applications in the complexity of algorithms, see [20, 40].
Some continuity properties of linear functionals in the symmetric case have their analogs in the asymmetric one.
Proposition 1.2. [21] Let (X, p) be a space with asymmetric seminorm and ϕ:X →R a linear functional. Then the following are equivalent.
(1) ϕis τp-τu-continuous at 0∈X.
(2) ϕis τp-τu-continuous on X.
(3) There existsL≥0 such that
(1.4) ∀x∈X, ϕ(x)≤L p(x).
(4) ϕis upper semi-continuous from (X, τp) to (R,| |).
A linear functional satisfying (1.4) is called semi-Lipschitz (or p-bounded) and L a semi-Lipschitz constant. Denote by Xp[ the set of all bounded linear functionals on the space with asymmetric seminorm (X, p).As we did mention, Xp[ is a convex cone inX#.
One can define a normk |p on Xp[ by
(1.5) kϕ|p = sup{ϕ(x) :x∈Bp}, ϕ∈Xp[.
Some useful properties of this norm, whose proofs can be found in [9, 12], are collected in the following proposition. We agree to call a linear functional ϕ on (X, p), (p,p)-bounded if it is both¯ p- and ¯p-bounded, where ¯p is the seminorm conjugate to p.
Proposition1.3. Ifϕis a bounded linear functional on a space with asym- metric seminorm (X, p), p6= 0, then the following assertions hold.
(1) kϕ|p is the smallest of the numbersL≥0for which the inequality (1.4) holds.
(2) We have
kϕ|p = sup{ϕ(x)/p(x) :x∈X, p(x)>0}
(1.6)
= sup{ϕ(x) :x∈X, p(x)<1}
(1.7)
= sup{ϕ(x) :x∈X, p(x) = 1}.
(1.8)
(3) If ϕ 6= 0, then kϕ|p > 0. Also, if ϕ 6= 0 and ϕ(x0) = kϕ|p for some x0 ∈Bp,thenp(x0) = 1.
(4) If ϕis (p,p)-bounded, then¯
ϕ(rB0p) = (−rkϕ|p¯, rkϕ|p) and ϕ(rBp0¯) = (−rkϕ|p, rkϕ|p¯)
where Bp0 ={x∈X:p(x)<1}, Bp0¯={x∈X : ¯p(x)<1} and r >0.
(5) If ϕis p-bounded but not p-bounded, then¯ ϕ(rBp0) = (−∞, rkϕ|p).
Remark 1.4. A linear functional ϕ:X →R is (p,p)-bounded if and only¯ if
(1.9) ∀x∈X, |ϕ(x)| ≤Lp(x),
for someL≥0.
Indeed, if L1, L2 ≥0 are such that
ϕ(x)≤L1p(x) and ϕ(x)≤L2p(−x),
for all x ∈ X, then −ϕ(x) = ϕ(−x) ≤ L2p(x), x ∈ X, so (1.9) holds with L= max{L1, L2}.
Denote by Xp[¯ the dual cone to (X,p) and let¯ X∗ be the conjugate of the seminormed space (X, ps),whereps is the symmetric seminorm associated to p and ¯p(see (1.1)).
Since
ϕ(x)≤Lp(x)≤Lps(x), x∈X,
implies|ϕ(x)| ≤Lp(x), x∈X, it follows thatXp[ is contained in the dualX∗ of (X, ps).Similarly,Xp[¯ is contained in X∗ too.
Forx∗∈X∗ put
kx∗k= sup{x∗(x) :x∈X, ps(x)≤1}.
Then k k is a norm on X∗ and X∗ is complete with respect to this norm, i.e.
is a Banach space (even if ps is not a norm, see [11]).
Proposition 1.5. Let (X, p) be a space with asymmetric seminorm.
(1) The conesXp[ and Xp¯[ are contained in X∗ and
kϕ|p=kϕk, ϕ∈Xp[ and kψ|p¯ =kψk, ψ∈Xp¯[. (2) We have kϕ|p =k −ϕ|p¯, so that
ϕ∈Xp[ andkϕ|p ≤r ⇐⇒ −ϕ∈Xp[¯andk −ϕ|p¯ ≤r.
The properties of the dual spaceXp[ were investigated in [21] where, among other things, the analog of the weak∗ topology of X was defined. This is denoted by w[ and has a neighborhood basis at a point ϕ∈Xp[, the family
Vx1,...,xn;(ϕ) ={ψ∈Xp[ :ψ(xk)−ϕ(xk)< , k= 1, ..., n},
for n∈N, x1, ..., xn∈X and >0. The w[-convergence of a net (ϕi :i∈I) inXp[ toϕ∈Xp[ can be characterized in the following way
ϕi −→w[ ϕ ⇐⇒ ∀x∈X, ϕi(x)→ϕ(x) in (R, u).
It was shown thatw[is the restriction of the topologyw∗=σ(X∗, X) onX∗ toXp[ (see [21]). This study was continued in [9] where separation theorems for convex sets and a Krein-Milman type theorem were proved. In [10] asymmetric locally convex spaces were introduced and their basic properties were studied.
Another direction of investigation is that of best approximation in spaces with asymmetric seminorm. Due to the asymmetry of the seminorm we have two distances. For a nonempty subsetY of a space with asymmetric seminorm (X, p) and x∈X put
(1.10) dp(x, Y) = inf{p(y−x) :y∈Y}, and
(1.11) dp(Y, x) = inf{p(x−y) :y∈Y}.
Note that dp(Y, x) =dp¯(x, Y).
Duality formulae and characterization results for best approximation in spaces with asymmetric norm were obtained in [5, 6, 9, 12, 34, 35]. The papers [32, 33, 39] are concerned with best approximation in spaces of semi- Lipschitz functions defined on asymmetric metric spaces (called quasi-metric spaces) in connection with the extension properties of these functions. In the papers [13, 24, 25, 36], supposing that p is the Minkowski functional pC of a bounded convex body C in a normed space (X,k k), some generic existence results for best approximation with respect to the asymmetric norm pC were proved, extending similar results from the normed case. As in the symmetric case, the geometric properties of the body C (or, equivalently, of the func- tional pC) are essential. A study of the moduli of convexity and smoothness corresponding topC is done in [43].
Best approximation with respect to some asymmetric norms in concrete function spaces of continuous or of integrable functions, called sign-sensitive approximation, was also studied in a series of papers, see [14, 15, 16, 41], the references quoted therein, and the monograph by Krein and Nudelman [23, Ch. 9, §5]).
The present paper, which can be viewed as a sequel to [12] and [9], is concerned mainly with characterizations of the elements of best approximation in a subspace Y of a space with asymmetric norm (X, p) and duality results for best approximation. As in the case of (symmetric) normed spaces the characterizations will be done in terms of some linear bounded functionals vanishing on Y. The duality results will involve the annihilator in Xp[ of the subspace Y. For this reason we start by recalling some extension results for bounded linear functionals on spaces with asymmetric seminorm. For proofs, all resorting to the classical Hahn-Banach extension theorem, see [9, 12].
Theorem 1.6. Let (X, p) be a space with asymmetric seminorm and Y a linear subspace of X. If ϕ0 : Y → R is a linear p-bounded functional on Y then there exists ap-bounded linear functionalϕ defined on the wholeX such that
ϕ|Y =ϕ0 and kϕ|p =kϕ0|p.
We agree to call a functional ϕ satisfying the conclusions of the above theorem anorm preserving extension of ϕ0.
Based on this extension result one can prove the following existence result.
Proposition 1.7. Let (X, p) be a space with asymmetric seminorm and x0 ∈X such that p(x0) >0. Then there exists a p-bounded linear functional ϕ:X →Rsuch that
kϕ|p = 1 and ϕ(x0) =p(x0).
In its turn, this proposition has the following corollary.
Corollary 1.8. If p(x0)>0 then
p(x0) = sup{ϕ(x0) :ϕ∈Xp[, kϕ|p ≤1}.
Moreover, there exists ϕ0 ∈Xp[, kϕ0|p = 1, such that ϕ0(x0) =p(x0).
The following proposition is the asymmetric analog of a well known result of Hahn.
Proposition 1.9. ([12])Let Y be a subspace of a space with asymmetric seminorm (X, p) and x0∈X.
(1) If d:=dp(x0, Y)>0,then there exists ϕ∈Xp[ such that (i)ϕ|Y = 0, (ii)kϕ|p = 1, and (iii)ϕ(−x0) =d.
(2) If d¯:=dp(Y, x0)>0, then there exists ψ∈Xp[ such that (j)ψ|Y = 0, (jj)kψ|p = 1, and (jjj)ψ(x0) = ¯d.
2. BEST APPROXIMATION IN SPACES WITH ASYMMETRIC SEMINORM
Let (X, p) be a space with asymmetric seminorm, ¯pthe seminorm conjugate topandY a nonempty subset ofX.The distancesdp(x, Y) anddp(Y, x) from an element x ∈ X to Y are defined by the formulae (1.10) and (1.11). An elementy0 ∈Y such thatp(y0−x) =dp(x, Y) will be called ap-nearest point tox inY, and an element y1 ∈Y such thatp(x−y1) = ¯p(y1−x) =dp¯(x, Y) is called a ¯p-nearest point tox inY.
Denote by
(2.1) PY(x) ={y∈Y :p(y−x) =dp(x, Y)}, and P¯Y(x) ={y∈Y :p(x−y) =dp(Y, x)},
the possibly empty sets of p-nearest points, respectively ¯p-nearest points, to x in Y. The set Y is called p-proximinal, p-semi-Chebyshev, p-Chebyshev if
for every x ∈ X the set PY(x) is nonempty, contains at most one element, contains exactly one element, respectively. Similar definitions are given in the case of ¯p-nearest points. A semi-Chebyshev set is called also auniqueness set.
For a nonempty subset Y of a space with asymmetric seminorm (X, p), denote by Yp⊥ the annihilatorofY inXp[, i.e.
Yp⊥ ={ϕ∈Xp[ :ϕ|Y = 0}.
We start by a characterization of nearest points given in [12] we shall need in the sequel.
Proposition 2.1. ([12]) Let (X, p)be a space with asymmetric seminorm, Y a subspace of X and x0 a point in X.
(1) Suppose that d := dp(x0, Y) > 0. An element y0 ∈ Y is a p-nearest point tox0 in Y if and only if there exists a bounded linear functional ϕ:X→R such that
(i)ϕ|Y = 0, (ii)kϕ|p= 1, (iii)ϕ(−x0) =p(y0−x0).
(2) Suppose that d¯:= dp(Y, x0) > 0. An element y0 ∈ Y is a p-nearest¯ point tox0 in Y if and only if there exists a bounded linear functional ψ:X →Rsuch that
(j)ψ|Y = 0, (jj)kψ|p = 1, (jjj)ψ(x0) =p(x0−y0).
From this theorem one can obtain characterizations of sets of nearest points.
Corollary 2.2. Let (X, p) be a space with asymmetric seminorm, Y a subspace of X, x∈X, and Z a nonempty subset of Y.
(1) If d = dp(x0, Y) > 0 then Z ⊂ PY(x) if and only if there exists a functional ϕ∈Xp[ such that
(i)ϕ|Y = 0, (ii)kϕ|p = 1, (iii)∀y ∈Z, ϕ(−x0) =p(y−x0).
(2) If d¯= dp(Y, x0) > 0 then Z ⊂ P¯Y(x) if and only if there exists a functional ψ∈Xp[ such that
(j)ψ|Y = 0, (jj)kψ|p = 1, (jjj)∀y ∈Z, ψ(x0) =p(x0−y).
In the next proposition we extend to the asymmetric case some charac- terization results for semi-Chebyshev subspaces (see [42, Chapter I, Theorem 3.2]).
Theorem2.3. LetY be a subspace of a space with asymmetric norm(X, p) such that p(x)>0 for every x6= 0.Then the following assertions are equiva- lent.
(1) Y is ap-semi-Chebyshev subspace of X.
(2) There are noϕ∈Yp⊥andx1, x2 ∈X withx1−x2 ∈Y\ {0}, such that (i) kϕ|p= 1 and (ii) ϕ(−xi) =p(−xi), i= 1,2.
(3) There are no ψ∈Yp⊥, x∈X, and y0∈Y \ {0} such that (j) kψ|p = 1 and (jj) ψ(−x) =p(−x) =p(y0−x).
Proof. (1)⇒ (2) Suppose that (2) does not hold. Letϕ∈Yp⊥andx1, x2 ∈ X withx1−x2 ∈Y \ {0}, such that the conditions (i) and (ii) of the assertion (2) are satisfied, and puty0 =x1−x2.Then
ϕ(−x2) =p(−x2) ⇐⇒ ϕ(y0−x1) =p(y0−x1), and
ϕ(−x1) =p(−x1) ⇐⇒ ϕ(0−x1) =p(0−x1).
By Proposition 2.1, it follows that 0 andy0 arep-nearest points tox1 inY. (2)⇒ (3) Suppose that (3) does not hold. Then there existψ∈Yp⊥, x∈ X, and y0 ∈Y \ {0}such that the conditions (j) and (jj) of the assertion (3) are fulfilled. It follows that the conditions (i) and (ii) of the assertion (2) are satisfied by ϕ=ψ, x1=x and x2 =y0−x, i.e. (2) does not hold.
(3)⇒ (1) Supposing that (1) does not hold, there exist z ∈ X\Y and y1, y2 ∈Y, y16=y2, such that
p(y1−z) =p(y2−z) =dp(z, Y).
If dp(z, Y) = 0, then y1 = y2 = z, a contradiction which shows that dp(z, Y)>0.
Ifx:=z−y1, then
dp(z−y1, Y) = inf{p(y+y1−z) :y∈Y}
= inf{p(y0−z) :y0 ∈Y}
=dp(z, Y)
=p(y1−z)
=p(y2−z).
By Proposition 2.1, there exists ψ∈Yp⊥, kψ|p= 1, such that ψ(y1−z) =p(y1−z) =p(y2−z),
or, denoting y0 :=y2−y1,this is equivalent to ψ(−x) =p(−x) =p(y0−x),
showing that (3) does not hold.
Remark 2.4. Obviously that a similar characterization result holds for ¯p-
semi-Chebyshev subspaces.
Using Corollary 2.2, one can extend Theorem 2.3 to obtain characterizations of pseudo-Chebyshev subspaces, a notion introduced by Mohebi [28] in the case of normed spaces. Concerning other weaker notions of Chebyshev spaces – quasi-Chebyshev subspaces, weak-Chebyshev subspaces, as well as for their behaviour in concrete function spaces, see the papers [26, 27, 29, 31]. For a subsetZ of a vector spaceX denote by aff(Z) the affine hull of the set Z, i.e.
aff(Z) = {x ∈ X : ∃n ∈ N,∃z1, ..., zn ∈ Z,∃a1, ..., an ∈ R, a1+...+an = 1 such that x = a1z1 +...+anzn}. There exists a unique subspace Y of X such that aff(Z) = z+Y, for an arbitrary z ∈ Z. By definition, the affine dimensionof the set Z is the dimension of this subspace Y of X.
A subspace Y of a space with asymmetric norm (X, p) is calledp-pseudo- Chebyshev if it is p-proximinal and the set PY(x) has finite affine dimension for everyx∈X.
The following theorem extends a result proved by Mohebi [28] in normed spaces.
Theorem 2.5. Let Y be a subspace of an asymmetric normed space (X, p) such that p(x)>0 for everyx6= 0.The following assertions are equivalent.
(1) The subspace Y isp-pseudo-Chebyshev.
(2) There do not exist ϕ ∈ Yp⊥, x0 ∈ X, and infinitely many linearly independent elements xn ∈X, n∈N, withx0−xn∈Y, n∈N, such that
(i)kϕ|p= 1 and (ii)ϕ(−xn) =p(−xn), n= 0,1, . . . .
(3) There do not exist ψ ∈ Yp⊥, x0 ∈ X, and infinitely many linearly independent elements yn∈Y, n∈N, such that
(j)kψ|p = 1 and (jj)ψ(−x0) =p(−x0) =p(yn−x0), n= 1,2, . . . . Proof. (1)⇒(2) Suppose that (2) does not hold. Then there exist ϕ ∈ Yp⊥, x0 ∈X, and infinitely many linearly independent elementsxn∈X, with x0 −xn ∈ Y, n ∈ N, satisfying the conditions (i) and (ii). The elements yn:=x0−xn, n∈N, all belong toY, are linearly independent, and
ϕ(yn−x0) =ϕ(−xn) =p(−xn) =p(yn−x0),
so that, by Corollary 2.2, they are all contained inPY(x0),showing thatY is notp-pseudo-Chebyshev.
(2)⇒(3) Suppose again that (3) does not hold, and let ψ∈Yp⊥, x0 ∈X, and the linearly independent elements {yn : n = 1,2, ...} ⊂ Y fulfilling the conditions (j) and (jj).
Then xn := x0−yn, n = 1,2, ..., are linearly independent elements in X and
ψ(−xn) =ψ(yn−x0) =p(−xn), n= 0,1,2, ..., showing that (2) does not hold.
(3)⇒(1) Supposing that (1) does not hold, there exist an element z ∈ X and an infinite set {yn : n = 1,2, ...} of linearly independent elements contained inPY(z).
By Corollary 2.2, there existsϕ∈Yp⊥, kϕ|p= 1, such that ϕ(yn−z) =dp(z, Y) =p(yn−z), n= 1,2, . . . .
Puttingx:=z−y1 we have
dp(x, Y) = inf{p(y+y1−z) :y ∈Y}= inf{p(y0−z) :y0∈Y}=
=dp(z, Y) =p(yn−z) =p(yn−y1−x), n= 2,3, ...,
showing that {yn−y1 :n = 2,3, ...} ⊂ PY(x). By Corollary 2.2, there exists ψ∈Yp⊥ with kψ|p= 1 such that
ψ(yn−yy−x0) =p(yn−y1−x0), n= 2,3, ...,
showing that (3) does not hold.
Phelps [37] emphasized for the first time some close connections existing between the approximation properties of the annihilator Y⊥ of a subspace Y of a normed space X and the extension properties of the subspace Y. A presentation of various situations in which such a connection occurs is done in [8]. The case of spaces with asymmetric norms was considered in [34, 35].
Let (X, p) be a space with asymmetric seminorm and Y a subspace of X.
For ap-bounded linear functional ϕ:Y →Rdenote by Ep(ϕ) ={ψ∈Xp[ :ψ|Y =ϕ, kψ|p =kϕ|p},
the set of all norm-preserving extensions of the functional ϕ. By the Hahn- Banach theorem (Theorem 1.6) the setEp(ϕ) is always nonempty.
Forϕ∈Xp[ consider the following minimization problem (2.2) γ(ϕ, Yp⊥) := inf{kϕ+ψ|p:ψ∈Yp⊥}.
A solution to this problem is an element ψ0 ∈Yp⊥ such that kϕ+ψ0|p = γ(ϕ, Yp⊥).Denote by ΠY⊥
p (ϕ) the set of all these solutions.
Theorem2.6. If the linear functionalϕ:X →Ris(p,p)-bounded then the¯ minimization problem (2.2) has a solution and the following formulae hold (2.3) γ(ϕ, Yp⊥) =kϕ|Y|p and ΠY⊥
p (ϕ) =Ep(ϕ|Y)−ϕ.
Proof. Let ϕ∈Xp[∩Xp[¯ and ψ∈Yp⊥. Then
kϕ+ψ|p ≥ k(ϕ+ψ)|Y|p =kϕ|Y|p, implying γ(ϕ, Yp⊥)≥ kϕ|Y|p.
If Φ ∈ Ep(ϕ|Y) then, because ϕ is (p,p)-bounded,¯ −ϕ ∈ Xp[ (see Propo- sition 1.5, ψ := Φ−ϕ∈ Yp⊥, and γ(ϕ, Yp⊥) ≤ kϕ+ψ|p = kΦ|p. Therefore γ(ϕ, Yp⊥) =kϕ|Y|p and
Ep(ϕ|Y)−ϕ⊂ΠY⊥ p (ϕ).
Conversely, if ψ ∈ ΠY⊥
p (ϕ), then Φ := ϕ+ψ satisfies Φ|Y = ϕ|Y and kΦ|p=kϕ+ψ|p =γ(ϕ, Yp⊥) =kϕ|Y|p, i.e. Φ∈Ep(ϕ|Y) and
ϕ+ ΠY⊥
p (ϕ)⊂Ep(ϕ|Y) ⇐⇒ ΠY⊥
p (ϕ)⊂Ep(ϕ|Y)−ϕ.
Denoting by
(2.4) Y⊥={ψ∈X∗ :ψ|Y = 0},
the annihilator Y⊥ of a subspace Y of X in the symmetric dual X∗ of the seminormed space (X, ps)∗, it follows that Y⊥ is a subspace of X∗. Consider on X∗ the asymmetric extended norm k |∗p :X∗→[0;∞] defined by
kϕ|∗p = supϕ(Bp).
We have for any ϕ∈X∗
ϕ∈Xp[ ⇐⇒ kϕ|∗p<∞, and kϕ|∗p =kϕ|p forϕ∈Xp[ (see Proposition 1.5).
Forϕ∈Xp[ consider the distance fromϕtoY⊥ defined by dp(Y⊥, ϕ) = inf{kϕ−ψ|∗p:ψ∈Y⊥}.
Because kϕ−0|∗p =kϕ|p<∞this distance is always finite. Put PY⊥(ϕ) ={ψ∈Y⊥ :kϕ−ψ|p =dp(Y⊥, ϕ)}.
Theorem2.7. Everyϕ∈Xp[ has ap-nearest point in¯ Y⊥and the following formulae hold
dp(Y⊥, ϕ) =kϕ|Y|p and PY⊥(ϕ) =ϕ−Ep(ϕ|Y).
Proof. Forψ∈Y⊥ we have
kϕ−ψ|∗p ≥ k(ϕ−ψ)|Y|∗p =kϕ|Y|p,
implying dp(Y⊥, ϕ)≥ kϕ|Y|p.If Φ∈Ep(ϕ|Y), thenψ:=ϕ−Φ∈Y⊥ and dp(Y⊥, ϕ)≤ kϕ−ψ|∗p =kΦ|p=kϕ|Y|p.
Therefore dp(Y⊥, ϕ) =kϕ|Y|p and ϕ−Ep(ϕ|Y)⊂PY⊥(ϕ).
If ψ∈ PY⊥(ϕ) and Φ :=ϕ−ψ, then Φ|Y =ϕ|Y and kΦ|p =kϕ−ψ|p = dp(Y⊥, ϕ) =kϕ|Y|p, i.e. Φ∈Ep(ϕ|Y), showing that ϕ−PY⊥(ϕ)⊂Ep(ϕ|Y),
or equivalently, PY⊥(ϕ)⊂ϕ−Ep(ϕ|Y).
From these theorems we obtain some uniqueness conditions for the mini- mization problems we have considered, in terms of the uniqueness of norm- preserving extensions.
Corollary 2.8. Let(X, p) be a space with asymmetric seminorm andY a subspace of X.
(1) If every f ∈ Yp[ has a unique norm preserving extension F ∈ Xp[, then the minimization problem (2.2) has a unique solution for every ϕ∈Xp[.
(2) Every point ϕ∈Xp[ has a unique p-nearest point in¯ Y⊥ if and only if every f ∈Yp[ has a unique norm-preserving extension F ∈Xp[. Proof. (1) If everyf ∈Yp[has a unique norm-preserving extensionF ∈Xp[, then for every ϕ ∈ Xp[ the set ΠY⊥
p (ϕ) = ϕ+Ep(ϕ|Y) contains exactly one element.
(2) Similarly,PY⊥(ϕ) =ϕ−Ep(ϕ|Y) contains exactly one element, provided everyf ∈Yp[ has exactly one norm-preserving extension F ∈Xp[.
Conversely, suppose that there exists f ∈ Yp[ having two distinct norm- preserving extensions F1, F2 ∈Xp[.Then
PY⊥(F1) =F1−Ep(F1|Y) =F1−Ep(f)⊃ {0, F1−F2}.
Remark2.9. We can not prove the reverse implication in the assertion (1) of the above corollary. To do this we would need an extension theorem for (p,p)-bounded linear functionals, preserving both¯ p- and ¯p-norm, and we are
not aware of such a result.
Some results connecting the -approximations and -extensions were ob- tained by Rezapour [38]. In the next proposition we transpose these results to the asymmetric case.
Let (X, p) be a space with asymmetric seminorm and Y a subspace of X.
Forx∈X and >0 let
PY(x) ={y∈Y :p(y−x)≤dp(x, Y) +} and
P¯Y(x) ={y∈Y :p(x−y)≤dp(Y, x) +}
denote the nonempty sets of -p-, respectively -¯p-nearest points to x in Y. Forϕ∈Xp[ consider the set of-solutions of the minimization problem (2.2)
ΠY⊥
p (ϕ) ={ψ∈Yp⊥ :kϕ+ψ|p ≤γ(ϕ, Yp⊥) +} and, finally, denote by
Ep(f) ={F ∈Xp[ :F|Y =f andkF|p≤ kf|p+}, the set of-extensions of a functionalf ∈Yp[.
These two sets are related in the following way.
Proposition 2.10. Let (X, p) be a space with asymmetric seminorm, Y a subspace of X andϕ∈Xp[. Then
ΠY⊥
p (ϕ) =Ep(ϕ|Y)−ϕ.
Proof. Indeed, by Theorem 2.6, ψ∈ΠY⊥
p (ϕ) ⇐⇒ ψ∈Yp⊥andkϕ+ψ|p ≤γ(ϕ, Yp⊥) +=kϕ|Y|p+
⇐⇒ ϕ+ψ∈Ep(ϕ|Y).
Working with the annihilator Y⊥ of the subspaceY in the symmetric dual X∗ = (X, ps)∗ given by (2.4) and putting
P¯Y⊥(ϕ) ={ψ∈Y⊥:kϕ−ψ|p ≤dp(ϕ, Y⊥) +}, we have
Proposition 2.11. Let Y be a subspace of a space with asymmetric semi- norm (X, p), >0, andϕ∈Xp[. Then
P¯Y(ϕ) =ϕ−Ep(ϕ|Y).
Proof. Indeed, by Theorem 2.7,
ψ∈P¯Y⊥(ϕ) ⇐⇒ ψ∈Y⊥andkϕ−ψ|p ≤dp(Y⊥, ϕ) +=kϕ|Y|p+
⇐⇒ ϕ−ψ∈Ep(ϕ|Y).
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Received by the editors: March 9, 2006.
