Rev. Anal. Num´er. Th´eor. Approx., vol. 33 (2004) no. 2, pp. 203–208 ictp.acad.ro/jnaat
CHARACTERIZATION OF ε-NEAREST POINTS IN SPACES WITH ASYMMETRIC SEMINORM∗
COSTIC ˘A MUST ˘AT¸ A†
Dedicated to professor Elena Popoviciu on the occasion of her 80th anniversary.
Abstract. In this note we are concerned with the characterization of the ele- ments of ε-best approximation (ε-nearest points) in a subspaceY of space X with asymmetric seminorm. For this we use functionals in the asymmetric dual Xb defined and studied in some recent papers [1], [3], [5].
MSC 2000. 41A65.
Keywords. Asymmetric seminormed spaces,ε-nearest points, characterization.
1. INTRODUCTION
LetX be a real linear space. A functional p:X→[0,∞) with the proper- ties:
(1) p(x)≥0,for allx∈X,
(2) p(tx) =tp(x),for allx∈X and t≥0, (3) p(x+y)≤p(x) +p(y),for all x, y∈X,
is called asymmetric seminorm onX,and the pair (X, p) is called a space with asymmetric seminorm.
The functionalp:X →[0,∞),defined byp(x) =p(−x), x∈X is another asymmetric seminorm onX, called the conjugate of p.
The functionalps:X→[0,∞), defined by
ps(x) = max{p(x), p(−x)}, x∈X,
is a seminorm on X. If ps satisfies the axioms of a norm, then p is called an asymmetric norm on X. It follows thatp satisfies the properties (1), (2), (3), and
(4) p(x) = 0 and p(−x) = 0 imply x= 0.
The asymmetric seminormponX generates a topologyτp onX, having as a basis of neighborhoods of a point x∈X the open p-balls
Bp0 (x, r) =x0∈X:p x0−x< r , r >0.
∗This work has been supported by the Romanian Academy under Grant GAR 13/2004.
†“T. Popoviciu” Institute of Numerical Analysis, P.O. Box 68-1, Cluj-Napoca, Romania, e-mail: [email protected].
The family of closed p-balls
Bp(x, r) =x0∈X:p x0−x≤r , r >0,
generates the same topology. This topology τp could not be Hausdorff (see [5]), and could not be linear (the multiplication by scalars is not continuous in general, see [1]).
Let R be the set of real numbers and u : R→[0,∞), u(a) = max{a,0}, a∈R.Then the function u is an asymmetric seminorm on Rand, for a∈R, the intervals (−∞, a+ε), ε > 0, form a basis of neighborhoods of a∈ R in the topology τu. The conjugate asymmetric seminorm of u is u :R→[0,∞), u(a) =u(−a), a∈R,and us(a) = max{u(a), u(−a)}=|a|is a norm onR. Consequently, u is an asymmetric norm onR.
Let ϕ:X →R be a linear functional. The continuity ofϕwith respect to the topologies τp and τu is called (p, u)-continuity, and it is equivalent to the upper semicontinuity of ϕas a functional from (X, τp) to (R,|.|).
The linear functionalϕ: (X, τp)→(R, u) is (p, u)-continuous if and only if it is p-bounded, i.e. there existsL≥0 such that
ϕ(x)≤Lp(x), for allx∈X.
The set of all (p, u)-continuous functionals is denoted by Xpb.With respect to pointwise addition and multiplication by real scalars, the set Xpb is a cone, i.e. λ≥0 andϕ, ψ∈Xpb imply ϕ+ψ∈Xb and λϕ∈Xpb.
The functionalk.|:Xpb →[0,∞) defined by
kϕ|p = sup{ϕ(x) :x∈X, p(x)≤1}, ϕ∈Xpb
satisfies the properties of an asymmetric seminorm, and the pair (Xpb,k.|p) is called the asymmetric dual of the asymmetric seminormed space (X, p) (see [5]). Some properties of this dual are presented in [1], [3], [5]. If there is no danger of confusion we shall use the notation Xb and kϕ| instead of Xpb and kϕ|p, respectively.
Let (X, p) be an asymmetric seminormed space andY a subspace ofX.Let Yb be the asymmetric dual of (Y, p).
The following result is the analog of a well known extension result for linear functionals in normed spaces.
Theorem 1 (Hahn-Banach). Let (Y, p) be a subspace of asymmetric semi- normed space(X, p). Then for every ϕ0∈Yb there existsϕ∈Xb such that
ϕ|Y =ϕ0, kϕ|=kϕ0|.
Proof. We consider the functional q : X → [0,∞), q(x) = kϕ0| · p(x), x ∈ X. Obviously q is subadditive and positive homogeneous and for every y∈Y we have
ϕ0(y)≤ kϕ0| ·p(y) =q(y)
i.e. ϕ0 is majorized byq onY.
By Hahn-Banach extension theorem it results that there exists the linear functional ϕ:X →Rwith properties:
ϕ|Y =ϕ0 and
ϕ(x)≤ kϕ0| ·p(x) , for everyx∈X.
It follows kϕ| ≤ kϕ0|,and, because
kϕ|= sup{ϕ(x) :x∈X, p(x)≤1}
≥sup{ϕ(y) :y ∈Y, p(y)≤1}
= sup{ϕ0(y) :y ∈Y, p(y)≤1}
=kϕ0|
we have kϕ|=kϕ0|.
2. THEε-BEST APPROXIMATION IN(X, p)
LetY be a nonvoid subset of the asymmetric seminormed space (X, p). The problem of best approximation of the elementx∈X by elements inY is: find an element y0∈Y such that
(1) dp(x, Y) := inf{p(y−x) :y∈Y}=p(y0−x).
Letε >0.The problem ofε-best approximation ofx∈Xby elements in Y is: find y0∈Y such that
(2) p(y0−x)≤dp(x, Y) +ε.
Obviously, the problem of ε-best approximation always admits a solution, because for every number n ∈ N there exists yn ∈ Y such that p(yn−x) ≤ dp(x, Y) +1n,so that p(yn−x)≤dp(x, Y) +ε, forn >h1εi+ 1.
In the following we denote by
(3) PY,ε(x) ={y∈Y :p(y−x)≤dp(x, Y) +ε}, x∈X the nonvoid set of the elements of ε-best approximation for x∈X inY.
The paper [3] contains characterizations, in terms of functionals inXb van- ishing onY, of the elements of best approximation ofx∈X by elements in a subspaceY ofX.Let us observe firstly that, one can consider also the problem of ε-best approximation by using the conjugate p ofp.In this case, for (4) dp(x, Y) = inf{p(x−y) :y∈Y}
one looks for y0 ∈Y such that
(5) p(x−y0)≤dp(x, Y) +ε.
Let us denote by
(6) PY,ε(x) ={y∈Y :p(y−x) =p(x−y)≤dp(x, Y) +ε}
the set of ε-best approximation of x∈X with respect to the conjugate asym- metric seminorm p.
In the following we obtain characterizations of elements of ε-best approxi- mation ofx∈X by elements of a subspaceY, with respect to the asymmetric seminormsp and p.
Results of this type, for elements of best approximations in a normed space X, using the elements of dual X∗ are obtained in [17] (see also [3], [9], [11], [13], [16], [18]).
Concerning the characterizations of elements of ε-best approximation in normed space, see papers [14], [15].
Theorem2. Let(X, p) be an asymmetric seminormed space,Y a subspace of X and x0 ∈ X \ Y, such that d=dp(x0, Y) > 0 and d= dp(x0, Y) >0.
Then
(a) An element y0 ∈ Y is in PY,ε(x0) if and only if there exists ϕ ∈ Xpb with the properties:
(i) ϕ(y) = 0, for all y∈Y, (iii) kϕ|p = 1,
(iii) ϕ(−x0)≥p(y0−x0)−ε.
(b) An element y0 is in PY,ε(x0) if and only if there exists ψ ∈ Xpb with the properties:
(j) ψ(y) = 0,for all y∈Y, (jj) kψ|p = 1,
(jjj) ψ(x0)≥p(x0−y0)−ε.
Proof. Letx0 ∈X \Y and Z =Y +hx0i be the direct sum of Y with the space generated byx0.Consider the functionalϕ0:Z →Rdefined by
ϕ0(z) =ϕ(y+λx0) =−λ,
wherez∈Z,and z is uniquely represented in the form z=y+λx0. The functionalϕ0 is linear onZ.
Observe thatϕ0 |Y = 0,and for everyλ >0 we have p(y−λx0) =λp1λy−x0
≥λd=d·ϕ0(y−λx0).
It follows that
ϕ0(y−λx0)≤ 1d·p(y−λx0), for everyλ >0.
Because the last inequality is also valid if ϕ0(y−tx0) =t≤0,it follows kϕ0|p ≤ 1d, and consequently ϕ0 ∈Zpb.
Now, let (yn)n≥1 be a sequence in Y such thatp(yn−x0) → d, forn → ∞, and such thatp(yn−x0)>0 for every n∈N.Then
kϕ0|p ≥ϕ0p(yyn−x0
n−x0)
= p(y 1
n−x0) → 1d,
and, consequently, kϕ0|p = 1d.
By Theorem 1, there existsϕ1∈Xb such that ϕ1 |Z =ϕ0, kϕ1|=kϕ0|p= d1.
Then, the functional ϕ = d·ϕ1 satisfies the properties: ϕ ∈ Xpb, ϕ | Y = d·ϕ1|Y = 0,
ϕ(−x0) =ϕ(y0) +ϕ(−x0)
=ϕ(y0−x0)
≥p(y0−x0)
≥p(y0−x0)−ε.
Conversely, if y0 ∈ Y and there exists ϕ ∈ Xpb with the properties (a) (i)-(iii), then for every y∈Y we have
p(y0−x0)≤ϕ(−x0) +ε
=ϕ(y−x0) +ε
≤ kϕ|p·p(y−x0) +ε
≤p(y−x0) +ε.
Taking the infimum with respect toy∈Y,one finds p(y0−x0)≤dp(x0, Y) +ε;
so that y0∈PY,ε(x0).
Similarly, defining ψ0 : Z = Y +hx0i → R by ψ(z) = ψ0(y+λx0) = λ, y∈Y andλ∈R,and proceeding in the same way, one obtains the claim
(b) of the theorem.
Theorem 2 has the following consequence:
Corollary 3. In the hypothesis of Theorem 2 we have:
(a0) M ⊂PY,ε(x0), if and only if there existsϕ∈Xb verifying (a) (i)-(ii) and the condition:
ϕ(−x0)≥p(u−x0)−ε, for all u∈M;
(b0) M ⊂ PY,ε(x0) if and only if there exists ψ ∈ Xb with properties (b) (j)-(jj), and verifying the condition:
ψ(x0)≥p(x0−u)−ε, for all u∈M.
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Received by the editors: June 11, 2003.