The present paper is concerned with the characterization of the elements of best approximation in a subspace \(Y\) of a space with asymmetric norm, in terms of some linear functionals vanishing on \(Y\). The approach is based on some extension results, proved in Section 3, for bounded linear functionals on such spaces. Also, the well known formula for the distance to a hyperplane in a normed space is extended to the nonsymmetric case.
Costica Mustata
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania
Keywords
Spaces with asymmetric norm; best approximation; Hahn-Banach theorem; characterization of best approximation.
Paper coordinates
C. Mustăţa and Şt. Cobzaş, Extension of bounded linear functionals and best approximation in space with asymmetric norm, Rev. Anal. Numer. Theor. Approx.. 33 (2004) no. 1, 39-50.
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Paper (preprint) in HTML form
2004-Mustata-Extension of bounded linear functionals-Jnaat
EXTENSION OF BOUNDED LINEAR FUNCTIONALS AND BEST APPROXIMATION IN SPACES WITH ASYMMETRIC NORM
S. COBZAŞ* and C. MUSTĂŢA ^(†){ }^{\dagger}
Abstract
The present paper is concerned with the characterization of the elements of best approximation in a subspace YY of a space with asymmetric norm, in terms of some linear functionals vanishing on YY. The approach is based on some extension results, proved in Section 3, for bounded linear functionals on such spaces. Also, the well known formula for the distance to a hyperplane in a normed space is extended to the nonsymmetric case.
MSC 2000. 41A65.
Keywords. Spaces with asymmetric norm, best approximation, Hahn-Banach theorem, characterization of best approximation.
1. INTRODUCTION
Let XX be a real vector space. An asymmetric seminorm on XX is a positive sublinear functional p:X rarr[0,oo)p: X \rightarrow[0, \infty), i.e. pp satisfies the conditions:
(AN1) quad p(x) >= 0\quad p(x) \geq 0,
(AN2) quad p(tx)=tp(x),t >= 0\quad p(t x)=t p(x), t \geq 0,
(AN3) quad p(x+y) <= p(x)+p(y)\quad p(x+y) \leq p(x)+p(y),
for all x,y in Xx, y \in X. The function bar(p):X rarr[0,oo)\bar{p}: X \rightarrow[0, \infty) defined by bar(p)(x)=p(-x),x in X\bar{p}(x)=p(-x), x \in X, is another positive sublinear functional on XX, called the conjugate of pp, and
p^(s)(x)=max{p(x),p(-x)},x in Xp^{s}(x)=\max \{p(x), p(-x)\}, x \in X
is a seminorm on XX. The inequalities
|p(x)-p(y)| <= p^(s)(x-y)quad" and "quad| bar(p)(x)- bar(p)(y)| <= p^(s)(x-y)|p(x)-p(y)| \leq p^{s}(x-y) \quad \text { and } \quad|\bar{p}(x)-\bar{p}(y)| \leq p^{s}(x-y)
hold for all x,y in Xx, y \in X. If the seminorm p^(s)p^{s} is a norm on XX then we say that pp is an asymmetric norm on XX. This means that, beside (AN1)-(AN3), it satisfies also the condition
(AN4) quad p(x)=0quad\quad p(x)=0 \quad and p(-x)=0p(-x)=0 imply quad x=0\quad x=0.
The pair ( X,pX, p ), where XX is a linear space and pp is an asymmetric seminorm on XX is called a space with asymmetric seminorm, respectively a space with asymmetric norm, if pp is an asymmetric norm.
An asymmetric seminorm pp generates a topology tau_(p)\tau_{p} on XX, having as a basis of neighborhoods of a point x in Xx \in X the open pp-balls
generates the same topology.
Denote by B_(p)=B_(p)(0,1)B_{p}=B_{p}(0,1) the closed unit ball of ( X,pX, p ) and by B_(p)^(')=B_(p)^(')(0,1)B_{p}^{\prime}=B_{p}^{\prime}(0,1) its open unit ball.
The topology tau_(p)\tau_{p} is translation invariant, i.e. the addition +:X xx X rarr X+: X \times X \rightarrow X is continuous, but the multiplication by scalars *:Rxx X rarr X\cdot: \mathbb{R} \times X \rightarrow X need not be continuous. For instance, in the space
C_(0)[0,1]={x in C[0,1]:int_(0)^(1)x(t)dt=0}C_{0}[0,1]=\left\{x \in C[0,1]: \int_{0}^{1} x(t) \mathrm{d} t=0\right\}
with the asymmetric seminorm p(x)=max x([0,1])p(x)=\max x([0,1]), the multiplication by scalars is not continuous at t_(0)=-1t_{0}=-1 and x_(0)=0x_{0}=0. Indeed, the ball B_(p)(0,1)B_{p}(0,1) is a neighborhood of 0=(-1)00=(-1) 0, but -B(0,r)⊈B(0,1)-B(0, r) \nsubseteq B(0,1) for any r > 0r>0, because the functions x_(n)x_{n} defined by
x_(n)(t)={[(n-1)(nt-1)","," for "0 <= t <= (1)/(n)],[(n)/(n-1)(t-(1)/(n))","," for "(1)/(n) <= t <= 1]:}x_{n}(t)= \begin{cases}(n-1)(n t-1), & \text { for } 0 \leq t \leq \frac{1}{n} \\ \frac{n}{n-1}\left(t-\frac{1}{n}\right), & \text { for } \frac{1}{n} \leq t \leq 1\end{cases}
is in B_(p)(0,1)B_{p}(0,1) for all nn, while p(-x_(n))=n-1 > rp\left(-x_{n}\right)=n-1>r for large nn (see [2]).
The topology tau_(p)\tau_{p} could not be Hausdorff even if pp is an asymmetric norm on XX. Necessary and sufficient conditions in order that tau_(p)\tau_{p} be Hausdorff were given in [8].
In this paper we shall study some best approximation problems in spaces with asymmetric seminorm. The significance of asymmetric norms for best approximation problems was first emphasized by Krein and Nudel'man (see [10, Ch. 9, § 5]). In the spaces C(T)C(T) and L_(r),1 <= r < ooL_{r}, 1 \leq r<\infty, one considers asymmetric norms defined through a pair w=(w_(+),w_(-))w=\left(w_{+}, w_{-}\right)of nonnegative upper semicontinuous functions, called weight functions, via the formula ||f|_(w)=max{w_(+)(t)f_(+)(t)-w_(-)(t)f_(-)(t):t in T}\|\left. f\right|_{w}=\max \left\{w_{+}(t) f_{+}(t)-w_{-}(t) f_{-}(t): t \in T\right\}, where f_(+),f_(-)f_{+}, f_{-}are the positive, respectively negative part of ff. In the case of the spaces L_(r)L_{r} the above formula is adapted to the corresponding integral norm. The approximation in such spaces is called sign-sensitive approximation and it is studied in a lot of papers, following the ideas from the symmetric case (see [1, 4, 5, 9, 18, 19, 20] and the references given in these papers). There are also papers concerning existence results, mainly generic, for best approximation in abstract spaces with asymmetric norms, see [3, 11, 12, 17.
In [14, 16, there were studied the relations between the existence of best approximation and uniqueness of the extension of bounded linear functionals on spaces with asymmetric norm. In 13, 15] similar problems were considered
within the framework of spaces of semi-Lipschitz functions on an asymmetric metric space (called quasi-metric space).
The present paper is concerned with the characterization of the elements of best approximation in a subspace YY of a space with asymmetric norm in terms of some linear functionals vanishing on YY. The approach is based on some extension results, proved in Section 3, for bounded linear functionals on such spaces. Also, the well known formula for the distance to a hyperplane in a normed space is extended to the nonsymmetric case. For the case of normed spaces see 21.
2. BOUNDED LINEAR MAPPINGS AND THE DUAL OF A SPACE WITH ASYMMETRIC SEMINORM
Let (X,p)(X, p) and (Y,q)(Y, q) be spaces with asymmetric seminorms and A:X rarr YA: X \rightarrow Y a linear mapping. The mapping AA is called bounded (or semi-Lipschitz) if there exists L >= 0L \geq 0 such that
{:(2.1)q(Ax) <= Lp(x)","quad" for all "x in X.:}\begin{equation*}
q(A x) \leq L p(x), \quad \text { for all } x \in X . \tag{2.1}
\end{equation*}
It was shown in [6] (see also [7]) that the boundedness of the linear mapping AA is equivalent to its continuity with respect to the topologies tau_(p)\tau_{p} and tau_(q)\tau_{q}. Denoting by L_(b)(X,Y)L_{b}(X, Y) the set of all bounded linear mapping from ( X,pX, p ) to ( Y,qY, q ), it turns out that L_(b)(X,Y)L_{b}(X, Y) is not necessarily a linear space but rather a convex cone in the vector space L_(a)(X,Y)L_{a}(X, Y) of all linear mappings from XX to YY, i.e.
lambda >= 0" and "A,B inL_(b)(X,Y)=>A+B inL_(b)(X,Y)" and "lambda A inL_(b)(X,Y)\lambda \geq 0 \text { and } A, B \in L_{b}(X, Y) \Rightarrow A+B \in L_{b}(X, Y) \text { and } \lambda A \in L_{b}(X, Y)
For instance, in the space X=C_(0)[0,1]X=C_{0}[0,1] considered in the previous section, the linear functional varphi(x)=x(1),x inC_(0)[0,1]\varphi(x)=x(1), x \in C_{0}[0,1], is bounded because varphi(x) <= p(x),x in X\varphi(x) \leq p(x), x \in X, but the functional -varphi-\varphi is not bounded. Taking x_(n)(t)=1-nt^(n-1)x_{n}(t)=1-n t^{n-1} we have p(x_(n))=1p\left(x_{n}\right)=1 for all nn, but -varphi(x_(n))=n-1rarr oo-\varphi\left(x_{n}\right)=n-1 \rightarrow \infty for n rarr oon \rightarrow \infty (see [2]).
As in the case of bounded linear mapping between normed linear spaces, one can define an asymmetric seminorm on L_(b)(X,Y)L_{b}(X, Y) by the formula
{:(2.2)||A∣=s u p{q(Ax):x in X","p(x) <= 1}:}\begin{equation*}
\| A \mid=\sup \{q(A x): x \in X, p(x) \leq 1\} \tag{2.2}
\end{equation*}
It is not difficult to see that ||*∣\| \cdot \mid is an asymmetric seminorm on the cone L_(b)(X,Y)L_{b}(X, Y) which has properties similar to those of the usual norm:
Proposition 2.1. Let ( X,pX, p ) and ( Y,qY, q ) be spaces with asymmetric seminorms and A inL_(b)(X,Y)A \in L_{b}(X, Y). Then
AA x in X quad q(Ax) <= ||A∣*p(x)\forall x \in X \quad q(A x) \leq \| A \mid \cdot p(x),
and ||A∣\| A \mid is the smallest number L >= 0L \geq 0 for which the inequality (2.1) holds.
||A|=s u p{(q(Ax))/(p(x)):x in X,p(x) > 0}:}\| A \left\lvert\,=\sup \left\{\frac{q(A x)}{p(x)}: x \in X, p(x)>0\right\}\right..
Proof. 1) If p(x)=0p(x)=0 then, by the boundedness of A,q(Ax)=0=||A∣p(x)A, q(A x)=0=\| A \mid p(x). If p(x) > 0p(x)>0 then p((1//p(x))x)=1p((1 / p(x)) x)=1 and
If q(Ax) <= Lp(x),AA x in Xq(A x) \leq L p(x), \forall x \in X, for some L >= 0L \geq 0, then q(Ax) <= Lq(A x) \leq L for all x in Xx \in X with p(x) <= 1p(x) \leq 1, implying ||A∣≤L\| A \mid \leq L.
2) Follows from the facts that q(Ax)=0q(A x)=0 if p(x)=0p(x)=0 and
Bounded linear functionals on a space with asymmetric norm
As in the case of normed spaces, the cone of bounded linear functional on a space with asymmetric seminorm will play a key role in various problems concerning these spaces.
On the space R\mathbb{R} of real numbers, consider the asymmetric seminorm u(alpha)=max{alpha,0}u(\alpha)= \max \{\alpha, 0\} and denote by R_(u)\mathbb{R}_{u} the space R\mathbb{R} equipped with the topology tau_(u)\tau_{u} generated by uu. It is the topology generated by the intervals of the form (-oo,a),a inR(-\infty, a), a \in \mathbb{R}. A neighborhood basis of a point a inR_(u)a \in \mathbb{R}_{u} is formed by the intervals (-oo,a+epsilon),epsilon > 0(-\infty, a+\epsilon), \epsilon>0. The seminorm conjugate to uu is bar(u)(alpha)=u(-alpha)=max{-alpha,0}\bar{u}(\alpha)=u(-\alpha)= \max \{-\alpha, 0\}, and u^(s)(alpha)=max{u(alpha),u(-alpha)}=|alpha|u^{s}(\alpha)=\max \{u(\alpha), u(-\alpha)\}=|\alpha|. The continuity of a linear functional varphi:(X,p)rarr(R,u)\varphi:(X, p) \rightarrow(\mathbb{R}, u) with respect to the topologies tau_(p)\tau_{p} and tau_(u)\tau_{u} will be called ( p,up, u )-continuity. It is easily seen that the ( p,up, u )-continuity of a linear functional varphi:(X,tau_(p))rarr(R,u)\varphi:\left(X, \tau_{p}\right) \rightarrow(\mathbb{R}, u) is equivalent to its upper semi-continuity as a functional from (X,tau_(p))\left(X, \tau_{p}\right) to (R,||)(\mathbb{R},| |). This is equivalent to the fact that for every alpha inR\alpha \in \mathbb{R} the set {x in X:varphi(x) >= alpha}\{x \in X: \varphi(x) \geq \alpha\} is closed in ( X,tau_(p)X, \tau_{p} ) and has consequence the fact that, for every tau_(p)\tau_{p}-compact subset YY of XX, the functional varphi\varphi is upper bounded on YY and there exists y_(0)in Yy_{0} \in Y such that varphi(y_(0))=s u p varphi(Y)\varphi\left(y_{0}\right)=\sup \varphi(Y). Also, the linear functional varphi\varphi is ( p,up, u )-continuous if and only if it is pp-bounded, i.e. there exists L >= 0L \geq 0 such that
{:(2.3)AA x in X quad varphi(x) <= Lp(x).:}\begin{equation*}
\forall x \in X \quad \varphi(x) \leq L p(x) . \tag{2.3}
\end{equation*}
Denote by X_(p)^(b)X_{p}^{b} ( X^(b)X^{b} when it is no danger of confusion) the cone of all bounded linear functionals on the space with asymmetric seminorm ( X,pX, p ) and call it the asymmetric dual of ( X,pX, p ). It follows that the functional
||varphi|=||varphi|_(p)=s u p{varphi(x):x in X,p(x) <= 1}:}\left\|\varphi|=\| \varphi|_{p}=\sup \{\varphi(x): x \in X, p(x) \leq 1\}\right.
is an asymmetric seminorm on X^(b)X^{b}.
We shall need the following simple properties of this seminorm.
Proposition 2.2. If varphi\varphi is a bounded linear functional on a space with asymmetric seminorm ( X,pX, p ), with p!=0p \neq 0, then:
||varphi∣\| \varphi \mid is the smallest of the numbers L >= 0L \geq 0 for which the inequality (2.3) holds;
We have:
{:[||varphi∣=s u p{varphi(x)//p(x):x in X","p(x) > 0}],[=s u p{varphi(x):x in X","p(x) < 1}],[=s u p{varphi(x):x in X","p(x)=1}]:}\begin{aligned}
\| \varphi \mid & =\sup \{\varphi(x) / p(x): x \in X, p(x)>0\} \\
& =\sup \{\varphi(x): x \in X, p(x)<1\} \\
& =\sup \{\varphi(x): x \in X, p(x)=1\}
\end{aligned}
If varphi!=0\varphi \neq 0 then ||varphi∣>0\| \varphi \mid>0.
Also, if varphi!=0\varphi \neq 0 and varphi(x_(0))=||varphi∣\varphi\left(x_{0}\right)=\| \varphi \mid for some x_(0)inB_(p)x_{0} \in B_{p}, then p(x_(0))=1p\left(x_{0}\right)=1.
Proof. We shall prove the assertions 2) and 3), the first one being a particular case of the corresponding result for linear mappings.
Supposing c:=s u p{varphi(x):p(x) < 1} < ||varphi∣c:=\sup \{\varphi(x): p(x)<1\}<\| \varphi \mid, let x_(0)in X,p(x_(0))=1x_{0} \in X, p\left(x_{0}\right)=1, be such that c < varphi(x_(0)) <= ||varphi∣c<\varphi\left(x_{0}\right) \leq \| \varphi \mid. Then there is a number alpha,0 < alpha < 1\alpha, 0<\alpha<1, such that varphi(alphax_(0))=alpha varphi(x_(0)) > c\varphi\left(\alpha x_{0}\right)=\alpha \varphi\left(x_{0}\right)>c, in contradiction to the definition of cc.
Let's show now that ||varphi∣=s u p{varphi(x):p(x)=1}\| \varphi \mid=\sup \{\varphi(x): p(x)=1\}. Suppose again that beta:=s u p{varphi(x):p(x)=1} < ||varphi∣\beta:=\sup \{\varphi(x): p(x)=1\}<\| \varphi \mid, and choose x_(0)in Xx_{0} \in X such that p(x_(0)) < 1p\left(x_{0}\right)<1 and varphi(x_(0)) > beta\varphi\left(x_{0}\right)>\beta. Putting x_(1)=(1//p(x_(0)))x_(0)x_{1}=\left(1 / p\left(x_{0}\right)\right) x_{0}, it follows p(x_(1))=1p\left(x_{1}\right)=1 and
a contradiction.
3) Because varphi(x) <= ||varphi∣p(x)\varphi(x) \leq \| \varphi \mid p(x), the equality ||varphi∣=0\| \varphi \mid=0 implies varphi(x) <= 0\varphi(x) \leq 0 and -varphi(x)=varphi(-x) <= 0-\varphi(x)=\varphi(-x) \leq 0, i.e. varphi(x)=0\varphi(x)=0 for all x in Xx \in X.
Suppose now that that for varphi!=0\varphi \neq 0 there exists x_(0)in Xx_{0} \in X, with 0 < p(x_(0)) < 10<p\left(x_{0}\right)<1, such that varphi(x_(0))=||varphi∣\varphi\left(x_{0}\right)=\| \varphi \mid. Then alpha:=1//p(x_(0)) > 1,x_(1)=alphax_(0)inB_(p)\alpha:=1 / p\left(x_{0}\right)>1, x_{1}=\alpha x_{0} \in B_{p} and
a contradiction, because ||varphi∣>0\| \varphi \mid>0.
An immediate consequence of the preceding result is the following one. We agree to call a linear functional ( p, bar(p)p, \bar{p} )-bounded if it is both pp - and bar(p)\bar{p}-bounded.
Proposition 2.3. Let varphi!=0\varphi \neq 0 be a linear functional on a space with asymmetric seminorm ( X,pX, p ).
If varphi\varphi is ( p, bar(p)p, \bar{p} )-bounded then
where B_(p)^(')={x in X:p(x) < 1},B_( bar(p))^(')={x in X: bar(p)(x) < 1}B_{p}^{\prime}=\{x \in X: p(x)<1\}, B_{\bar{p}}^{\prime}=\{x \in X: \bar{p}(x)<1\} and r > 0r>0.
2) If varphi\varphi is pp-bounded but not bar(p)\bar{p}-bounded then
varphi(rB_(p)^('))=(-oo,r|| varphi|_(p)).\varphi\left(r B_{p}^{\prime}\right)=\left(-\infty, r \|\left.\varphi\right|_{p}\right) .
Proof. Obviously that it is sufficient to give the proof only for r=1r=1.
Suppose that varphi\varphi is (p, bar(p))(p, \bar{p})-bounded. Then, by Proposition 2.2,
s u p varphi(B_(p)^('))=|| varphi|_(p)\sup \varphi\left(B_{p}^{\prime}\right)=\|\left.\varphi\right|_{p}
and
i n f varphi(B_(p)^('))=-s u p{varphi(-x):p(x) < 1}=-s u p{varphi(x^(')):p(-x^(')) < 1}=-|| varphi|_( bar(p))\inf \varphi\left(B_{p}^{\prime}\right)=-\sup \{\varphi(-x): p(x)<1\}=-\sup \left\{\varphi\left(x^{\prime}\right): p\left(-x^{\prime}\right)<1\right\}=-\|\left.\varphi\right|_{\bar{p}}
Also, by the assertion 3) of Proposition 2.2, varphi(x) < || varphi|_(p)\varphi(x)<\|\left.\varphi\right|_{p} and varphi(x) > -|| varphi|_( bar(p))\varphi(x)>-\|\left.\varphi\right|_{\bar{p}} for any x inB_(p)^(')x \in B_{p}^{\prime}.
Because B_(p)^(')B_{p}^{\prime} is convex, varphi(B_(p)^('))\varphi\left(B_{p}^{\prime}\right) will be a convex subset of R\mathbb{R}, that is an interval, and the above considerations show that
varphi(B_(p)^('))=(i n f varphi(B_(p)^(')),s u p varphi(B_(p)^(')))=(-|| varphi|_( bar(p)),||varphi|_(p)).\varphi\left(B_{p}^{\prime}\right)=\left(\inf \varphi\left(B_{p}^{\prime}\right), \sup \varphi\left(B_{p}^{\prime}\right)\right)=\left(-\left.\left\|\left.\varphi\right|_{\bar{p}},\right\| \varphi\right|_{p}\right) .
If varphi\varphi is pp-bounded and
s u p{varphi(x); bar(p)(x) < 1}=oo.\sup \{\varphi(x) ; \bar{p}(x)<1\}=\infty .
then
i n f{varphi(x^(')):p(x^(')) < 1}=-s u p{varphi(x):p(-x) < 1}=-oo\inf \left\{\varphi\left(x^{\prime}\right): p\left(x^{\prime}\right)<1\right\}=-\sup \{\varphi(x): p(-x)<1\}=-\infty
3. EXTENSION RESULTS FOR BOUNDED LINEAR FUNCTIONALS
In this section we shall prove the analogs of some well known extension results for linear functional in normed spaces. The main tool is the Hahn-Banach extension theorem for linear functionals dominated by sublinear functionals.
Throughout this section ( X,pX, p ) will be a space with asymmetric seminorm.
Proposition 3.1. Let YY be a subspace of XX and varphi_(0):Y rarrR\varphi_{0}: Y \rightarrow \mathbb{R} a bounded linear functional. Then there exists a bounded linear functional varphi:X rarrR\varphi: X \rightarrow \mathbb{R} such that
varphi|_(Y)=varphi_(0)quad" and "quad||varphi|=||varphi_(0)|.:}\left.\varphi\right|_{Y}=\varphi_{0} \quad \text { and } \quad\left\|\varphi\left|=\| \varphi_{0}\right| .\right.
Proof. The functional q(x)=||varphi_(0)∣p(x),x in Xq(x)=\| \varphi_{0} \mid p(x), x \in X, is sublinear and varphi_(0)(y) <= q(y),y in Y\varphi_{0}(y) \leq q(y), y \in Y. By the Hahn-Banach extension theorem there exists a linear functional varphi:X rarrR\varphi: X \rightarrow \mathbb{R} such that
varphi|_(Y)=varphi_(0)quad" and "quad AA x in X varphi(x) <= ||varphi_(0)∣p(x).\left.\varphi\right|_{Y}=\varphi_{0} \quad \text { and } \quad \forall x \in X \varphi(x) \leq \| \varphi_{0} \mid p(x) .
The second of the above relations implies that varphi\varphi is bounded and ||varphi| <= ||varphi_(0)|:}\left\|\varphi\left|\leq \| \varphi_{0}\right|\right.. Since
||varphi|=s u p{varphi(x):x in X,p(x) <= 1} >= s u p{varphi(y):y in Y,p(y) <= 1}=||varphi_(0)|:}\left\|\varphi\left|=\sup \{\varphi(x): x \in X, p(x) \leq 1\} \geq \sup \{\varphi(y): y \in Y, p(y) \leq 1\}=\| \varphi_{0}\right|\right.
it follows ||varphi|=||varphi_(0)|:}\left\|\varphi\left|=\| \varphi_{0}\right|\right..
We agree to call a functional varphi\varphi satisfying the conclusions of Proposition 3.1 a norm preserving extension of varphi_(0)\varphi_{0}.
Proposition 3.2. If x_(0)x_{0} is a point in XX such that p(x_(0)) > 0p\left(x_{0}\right)>0 then there exists a bounded linear functional varphi:X rarrR\varphi: X \rightarrow \mathbb{R} such that
||varphi∣=1quad" and "quad varphi(x_(0))=p(x_(0))\| \varphi \mid=1 \quad \text { and } \quad \varphi\left(x_{0}\right)=p\left(x_{0}\right)
Proof. Let Y:=Rx_(0)Y:=\mathbb{R} x_{0} and let varphi_(0):Y rarrR\varphi_{0}: Y \rightarrow \mathbb{R} be defined by varphi_(0)(tx_(0))=tp(x_(0))\varphi_{0}\left(t x_{0}\right)=t p\left(x_{0}\right), t inRt \in \mathbb{R}. It follows that varphi_(0)\varphi_{0} is linear and
for t <= 0t \leq 0. Again, the Hahn-Banach extension theorem yields a linear functional varphi:X rarrR\varphi: X \rightarrow \mathbb{R}, such that
varphi|_(Y)=varphi_(0)quad" and "quad AA x in X varphi(x) <= p(x).\left.\varphi\right|_{Y}=\varphi_{0} \quad \text { and } \quad \forall x \in X \varphi(x) \leq p(x) .
It follows ||varphi∣≤1,varphi(x_(0))=p(x_(0))\| \varphi \mid \leq 1, \varphi\left(x_{0}\right)=p\left(x_{0}\right), and, since p((1//p(x_(0)))x_(0))=1p\left(\left(1 / p\left(x_{0}\right)\right) x_{0}\right)=1,
i.e. ||varphi∣=1\| \varphi \mid=1.
This last proposition has as consequence the following useful result.
Corollary 3.3. If p(x_(0)) > 0p\left(x_{0}\right)>0, then
Proof. Denote by ss the supremum in the right hand side of the above formula. Since varphi(x_(0)) <= ||varphi∣p(x_(0)) <= p(x_(0))\varphi\left(x_{0}\right) \leq \| \varphi \mid p\left(x_{0}\right) \leq p\left(x_{0}\right) for every varphi inX^(b),||varphi∣≤1\varphi \in X^{b}, \| \varphi \mid \leq 1, it follows s <= p(x_(0))s \leq p\left(x_{0}\right). Choosing varphi inX_(p)^(b)\varphi \in X_{p}^{b} as in Proposition 3.2, it follows p(x_(0))=varphi(x_(0)) <= sp\left(x_{0}\right)=\varphi\left(x_{0}\right) \leq s.
The next extension result involves the distance from a point to a set in an asymmetric seminormed space. Let YY be a nonempty subset of an asymmetric seminormed space ( X,pX, p ). Due to the asymmetry of the seminorm pp we have to consider two distances from a point x in Xx \in X to YY, namely
{:(3.1)d_(p)(x","Y)=i n f{p(y-x):y in Y}:}\begin{equation*}
d_{p}(x, Y)=\inf \{p(y-x): y \in Y\} \tag{3.1}
\end{equation*}
and
{:(3.2)d_(p)(Y","x)=i n f{p(x-y):y in Y}:}\begin{equation*}
d_{p}(Y, x)=\inf \{p(x-y): y \in Y\} \tag{3.2}
\end{equation*}
Observe that d_(p)(Y,x)=d_( bar(p))(x,Y)d_{p}(Y, x)=d_{\bar{p}}(x, Y), where bar(p)\bar{p} is the seminorm conjugate to pp.
Proposition 3.4. Let YY be a subspace of a space with asymmetric seminorm ( X,pX, p ) and x_(0)in Xx_{0} \in X. Denote by bar(d)\bar{d} the distance d_( bar(p))(x_(0),Y)d_{\bar{p}}\left(x_{0}, Y\right) and suppose bar(d) > 0\bar{d}>0.
Then there exists a pp-bounded linear functional varphi:X rarrR\varphi: X \rightarrow \mathbb{R} such that
(i)varphi|_(Y)=0,quad(ii)||varphi∣=1,quad" and "quad(iii)varphi(x_(0))= bar(d)\left.(i) \varphi\right|_{Y}=0, \quad(i i) \| \varphi \mid=1, \quad \text { and } \quad(i i i) \varphi\left(x_{0}\right)=\bar{d}
If d=d_(p)(x_(0),Y) > 0d=d_{p}\left(x_{0}, Y\right)>0 then there exists psi inX_(p)^(b)\psi \in X_{p}^{b} such that
Proof. Suppose that bar(d)=d_( bar(p))(x_(0),Y) > 0\bar{d}=d_{\bar{p}}\left(x_{0}, Y\right)>0, so that x_(0)!in Yx_{0} \notin Y. Let Z:=Y+Rx_(0)((+^(˙)):}Z:=Y+\mathbb{R} x_{0} \left(\dot{+}\right. stands for the direct sum) and let psi_(0):Z rarrR\psi_{0}: Z \rightarrow \mathbb{R} be defined by
psi_(0)(y+tx_(0))=t,y in Y,t inR.\psi_{0}\left(y+t x_{0}\right)=t, y \in Y, t \in \mathbb{R} .
Then psi_(0)\psi_{0} is linear, psi_(0)(y)=0,AA y in Y\psi_{0}(y)=0, \forall y \in Y, and psi_(0)(x_(0))=1\psi_{0}\left(x_{0}\right)=1. For t > 0t>0 we have
p(y+tx_(0))=tp(x_(0)+t^(-1)y) >= t bar(d)= bar(d)*psi_(0)(y+tx_(0))p\left(y+t x_{0}\right)=t p\left(x_{0}+t^{-1} y\right) \geq t \bar{d}=\bar{d} \cdot \psi_{0}\left(y+t x_{0}\right)
Since this inequality obviously holds for t <= 0t \leq 0, it follows ||psi_(0)∣≤1// bar(d)\| \psi_{0} \mid \leq 1 / \bar{d}. Let (y_(n))\left(y_{n}\right) be a sequence in YY such that p(x_(0)-y_(n))rarr bar(d)p\left(x_{0}-y_{n}\right) \rightarrow \bar{d} for n rarr oon \rightarrow \infty and p(x_(0)-y_(n)) > 0p\left(x_{0}-y_{n}\right)>0 for all n inNn \in \mathbb{N}. Then
implying ||psi_(0)∣≥1// bar(d)\| \psi_{0} \mid \geq 1 / \bar{d}. Therefore ||psi_(0)∣=1// bar(d)\| \psi_{0} \mid=1 / \bar{d}.
If bar(psi):X rarrR\bar{\psi}: X \rightarrow \mathbb{R} is a linear functional such that
( bar(psi))|_(Z)=psi_(0)quad" and "quad||( bar(psi))|=||psi_(0)|:}\left.\bar{\psi}\right|_{Z}=\psi_{0} \quad \text { and } \quad\left\|\bar{\psi}\left|=\| \psi_{0}\right|\right.
then the linear functional varphi= bar(d)* bar(psi)\varphi=\bar{d} \cdot \bar{\psi} fulfills all the requirements of the proposition.
Suppose now d=d_(p)(x_(0),Y) > 0d=d_{p}\left(x_{0}, Y\right)>0, and let Z:=Y+Rx_(0)Z:=Y+\mathbb{R} x_{0}. Define psi_(0):Z rarrR\psi_{0}: Z \rightarrow \mathbb{R} by
psi_(0)(y+tx_(0))=-t Longleftrightarrowpsi_(0)(y-tx_(0))=t quad" for "y in Y" and "t inR.\psi_{0}\left(y+t x_{0}\right)=-t \Longleftrightarrow \psi_{0}\left(y-t x_{0}\right)=t \quad \text { for } y \in Y \text { and } t \in \mathbb{R} .
Then psi_(0)\psi_{0} is linear and, for t > 0t>0, we have
for t > 0t>0. Since this inequality is obviously true if psi_(0)(y-tx_(0))=t <= 0\psi_{0}\left(y-t x_{0}\right)=t \leq 0, it follows that psi_(0)\psi_{0} is bounded and ||psi_(0)∣≤1//d\| \psi_{0} \mid \leq 1 / d. Choosing a sequence (y_(n)^('))\left(y_{n}^{\prime}\right) in YY such that p(y_(n)^(')-x_(0))rarr dp\left(y_{n}^{\prime}-x_{0}\right) \rightarrow d and p(y_(n)^(')-x_(0)) > 0p\left(y_{n}^{\prime}-x_{0}\right)>0 for all nn, and reasoning like above one obtains the inequality ||psi_(0)∣≥1//d\| \psi_{0} \mid \geq 1 / d, so that ||psi_(0)∣=1//d\| \psi_{0} \mid=1 / d. Extending psi_(0)\psi_{0} to a functional psi_(1)inX_(p)^(b)\psi_{1} \in X_{p}^{b} of the same norm, and letting psi=d*psi_(1)\psi=d \cdot \psi_{1}, one obtains the wanted functional psi\psi.
4. APPLICATIONS TO BEST APPROXIMATION
Let ( X,pX, p ) be a space with asymmetric seminorm and YY a nonempty subset of XX. By the asymmetry of the seminorm pp we have to distinct two "distances" from a point x in Xx \in X to the subset YY, as given by (3.1) and (3.2).
Since d_(p)(Y,x)=d_( bar(p))(x,Y)d_{p}(Y, x)=d_{\bar{p}}(x, Y), we shall use the notation d_( bar(p))(x,Y)d_{\bar{p}}(x, Y) for the distance (3.2).
An element y_(0)in Yy_{0} \in Y such that p(x-y_(0))= bar(p)(y_(0)-x)=d_( bar(p))(x,Y)p\left(x-y_{0}\right)=\bar{p}\left(y_{0}-x\right)=d_{\bar{p}}(x, Y) is called a bar(p)\bar{p}-nearest point to xx in YY, and an element y_(1)in Yy_{1} \in Y such that p(y_(1)-x)=d_(p)(x,Y)p\left(y_{1}-x\right)=d_{p}(x, Y) will be called a pp-nearest point to xx in YY.
By Proposition 3.4, we obtain the following characterization of bar(p)\bar{p}-nearest points.
Proposition 4.1. Let ( X,pX, p ) be a space with asymmetric seminorm, YY a subspace of XX and x_(0)x_{0} a point in XX such that bar(d)=d_( bar(p))(x_(0),Y) > 0\bar{d}=d_{\bar{p}}\left(x_{0}, Y\right)>0.
An element y_(0)in Yy_{0} \in Y is a bar(p)\bar{p}-nearest point to x_(0)x_{0} in YY if and only if there exists a bounded linear functional varphi:X rarrR\varphi: X \rightarrow \mathbb{R} such that
(i) varphi|_(Y)=0\left.\varphi\right|_{Y}=0,
(ii) ||varphi∣=1\| \varphi \mid=1,
(iii) varphi(x_(0))=p(x_(0)-y_(0))\varphi\left(x_{0}\right)=p\left(x_{0}-y_{0}\right).
Proof. Suppose that y_(0)in Yy_{0} \in Y is such that p(x_(0)-y_(0))=d=d_( bar(p))(x_(0),Y) > 0p\left(x_{0}-y_{0}\right)=d=d_{\bar{p}}\left(x_{0}, Y\right)>0. By Proposition 3.4, there exists varphi inX_(p)^(b),||varphi∣=1\varphi \in X_{p}^{b}, \| \varphi \mid=1, such that varphi|_(Y)=0\left.\varphi\right|_{Y}=0 and varphi(x_(0))=d=p(x_(0)-y_(0))\varphi\left(x_{0}\right)=d=p\left(x_{0}-y_{0}\right).
Conversely, if for y_(0)in Yy_{0} \in Y there exists varphi inX_(p)^(b)\varphi \in X_{p}^{b} satisfying the conditions (i)-(iii), then for every y in Yy \in Y,
implying p(x_(0)-y_(0))=d_( bar(p))(x_(0),Y)p\left(x_{0}-y_{0}\right)=d_{\bar{p}}\left(x_{0}, Y\right).
Another consequence of Proposition 3.4 is the following duality formula for best approximation:
Proposition 4.2. Let YY be a subspace of a space with asymmetric seminorm ( X,pX, p ). If d_(p)(Y,x_(0)) > 0d_{p}\left(Y, x_{0}\right)>0 then the following duality formula holds:
where Y^(_|_)={varphi inX_(p)^(b): varphi|_(Y)=0}Y^{\perp}=\left\{\varphi \in X_{p}^{b}:\left.\varphi\right|_{Y}=0\right\}.
Proof. For any psi inY^(_|_),||psi∣≤1\psi \in Y^{\perp}, \| \psi \mid \leq 1 and any y in Yy \in Y, we have:
implying s u p{psi(x_(0)):psi inY^(_|_),||psi∣≤1} <= d_(p)(Y,x_(0))\sup \left\{\psi\left(x_{0}\right): \psi \in Y^{\perp}, \| \psi \mid \leq 1\right\} \leq d_{p}\left(Y, x_{0}\right). If we choose psi\psi to be the functional varphi\varphi given by Proposition 3.4, then we obtain the reverse inequality: d_(p)(Y,x_(0))=varphi(x_(0)) <= s u p{psi(x_(0)):psi inY^(_|_),||psi|| <= 1}d_{p}\left(Y, x_{0}\right)=\varphi\left(x_{0}\right) \leq \sup \left\{\psi\left(x_{0}\right): \psi \in Y^{\perp},\|\psi\| \leq 1\right\}.
The distance to a hyperplane
The well known formula for the distance to a closed hyperplane in a normed space has an analog in spaces with asymmetric seminorm. Remark that in this case we have to work with both of the distances d_(p)d_{p} and d_( bar(p))d_{\bar{p}} given by (3.1) and (3.2).
Proposition 4.3. Let ( X,pX, p ) be a space with asymmetric seminorm, varphi inX_(p)^(b),varphi!=0,c inR\varphi \in X_{p}^{b}, \varphi \neq 0, c \in \mathbb{R},
H={x in X:varphi(x)=c}H=\{x \in X: \varphi(x)=c\}
the hyperplane corresponding to varphi\varphi and cc, and
H^( < )={x in X:varphi(x) < c}quad" and "quadH^( > )={x in X:varphi(x) > c}H^{<}=\{x \in X: \varphi(x)<c\} \quad \text { and } \quad H^{>}=\{x \in X: \varphi(x)>c\}
for every x_(0)inH^( < )x_{0} \in H^{<}.
2) If there exists an element z_(0)in Xz_{0} \in X with p(z_(0))=1p\left(z_{0}\right)=1 such that varphi(z_(0))=||varphi∣\varphi\left(z_{0}\right)= \| \varphi \mid, then every element in H^( > )H^{>}has a bar(p)\bar{p}-nearest point in HH and every element in H <H< has a p-nearest point in HH.
If there is an element x_(0)inH^( > )x_{0} \in H^{>}having a bar(p)\bar{p}-nearest point in HH, or there is an element x_(0)^(')inH^( < )x_{0}^{\prime} \in H^{<}having a pp-nearest point in HH, then there exists an element z_(0)in X,p(z_(0))=1z_{0} \in X, p\left(z_{0}\right)=1, such that varphi(z_(0))=||varphi∣\varphi\left(z_{0}\right)=\| \varphi \mid. It follows that, in this case, every element in H^( > )H^{>}has a bar(p)\bar{p}-nearest point in HH, and every element in H^( < )H^{<}has a pp-nearest point in HH.
Proof. Let x_(0)inH^( > )x_{0} \in H^{>}. Then, for every h in H,varphi(h)=ch \in H, \varphi(h)=c, so that
By the assertion 2) of Proposition 2.2, there exists a sequence ( z_(n)z_{n} ) in XX with p(z_(n))=1p\left(z_{n}\right)=1, such that varphi(z_(n))rarr||varphi∣\varphi\left(z_{n}\right) \rightarrow \| \varphi \mid and varphi(z_(n)) > 0\varphi\left(z_{n}\right)>0 for all n inNn \in \mathbb{N}. Then
It follows d_( bar(p))(x_(0),H) >= (varphi(x_(0))-c)//||varphi∣d_{\bar{p}}\left(x_{0}, H\right) \geq\left(\varphi\left(x_{0}\right)-c\right) / \| \varphi \mid, so that formula (4.1) holds.
To prove (4.2), observe that for h in Hh \in H,
so that d_(p)(x_(0)^('),H) >= (c-varphi(x_(0)^(')))//||varphi∣d_{p}\left(x_{0}^{\prime}, H\right) \geq\left(c-\varphi\left(x_{0}^{\prime}\right)\right) / \| \varphi \mid, and formula (4.2) holds too.
2) Let z_(0)in Xz_{0} \in X be such that p(z_(0))=1p\left(z_{0}\right)=1 and varphi(z_(0))=||varphi∣\varphi\left(z_{0}\right)=\| \varphi \mid. Then, for x_(0)inH^( > )x_{0} \in H^{>} and x_(0)^(')inH^( < )x_{0}^{\prime} \in H^{<}, the elements
h_(0):=x_(0)-(varphi(x_(0))-c)/(varphi(z_(0)))z_(0)quad" and "quadh_(0)^('):=(c-varphi(x_(0)^(')))/(varphi(z_(0)))z_(0)+x_(0)^(')h_{0}:=x_{0}-\frac{\varphi\left(x_{0}\right)-c}{\varphi\left(z_{0}\right)} z_{0} \quad \text { and } \quad h_{0}^{\prime}:=\frac{c-\varphi\left(x_{0}^{\prime}\right)}{\varphi\left(z_{0}\right)} z_{0}+x_{0}^{\prime}
belong to HH,
p(x_(0)-h_(0))=(varphi(x_(0))-c)/(||varphi||)=d_( bar(p))(x_(0),H)quad" and "quad p(h_(n)^(')-x_(0)^('))=(c-varphi(x_(0)^(')))/(||varphi||)=d_(p)(x_(0)^('),H).p\left(x_{0}-h_{0}\right)=\frac{\varphi\left(x_{0}\right)-c}{\|\varphi\|}=d_{\bar{p}}\left(x_{0}, H\right) \quad \text { and } \quad p\left(h_{n}^{\prime}-x_{0}^{\prime}\right)=\frac{c-\varphi\left(x_{0}^{\prime}\right)}{\|\varphi\|}=d_{p}\left(x_{0}^{\prime}, H\right) .
If an element x_(0)inH^( > )x_{0} \in H^{>}has a bar(p)\bar{p}-nearest point h_(0)in Hh_{0} \in H, then
It follows that z_(0)=(x_(0)-h_(0))//p(x_(0)-h_(0))z_{0}=\left(x_{0}-h_{0}\right) / p\left(x_{0}-h_{0}\right) satisfies the conditions p(z_(0))=1p\left(z_{0}\right)=1 and varphi(z_(0))=||varphi∣\varphi\left(z_{0}\right)=\| \varphi \mid.
If an element x_(0)^(')inH^( < )x_{0}^{\prime} \in H^{<}has a pp-nearest point h_(0)^(')h_{0}^{\prime} in HH, then z_(0)^(')=(h_(0)^(')-:}{:x_(0)^('))//p(h_(0)^(')-x_(0)^('))z_{0}^{\prime}=\left(h_{0}^{\prime}-\right. \left.x_{0}^{\prime}\right) / p\left(h_{0}^{\prime}-x_{0}^{\prime}\right) satisfies p(z_(0)^('))=1p\left(z_{0}^{\prime}\right)=1 and varphi(z_(0)^('))=||varphi∣\varphi\left(z_{0}^{\prime}\right)=\| \varphi \mid.
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Received by the editors: March 28, 2004.
*"Babeş-Bolyai" University, Faculty of Mathematics and Computer Science, 400084 ClujNapoca, Romania, e-mail: scobzas@math.ubbcluj.ro. ^(†){ }^{\dagger} "T. Popoviciu" Institute of Numerical Analysis, O.P 1, C.P. 68, Cluj-Napoca, Romania, e-mail: cmustata@ictp.acad.ro.
