A CLASS OF GENERALIZED BEST APPROXIMATION PROBLEMS IN LOCALLY CONVEX LINEAR TOPOLOGICAL SPACES

In this paper a class of generalized best approximation problems is formulated in locally 
convex linear topological spaces and is solved, using standard results of locally convex linear topological 
spaces.


INTRODUCTION
The best approximation problem in normed linear spaces was considered by several authors including Burbu [1], Singer  [2].Also in locally convex spaces some results were obtained by Singer [3].The principle objective of this paper is to generalize the idea of best approximation problem in locally convex linear topological space setting and to find the best control.
2. STATEMENT OF TIIE PROBLEM Let E be a convex subset of a locally convex linear topological space X and X* be the conjugate space of all continuous linear functionals defined on X.
Consider the problem x sup [(m-x,f) +IE(m) fX" where x is the given element of X and IE is the indicator function such that To solve this problem let us consider the following definition and theorems DEFINITION 1.An element E E is called a best approximation to z X from E if sup I( z, f)l -< sup {1( m, f)l, for all m }.
TIOREM 1.An element 6 E is the best approximation to x 6 X from elements of the convex set E if and only ifthere exists x 6 X* such that (i) sup 1(5,)1 sup I(/-x, zeXcX'" rex" where X** is the conjugate space of X*. PROOF.Now we have l(, f)l (:r, m ), V m E. ( In particular, for m l, we obtain sup i(e-,/) which implies condition (i).Consequently from inequality (2.3) condition (ii) follows, as claimed.Conversely, it is clear that condition (i) and (ii) imply that is a best approximation, because we have sup I(/x, f)l (z, m z) and so, we must have (2.1).
COROLLARY 1.If 6 E is a best approximation of x 6 X by elements of the convex set E, then  PROOF.This follows clearly if we use the relationship between the solutions to the problem (P) and the existence of the saddle points ( [1]).To this end it suffices to remark that the point x0, the existence of which is ensured by Theorem 1, is just the solution of the dual problem.This completes the proof.
REMARK 1.Let d sup (l(z rn, f)l; m E E} be distance between the point z and the convex fX" set E.
Then we obtain a weak minimax relation by replacing "rain" by "inf' because in such a ease only the dual problem has solutions.
Next we shall notice several special cases in which conditions (i) and (ii) of Theorem have a simplified form.Namely, if E is a convex cone with vertex in the origin, then condition (ii) is equivalent to the following pair of conditions (ii') (z,m) < O, Vm .E,i .e. z E (ii") (z, z) sup Iz e; fl rex" where E is the polar set of K ( [4], p. 136).
Here is the argument.From condition (ii) replacing z by z, we obtain (z,z nrn) > sup I(z I, f)l2, Vrn E, Vn N rex.
because E is a cone.Therefore we cannot have (z, rn) > 0 for some element m E E, that is (ii') holds.
Moreover, from properties (ii) and (ii') it follows that sup I( t, f)! _< (z;, z t) leX" _< sup I(z,z)l sup zeXcX'" feX" sup I(z , I)1 , /'eX" hence (z, z l) sup I(/-z, f)l2.and (from (ii) if rn 0) (z,z) _> sup I(z-t,/)l reX" which implies property (ii').The reciprocal is obvious.When E is a linear space, condition (ii') is equivalent to (z, m)= 0, Vm E because in this case E E. It should be mentioned that the best approximation belongs to E f"l {z .X" sup <_ d} and it exists if and only if there exist separating hyperplanes which meet E.Moreover, the set of all best approximations is convex and coincides with the intersection of the set with any separating hyperplanes.When this intersection is non-empty the separating hyperplane is a supporting hyperplane and is given by the equation (z,m-z)=sup I(z-l, 1)l2, mEX.

IX"
Now, let us study the existence of the best approximation.Let We easily see that where if and only if it is lower-semicontinuous with respect to the weak topology on z.
PROOF.We have already seen in Proposition 2.5 ([1], p. 12) that a convex subset of a locally convex linear topological space is (strongly) closed if and only if it is closed in the corresponding weak topology on X.In particular we may infer that epi f is (strongly) closed if it is weakly closed.This establishes the theorem.THEOREM 3. If the convex set E is such that there exists an > 0 for which the set E N S(z; d + ) is weakly compact, then z has a best approximation in E.
PROOF.According to relation (2.6) it is sufficient to recall that a lower-semicontinuous funon on a compact set attains its infimum In our case, the function is obviously weakly lower-semicontinuous (see Theorem 2) on the weakly compact set E C S(m, d + ).COROLLARY 2. In a semireflexive loc, ally convex linear topological space every element possesses at least one best approximation with respect to every closed convex set.
PROOF.The set E C S(m; d + 1) is convex closed and bounded and hence it is wealdy compact by virtue ofthe Alaoglu Theorem ([5], p. 15).COROLLARY 3. In a locally convex linear topological space every element possesses at least one best approximation with respect to every closed, convex and finite dimensional set.PROOF.In a finite dimensional space the bounded closed convex sets are compact and hence weakly compact.DEFINITION 2. Let X, Y be locally linear topological spaces of the same nature.A linear operator T" X -, Y is continuous if and only if it is bounded.In other words there exists K > 0 such that sup I(Tt, m')l < Ksup I(t,r)l, Vt X.
rn" 6Y" l'X" The set L(X, Y) of all linear continuous operators defined on X with values in Y becomes a locally convex linear topological space by sup .(T,f),sup{ sup ,(Tl, m')l;sup ,(l,l'),< 1} If Y R, we find that X* L(X, R), called the dual of X, is locally convex linear topological space defined by sup [(l*, l)[ sup{ l l " (l); sup }.IfX is real locally convex linear topological space, then sup I(z',Z)l--supZ'(z);sup I(Z,Z')l _< IXcX'" t, I'X" THEOREM 4. Let f0 be a continuous linear functional on a linear subspace A of a locally convex linear topological space X.Then, there exists a continuous linear functional f on the whole of X, i.e., f X*, such that (i) PROOF.Since f0 is continuous on A, by relation (2.8) we have fo(m) < sup I(f0,Z)l sup I(m,t')l, Vrn A.
IXcX" I'X" By the Hahn-Banach Theorem ([ ], Theorem 1.10, p. 17) for f0 and for the convex function p(x) sup I(f0,/)l sup IXcX'" l'X" A specialization of this theorem yields a whole class of existence results.In this context we shall present a general and classical theorem concerning the existence of continuous linear funetionals with important consequence in the duality theory oflocally convex linear topological spaces.TItEOREM 5. Let m be a non negative number and let h B --, R be a given real ftmetion, where B is a non-empty set of the locally convex linear topological space X.Then, h has a continuous linear extension f on all ofX such that sup ](f,/)l < rn if and only ifthe following condition holds:  PROOF.From relation (2.7a) and (2.7b) it is clear that condition (2.9) is necessary.To prove the sufficiency we consider A span B and we define f0 on A by fo(m)=EA, h(at), if m= A,aaA, oB.t=l :=1 First, using condition (2.9) we observe that f0 is well defined on A. Moreover from condition (2.9) the continuity of f0 on A follows and sup I(fo, l)l < m.Thus any extension given under Theorem 4 has all lXcX.. the required properties.THEOREM 6.For any linear subspace A of a locally convex linear topological space X and X there exists f X* with the following properties i) //A 0 ii) f(l) inf sup I(/-m, l*)12 d(l, A) meA (.I'eX" iii) sup I(f,Z)I-inf sup I(z m,l')l d(l,A) IeXcX'" mA I,l*X" PROOF.We take B=AU{I} and h B---,R defined by h(m) O, rn E A, and h(l) d2(l,A).
We observe that for any A # 0 we have which is just inequality (2.9)The desired result then follows by applying Theorem 5 Indeed, we have properties (i) and (ii) and sup [(f,l)l < d(l,A).Since m d(l,A).On the other hand, if we leXcX'" consider a sequence {m} C A such that sup[(/+ m,,/*)[ d(l,A) we obtain PeA sup I(f,l)!>f Ii,,,Z')l sup I(/+rn, /') which implies sup I(f, l)l >_ d(1, A).Hence property (iii) also holds.
IXcX'" COROLLARY 4. In a locally convex linear topological space X for every E X there exists a continuous linear functional f E X" such that Moreover, if # 0, there exists g e X" such that (i') z() sup I'EX" (ii') sup lEXcX'" PROOF.By Theorem 6, d(l,A) sup I(,l*)l where A (0). Then the corollary completes the l'X" proof.
DEFINITION 3. The space X is strictly convex if every point of the polar set { X sup l(l, f)l l is an extreme point.
THEOREM 7. A locally convex linear topological space X is strictly convex if and only if the following equivalent properties hold: (i) if sup I(z + y, f)l sup I(z, f)l + sup I(Y, f)l and z # 0 there is > 0 such that y fx" fx" fx. (ii x y, then sup [((x + y), f) < 1; rex" rex" rex" (iv) the function z sup ](z, f)I , z X, is strictly convex. feX" PROOF.Let X be strictly convex and let z,y X\{O} be such that sup I(z / y,/)t sup I(x, f)l + sup I(x, f)l- rex" reX" rex" By Corollary 4 for every x E X, there exists a continuous linear functional z" E X* such that (x + y,x*) sup ](x + y,z')l EX Since (z,z*) < sup I(z, f)l, (Y, z) < sup I(Y, f)l rex" rex" we must have (x,x') sup I(x, f)l and (y, x') sup I(Y, f)l, i.e., fx. rex" sup I(x, f)l'z* sup I(Y, f)l \rex" rex. =1.
Because X is strictly convex it follows that sup I(, f)l rex" hence property (i) holds with sup I(/, f)l' fX sup I(, f)l rex" sup I(z, f)l" leX" sup I(z',)l . zXX" To prove that (i) (ii) we assume by contradiction that there exists z such that sup I(z, f)i sup I(u, f)l and sup I(,Xz + ( ,X)U, f)l 1, where ,X ]0, [.Therefore we have fX fX" fX sup I(Xz + (1 X)F, f)l sup I(,Xa:, f)l + sup I((1 X), f){.
is establish d the proofis compile.
EOM 8.If X is locly convex topoloc space wch is grimy conve then ch element z X possesses at most one best appromafion th resp to a convex s E C X.
PROOF.se by contradion that there est two distin best approtions/,/ Since e set ofbe approtions is convex, it follows that (l + l) is so a best appromation.I( 1=sup -(+l),] Iex" where X" is the eenjugate spa of X d ereby In ew f the cnve ( Theore 7) we have >sup (-l)+ (-l),I sup -(+ fX" wch is a contradion.Ts comples the proof. 2. Ts prope is caefisfic of the stfily convex spaces: if, in a lowly convex line topoloc space X, e element possses at most a best approbation th respt to eve convex t (it is ou for e seents), then X is fictly convex.
Hence the orion h at the be approtion th resp to e clo convex set [z, ] ev elemt of ts set, d ts clely contradicts the uqueness.
From Corofi 2 d Theorem 3 it follows that in a serefleve strictly loy conv topoloc space, for ev closed convex t g we c define the nction PX X by Pgz l, where is the best approbation of X by elemts of E. Ts nction is ed the projtion non ofthe space X into E. We note that Pgz g for eve z X. DEON 4. L us consider e foHog gener fy offion probls where X, Y e locy convex line topoloc spaces d F X x Y R. Let us denote by d H { (, a) Y R; there efigs X such that F(, ) THEOREM 9. Let X, Y are locally convex linear topological spaces and F" X x Y --, ]-oo, +oo] be a positively homogeneous and lower-semicontinuous function satisfying the following coercivity condition F@, O) > 0 for any a: e XI{O}.
Then, if epi F is locally compact, every problem P has an optimal solution whenever its value is finite.
PROOF.It is easy to observe that H Projy(epi F).
By hypothesis epi h is a closed cone and so (epi F)o epi F. Therefore, it is sufficient to use Corollary  (2.10).TBOREM 10.If E is a closed locally compact convex set of a strictly convex locally convex linear topological space X, then the projection function is continuous on X.
for all n > no ().Denote where X* is the dual space of X.
Thus i"] S(a:;d + )f]E -) and any subsequence of PEz,, has a cluster point which satisfies >0 sup I(a:l, f)l-d.Because X is strictly convex, this poim is unique and so PEa:,, 1 PEa: as fX" claimed.
DEFINITION 5. A set E is called proximinal if every elemem of X has a best approximation in E. That is, the set//7 is proximal ifthe problem rain{ sup I(z-m,f)l} meE feX" has a solution for every x 6 X.
THEOREM 11.A nonempty set E of a locally convex linear topological space X is proximinal if and only if epi sup (., f)[ + E x {0} is closed in X x R.
/x-Moreover, if E is a convex set which contains the origin we have x" 6 X*.