PHELPS TYPE DUALITY RESULTS IN BEST APPROXIMATION

. The aim of the present paper is to show that many Phelps type duality result, relating the extension properties of various classes of functions (continuous, linear continuous, bounded bilinear, H¨older-Lipschitz) with the approximation properties of some annihilating spaces, can be derived in a unitary and simple way from a formula for the distance to the kernel of a linear operator, extending the well-known distance formula to hyperplanes in normed spaces. The case of spaces c 0 and l ∞ is treated in details.

(1) be the annihilator of Y in X * .In the seminal paper [38] R. R. Phelps initiated the study of the relations between the extension properties of the space Y and the approximation properties of its annihilator Y ⊥ .Namely, Y ⊥ is Chebyshevian if and only if every functional y * ∈ Y * has a unique norm-preserving extension x * ∈ X * .It is known that, by Hahn-Banach theorem, every y * ∈ Y * has at least one norm-preserving extension.Since then, there have been found a lot of situations in which similar duality results hold, corresponding to variuos extension results -Helly extension theorem for linear functionals, Tietze's extension theorem for continuous functions, McShane's extension theorem for Lipschitz functions, Nachbin's extension theorem for continuous linear operators etc.The aim of the present paper is to show that all these results follow immediately from a formula for the distance to the kernel of a continuous linear operator, inspired by the well-known distance formula to hyperplanes in normed spaces.
For a continuous linear operator A : X 1 → X 2 , between two normed spaces be its kernel.Also, for x ∈ X 1 , put (3) E(x) = y ∈ X 1 : Ay = Ax and y = Ax A .
Proof. 1.For every z ∈ Z, we have and, letting n → ∞, one obtains which, combined with the point 1 of the theorem, yields (4).Conversely, if the equality (4) holds and (z n ) is a sequence in Z such that x − z n → d(x, Z), then the sequence (z n ) verifies (5).

Since
x Examples

The distance from a point to a hyperplane
Let X be a normed space, x * ∈ X * , x * = 0, and Z = ker x * .Take Observe now that condition (7) holds if and only if showing that x * attains its norm on the element u 0 = (x 0 − z 0 )/ x 0 − z 0 of B.
In fact we have shown that if x * supports the unit ball B of X then (7) holds for every x ∈ X, and if (7) holds for a single element x 0 ∈ X \ Z then x * supports the unit ball B and, therefore, (7) holds for every x ∈ X.It follows that the subspace Z = ker x * is proximinal if x * supports the unit ball of X, and antiproximinal if not.
If h 0 ∈ X and H = h 0 + Z = {x ∈ X : x * (x) = a} where a = x * (h 0 ), is a closed hyperplane paralel to Z then, by (8), x * a well-known formula.

Restriction operators
Let E be a normed space and S, T nonvoid sets with S ⊂ T. Consider two normed spaces X 1 = X 1 (T, E) and X 2 = X 2 (S, E) of mappings from T (respectively S) to E, the vector operations being defined pointwise.Suppose that there are verified the following conditions (9) x| S ∈ X 2 and x| S ≤ x for every x ∈ X 1 .For y ∈ X 2 denote by ( 10) the (possibly empty) set of norm-preserving extensions of y in X 1 .One says that the space X 2 has the extension property with respect to X 1 if E(y) = ∅, for every y ∈ X 2 .Let A : X 1 → X 2 be the restriction operator defined by ( 11) By (9), A is well defined, linear, continuous, and From Theorem 1 one obtains: Consequently, if X 2 has the extension property with respect to X 1 then A = 1, the space S ⊥ is proximinal in X 1 and the formulae ( 14), ( 15) hold.

Hahn-Banach extensions
Let X be a normed space and Y a closed subspace of X.Put X 1 = X * and X 2 = Y * .By Hahn-Banach theorem every y * ∈ Y * has a norm preserving extension in X * , i.e. the space Y * has the extension property with respect to X * .By Proposition 1, it follows that Y ⊥ is proximinal in X * , ( 16) From the second formula in ( 16) follows Phelps' result [38] that Y ⊥ is Chebyshevian in X * if and only if every y * ∈ Y * has a unique norm-preserving extension in X * , as well as the result of Xu Ji Hong [20], asserting that P Y ⊥ (x * ) has affine dimension at most k − 1, for every x * ∈ X * , if and only if every y * ∈ Y * has at most k linearly independent norm-preserving extensions in X * .For other results concerning the unicity in Hahn-Banach extension theorem see E. Oja's papers [34,36] and the monograph [35].
If E is a Banach space with the binary intersection property then, by a result of L. Nachbin [33], the space L(Y, E) has the extension property with respect to L(X, E).Here X, Y are normed spaces with Y ⊂ X and L(X, E) (L(Y, E)) denotes the space of all continuous linear operators from X, E) and the formulae ( 14) and ( 15) apply.
Using some extension results for bounded bilinear functionals and operators on 2-normed spaces one can prove similar duality results for spaces of bounded bilinear operators or functionals on 2-normed spaces (see [6,7]) Let (X, , ) be a 2-normed space in the sense of S. Gähler [16] and let E be a normed space.A bilinear operator A : . If E has the binary intersection property, then a similar extension result is valid for the spaces [3] or [6]).Denoting by E(f ) the set of all these extensions and by (see [7]).If E has the binary intersection property then the above results hold for the spaces of bounded bilinear operators [6]).

Helly extensions
Let X be a real normed space and J : X → X * * the canonical embedding operator of X in its bidual, defined by Let Y be a closed subspace of X and Y ⊥ its annihilator in X * .For x * * ∈ X * * define the set of Helly extensions of x * * by Helly extensions can not exist, i.e. it is possible that 1 , . . ., x * n are in X * and > 0 then there is x ∈ X such that x < x * * + and x * i (x) = x * * (x * i ), i = 1, . . ., n.This is Helly's theorem (see [13, p. 86]) justifying the denomination "Helly extension".Restricting to J(X) we have Observe that if x ∈ X is fixed and y ∈ Y is arbitrary then, denoting by B * the closed unit ball of X * , we have By a theorem of Hahn (see [13, Lemma II.3.12]),there exists y * 0 ∈ Y ⊥ such that y * 0 = 1 and Since Y is a closed subspace of X, it follows that for every x ∈ X \ Y there exists y * ∈ Y ⊥ such that y * (x) = 1 (see [13, Consequence II.3.13]),implying Also, by (17) and Proposition 1, and that Y is proximinal if and only if every element x ∈ X admits a Helly extension.For results of this kind see [37,11].

The spaces c 0 and l ∞
We illustrate the above considerations on the case of spaces c 0 and l ∞ .As usual, denote by c 0 (l ∞ ) the space of all converging to zero (respectively bounded) sequences of real numbers.Equipped with the sup-norms they are Banach spaces and c 0 ⊂ l ∞ .
1.The subspace c 0 is proximinal in l ∞ and the distance of an element x ∈ l ∞ to c 0 is given by the formula (18) d(x, c 0 ) = lim sup |x(n)|.

Every continuous linear functional y
The proof is immediate (see e.g.[4] for this result as well as for other distance formulae and proximinality results in Banach spaces of vectorvalued sequences).Let x * ∈ l * ∞ be such that (21) x * | c 0 = y * and x * = y * .
To prove the unicity of x * we shall follow the ideas in the proof of Helly's one step extension theorem (see the proof of Theorem II.3.20 in [13]).
Let x ∈ l ∞ \ c 0 .For z ∈ c 0 we have The inequalities ( 22) and ( 23) yield (24) sup Now, by (20), the inequalities ( 22) and (23) give Writing for all i ∈ N, it follows that the supremum for y ∈ c 0 in the left-hand side of ( 25) is ≤ 0. Let β = x > 0, and let y n (i) = x(i) + i (β + 1), for 1 ≤ i ≤ n, and y n (i) = 0, for i > n, n ∈ N. Then x − y n = β + 1 for n sufficiently large (such that at least one a i , 1 ≤ i ≤ n, be different from zero), so that the expression in the left-hand side of (25) becomes It follows that sup Reasoning similarly, one obtains inf Taking into account the inequalities (24) and the fact that x ∈ l ∞ was arbitrarily chosen, it follows for all x ∈ l ∞ , proving the unicity of the extension x * .Remark.The above proof of the unicity of the extension of linear functionals on c 0 is suggested in [41,Problem 12.20].
Using the representation of continuous linear functionals on l ∞ we can obtain more information on the behavior of c ⊥ 0 in l * ∞ .It is known that the dual space of l ∞ can be identified with the space ba(P(N)) of all finitely additive bounded measures on P(N) (see [13,Th.IV.8.16] or [2, C. 4.7.11]).Following [2] we shall call the elements of ba(P(N)) charges.The variation of a charge µ ∈ ba(P(N)) is defined by One shows that |µ| is in ba(P(N)) too, and is a norm on ba(P(N)) with respect to which ba(P(N)) is a Banach space.The space ca(P(N)) of countably additive finite measures on P(N)) is a closed subspace of ba((N)) and the correspondence is an isometric isomorphism between the Banach spaces ca(P(N)) and l 1 .
In our case and ( 29) for every A ⊂ N. It follows that µ p (A) = 0 for every finite subset A of N. In particular µ({i}) = 0, i ∈ N. so that, by (26), µ p (x) = 0, x ∈ c 0 , showing that pba(P(N)) ⊂ c ⊥ 0 .Conversely, if µ ∈ c ⊥ 0 then, by (26), implying µ({i}) = 0, i ∈ N, which by ( 28) yields µ c = 0 and µ = µ p ∈ pba(P(N)), i.e. c ⊥ 0 ⊂ pba (P(N)).Consequently (30) c ⊥ 0 = pba(P(N)).If µ = µ c + µ p then, by (28), For a thorough exposition of the present day situation in M-ideals theory we recommend the monograph [17] (see also [35]).In this language, the relations (31) and (32) tell us that c 0 is an M-ideal in l ∞ .From the general theory of M-ideals it follows that the space c 0 is proximinal in l ∞ and that every continuous linear functional on c 0 has a unique norm-preserving extension to l ∞ .Since µ p | c 0 = 0, for µ ∈ba(P(N)), the formulae (16) become This is another way to obtain the results in Proposition 2. Since c 0 is proximinal in l ∞ it follows that every x ∈ l ∞ admits a Helly extension with respect to c 0 .To obtain concrete representations in this situation we have to work with the bidual l * * ∞ = ba * (P(N)).But, as it is asserted in [2, p. 231], no satisfactory representation for the elements of ba * (P(N)) is known.

Tietze extensions
Let T be a locally compact Hausdorff topological space and S a nonvoid closed subset of T .Denote by C 0 (T ) (C 0 (S)) the space of all real-valued continuous functions on T (respectively S) vanishing at infinity.Equipped with the sup-norms C 0 (T ) and C 0 (S) are Banach algebras and is a closed ideal in C 0 (T ).By Tietze extension theorem (see [14]) every f ∈ C 0 (S) admits a norm-preserving extension F ∈ C 0 (T ).It follows that Proposition 1 can be applied to deduce that Z(S) is a proximinal subspace of C 0 (T ) and that the formulae Results of this kind have been obtained in [10,37].Using Dugundji's [12] vector-version of Tietze extension theorem one obtains the validity of the formulae (33) for C 0 (T, E), C 0 (S, E), with E a Banach space.

Spaces of Lipschitz and Hölder functions
Let (X, d) be a metric space.A function F : The quantity F is called the Lipschitz norm of the function F and the space of all real-valued Lipschitz functions on X is denoted by Lip(X).Since F = 0 for constant functions, (34) is in fact only a semi-norm on Lip(X).There are several ways to transform Lip(X) into a Banach space.One consists in fixing a point x 0 ∈ X and consider the space Lip 0 (X) of all functions in Lip(X) vanishing at x 0 .Other way consists in considering the space BLip(X) of all bounded functions in Lip(X), normed by ( 35) or by ( 36) where F is given by (34) and F ∞ is the sup-norm.
McShane [25] proved that every Lipschitz functions f , defined on a subset Y of X, admits a norm preserving extension to X (see also [8,26]).The case X = R n was considered by M. Kirszbraun [22].Based on this result one can show that the space Lip 0 (Y ) has the extension property in Lip 0 (X) so that, by Proposition 1, for every F ∈ Lip 0 (X), where Here Y is a subset of X containing the fixed point x 0 .Similar results hold for the spaces BLip(Y ) and BLip(X) with respect to the norms (35) or (36) (see [29]).In this case one can show that every f ∈ BLip(Y ) has an extension F ∈ BLip(X) preserving both the Lipschitz and uniform norms, implying that the space BLip(Y ) have the extension property with respect to BLip(X) for both of the norms (35) and (36).
For 0 < α ≤ 1, a function F : X → R is called Hölder of order α if Denote by Λ α (X) the space of all Hölder functions on X.To obtain Banach spaces of Hölder functions one can proceed like above, by considering the space Λ α 0 (X) of Hölder functions vanishing at a fixed point x 0 or the space BΛ α (X) of bounded Hölder functions on X. Duality results for these spaces have been obtained by C. Mustȃt ¸a [31,32].