Non-uniqueness of certain Hahn - Banach extensions

Suppose that К denotes a. non-Archimedean, nontrivially valued field, i. e., a. field with an absolute value | • | that satisfies the strong triangle inequality: for all a,b E K, a + b\ <  max(|a|, 6|). Let X be a. normed space over К in which the norm also satisfies the strong triangle inequality and X' denotes its continuous dual. We refer to X as a. non-Archimedean normed space. For a. subspace M of X, M = {/ E X' : /(ж) = 0, ж E Af}, the orthogonal of M; the orthogonal of M С X¹ is given by M = {ж E X : /(ж) = 0, f E Mf.

DEFINITION 1. If each nested sequence Bi D B^ D • • • D B_n D • • • of balls in К has nonempty intersection, К is called spherically complete.

Note the absence of any requirement that the diameters shrink to 0 in this stronger version of completeness. If / is a. continuous linear functional defined on M, an extension of f to F E X' of the same norm is called a. Hahn-Banach extension. In the context of non-Archimedean normed spaces the Hahn Banach theorem can fail — there exist spaces X and continuous linear functionals defined on a. subspace M of X that have no Hahn-Banach extension. If К is spherically complete, however, then any continuous linear functional / defined on any subspace M of X has a. Hahn-Banach extension (see [4; p. 78] or [5; p. 102]).

When are Hahn-Banach extensions unique? There are two principal, classical results, one (the Taylor-Foguel theorem) for any subspace M of X and another (Phelps’s theorem) that deals with one subspace at a. time:

Theorem 1 (Taylor-Foguel). If E is a. normed space over JR or C then the following conditions are equivalent:

• (a)    For any subspace M of E and any f E M', f has a. unique Hahn-Banach extension;

• (b)    E' is strictly convex (equivalently «strictly normed») in the sense that for any two unit vectors / and g and any t E (0,1), \\tf + (1 — t)g\\ < 1.

Theorem 2 (Phelps). If M is a. subspace of the normed space E over R orC then the following conditions are equivalent:

• (a)    For any / E M', / has a. unique Hahn-Banach extension;

• (b)    M has a. unique best approximation in E' in the sense that given any / E E' there exists a. unique m E M such that Ц/ — 7?i|| = inf{||/ — д\\ : g E AL^±} = d(f,M"^L).

• 2.    Unique Hahn Banach Extensions

What can be said about uniqueness in the non-Archimedean case. After proving a certain lemma ([5; Lemma 4.4, p. 100] van Rooij observes (p. 103) that Hahn Banach extensions are unique if and only if the subspace M is dense in X or f = 0. We prove the non-uniqueness by different means in the next section.

We obtain a version (Th. 5) of Phelps’s theorem concerning uniqueness of Hahn-Banach extensions on a subspace M of X and uniqueness of best approximations from M. We then show that the conditions for uniqueness are never satisfied in non-Archimedean spaces.

A subspace M of X is proximinal if for all ж E X there exists a «best approximation» m E M to ж, i. e., 77i E M with ||ж — 7?г|| = d(x,M). We denote the set of all best approximations of ж from M by

Рм(^х) = V™ С M : ||ж — m|| = d(x,M)Y

If Рм (ж) is a singleton for every ж E X then M is called Chebycheu. It is easy to verify that Рм (ж) is closed.

As in the real or complex case, for spherically compete K, conventional orthogonal facts are valid as well as orthogonals are proximinal.

Theorem 3. Let К be spherically complete and let ст (X¹ ,X) denotes the weak-* topology on X¹. Then

• (a)    [3; p. 211] For M С X', M^LL = с\^_х,M. Thus, if M is ст (X', X)-closed, M = M^.

• (b)    [3; p. 215] For M С X, (X/M)' is algebraically isomorphic to M ■ and M' is algebraically isomorphic to X') M.

Theorem 4. Let К be spherically complete, M be a. subspace of X and / E X’. If F is any extension of /|_M of the same norm, then F — / is a. best approximation to / from M and d(f,M"^L) = ||/|_M||, f e., M is proximal.

< Let / E X'. Then for every m' E M\

Ik 1м II ^{= SU}P Ш(^ж) ^{: x} e U A M^ = sup {]/ (ж) - m' (ж) : ж E U A A/}

< sup { / (ж) — m (ж) | : ж E U} = Ц/ — m'|| .

Since m' E M  is arbitrary, it follows that ||/|m|| ^ ^(/,Tf±). To obtain the reverse inequality, consider an extension F E X' of /\м with ||F|| = ||/|m||- Since / — F E M,

\\f\M\\ = \\F\\ = \\j-(j-F))\\^d(f,M±).

In other words, / — F is a best approximation to / from M and ||/ \m || = d ^,М^У >

Using a technique of Herrero’s [1], we now obtain a version of Phelps’s theorem that a subspace M of X has unique Hahn-Banach extensions if and only if M is Chebysev.

Theorem 5. For M С X over a spherically complete field К, the following assertions are equivalent:

• (a)    each / E M' has a unique Hahn-Banach extension;

• (b)    M^± is Chebychev.

• <    (a) =^ (b): Let / E X¹. By Theorem 4, M is proximinal, so it only remains to prove uniqueness of best approximations. If g, h E Pm¹ (/) , then / — g and / — h are extensions of /\_M i since g,h E P_Mx (/),

y-g^WJ-h^d^M^.

Since extensions of / \m of the same norm are unique, / — g = / — h which implies g = h.

• ( b) =^ (a): Suppose / E M' has extensions g,h E X' of the same norm as /. Then h is an extension of g \m to h of the same norm. Therefore, by Theorem 4, g — h is a. best approximation to g from M. Since ||/i|| = ||g|| = ||/|| and

\\g\\ = \\g - 0\\ = \\h\\ = \\g Чэ - h^

it follows that 0 E Pm (g) as well. By the uniqueness of best approximation, g — h = 0. > Since a weak-* closed subspace M of X' is the orthogonal of M\ it follows that:

Corollary 1. A weak-* closed subspace M of X' is Chebychev if and only if each bounded linear map / : M —> К has a unique extension F E X' of the same norm.

The following result establishes that non-Archimedean spaces are never Chebychev.

Theorem 6 (cf. [2]). Suppose M С X is a. closed subspace and x ^ M. If m E Рм (ж) and m' E M is such that ||?7i' — ?7i|| < ||ж — 77i|| , then m' E Рм (ж) .

• <    Since x ^ M and m' E M, it follows that ||ж — m'|| > 0. By the strong triangle inequality, ||ж — 77i'|| = ||ж — m||. [>

It follows from Corolary 1 and Theorem 6 that Hahn Banach extensions are never unique.