THE MACKEY-ARENS AND HAHN-BANACH THEOREMS FOR SPACES OVER VALUED FIELDS

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Ann. Math. Blaise Pascal, Vol. 2, N° 1, 1995, pp.147-153 Astract. Characterizations of the spherical completeness of a non-archimedean complete non-trivially valued field in terms of classical theorems of Functional Analysis are obtained. 1991 Mathematics subject classification: : ,~651 D Spherical completeness Throughout this paper K = (h', ( )) will denote a non-archimedean complete valued field with a non-trivial valuation ( . I. It is well-known that the absolute value function I . | of the field of the real numbers IR or the complex numbers. C satisfies the following properties : (i) 0 H, |x| = 0 iff x = 0, (ii) |x + |x| + |y|, (iii) |xy| = |x||y|, x, y E IR or x, y ~C. If K is a field, then by a valuation on K we will mean a map | . | of K into 1R, satisfying the above properties; in this case (.K, ~ . . )) will be called a valued field. We will assume that K is complete with respect to the natural metric of K.
It turns out that if K is not isomorphic to IR or C, then its valuation satisfies the following strong triangle inequality, cf. e.g. [12], (ii') I x + max{|x|,|y|}, x, y E K.
A valued field K whose valuation satisfies (ii') will be called non-archimedean and its valuation non-archimedean.
Let us first recall the following well-known result of Cantor Theorem 0 Let (X, p) be a metric space. Then it is complete iff every shrinking s equence of closed balls whose radii tend to zero has non-empty intersection.
Consider the set IN of the natural numbers endowed with the following metric p defined by p(m, n) = 0 if m = n and 1 + max(~, ~) if m ~ n.
Then the metric p is non-archimedean, i.e. p(m, n) = 0 iff either m = n, or p(m, n) for all m, n, k E IN. It is easy to see that every shrinking sequence of balls in IN whose radii tend to zero has non-empty intersection; note that every ball whose radius is smaller than 1 contains exactly one point. On the other hand, the balls Bl+ i ( 1), Bl+ z (2), ..., form a decreasing sequence and their intersection is empty. This suggests the following, see Ingleton [3] : A non-archimedean metric space (X, p) will be said to be spherically complete if the intersection of every shrinking sequence of its balls is non-empty.
Clearly spherical completeness implies completeness; the converse fails : The space (N, p) is complete but not spherically complete. We refer to [11] and [12] for more infomation concerning this property.
Theorem 1 Let (X, p) be a non-archimedean metric space. Then (X, p) is spherically complete iff given an arbitrary family B of balls in X, no two of which are disjoint, then the intersection of the elements of B is non-empty. The aim of this note is to collect a few characterizations of the spherical completeness of K in terms of the Mackey-Arens, Hahn-Banach and weak Schauder basis theorems, respectively, see [5], [6], [7], [12].
The Mackey-Arens and Hahn-Banach theorems The terms " K-space" , "topology","seminorm or norm" will mean a Hausdorff locally convex space (lcs) over K, a locally convex topology (in the sense of Monna) and a nonarchimedean seminorm (norm), respectively A seminorm on a vector space E over K is non-archimedean if it satisfies condition (ii'). Clearly the topology T generated by a norm is locally convex. . Recall that a topological vector space (tvs) E = {E, T) over K is locally co~ve~ [10] if T has a basis of absolutely convex neighbourhoods of zero. A subset U of E is absolutely convex (in the sense of Monna [10]) if ax + ,Qy E U, whenever x, y E U, E, 1, |03B2| 1. For the basic notions and properties concerning tvs and lcs over K we refer to [10], [11], [13].
A locally convex (lc) topology, on (E, T) is called compatible with T, if T and ~y have the same continuous linear functionals; {E, T)* = (E, ~y)*. (E, T) is dual-separating if (E, T)* separates points of E. If G is a vector subspace of E, ?~G and T IG denote the topology T restricted to G and the quotient topology of the quotient space E/G, respectively. If a is a finer l.c. topology on E/G, we denote by y := T V a the weakest I.e. topology on E such that T q, = a, y~G = T~G, cf. e.g. [1]. The sets U n compose a basis of neighbourhoods of zero for q, where U, V run over bases of neighbourhoods of zero for T and a, respectively, q := EE/G is the quotient map. By sup{, a} we denote the weakest l.c. topology on E which is finer than T and a. By the Mackey topology p.( E, E*) associated with a lcs E = (E, T) we mean the finest locally convex topology on E compatible with T. In [14] Van Tiel showed that every lcs over spherically complete K admits the Mackey topology.
In (3j Ingleton obtained a non-archimedean variant of the Hahn-Banach theorem for normed spaces, where K is spherically complete. Theorem 2 If E = (E, ~~) as a normed space over K and K is spherically complete and D is a subspace of E, then for every continuous linear functional g E D* there exists a continuous linear extension f E E* of g such that = ~f~. This suggests the following : A lcs E will be said to have the Hahn-Banach Extension Property (HBEP) [9] if for every subspace D every g E D* can be extended to f E E*. It is known that every lcs over spherically complete K has the HBEP, cf. e.g. ~11~.
The following theorem characterizes the spherical completeness of K in terms of classical theorems of Functional Analysis; cf. also [5], [6] and [12], Theorem 4.I5. The proof of our Theorem 3 uses some ideas of [4] extended to the non-archimedean case. I°° (resp. co) denotes the space of the bounded sequences ( resp. the sequences of limit 0} with coefficients in Ii'.
Theorem 3 The following conditions on K are equivalent : (i) K is spherically complete.
(ii) There exists g E (l°°)* such that g(x) = ~n xn for every x E co.
(v) Every Ics over K (resp. K .normed space~ has the HBEP. (vi) The completion of a dual-separating lcs over K (resp. K -normed space) is dual. separating.
(vii) Every closed subspace of a dual-separating lcs over K (resp. K-normed space~ is weakly closed. (viii) For every lcs over K (resp. K -normed space ) every weakly convergent sequence is convergent.
(ix) Every weak Schauder basis in a lcs over K (resp. K-normed space) is a Schauder basis.
Proof By Theorem 4.15 of [12]  (iv) implies {i) : Assume that K is not spherically complete and consider the space l~ of K-valued bounded sequences endowed with the topology T generated by the norm = supn x = (xn) E l~. Let f be a non-zero linear function on I°° with f |c0 = o. Set E := I°° and F := co. Define a linear functional h on the quotient space E/F by h(q(x)) = f {x), where q : E -+ E/F is the quotient map. Let a be the quotient topology of E/F. Since (E/F, a)* = 0, see (iii) implies (i), F is dense in the weak topology o(E, E*) (recall that E* = F, [12], Theorem 4.17). Observe that on E/F there exists a K-normed topology /3 such that (E/F, a) and (E/F,,Q) are isomorphic and h is continuous in the topology sup{ a, ~3~. Indeed, choose a-o E E/F such that h(xo) = 2 and define a linear map T : : E/F --~ E/F by T(x) := xh(x)xQ, x E E/F. Then T~ = id. Define ,Q :; T{a) (the image topology). Then h is continuous in the topology sup{03B1,03B2}. Then there exists m E N such that Gm is dense in E. Hence we obtain a subspace G as required. Let a be a K-normed topology on E/G such that the spaces (E/G, a) and (E/F, T~F) are isomorphic. Then the topology, := T V a is compatible with T and strictly finer than T. Let Eo be the completion of the dual-separating K-normed space (E, ~). Choose x E EoBE. There exists a sequence (xn) in E and y E E such that (vii) implies (i) : Assume that K is not spherically complete. The space G constructed in the previous case is closed in (E, ~y) and dense in (E, o(E, E*)), where E* := (E, ~y)*.
(v) implies (i) : Assume that K is not spherically complete. Let (en) be the sequence of the unit vectors in E, where E is as above. Then en --~ 0 in o(E, E*), [13]. Clearly  (ix) implies ( i ) : Assume that K is not spherically complete. The sequence ( e n ) is a Schauder basis in (E, o(E, E*)) but it is not a Schauder basis in the original topology of E. The second part of this sentence follows from the fact that E is not of countable type, cf. e.g. [12]. On the other hand, by Theorem 4.17 of [12] (and its proof) the space E is reflexive and for every g E E* there exists (an) E F such that g( x) = En xnan for every x = (xn) E E. Since ( E, ~( E, E* ) ) is a sequentially complete lcs [12], Theorem 9.6,then Zkek weakly converges to x = (xn).
Remark In [9] Martinez-Maurica and Perez-Garcia proved that whenever K is spherically complete, then the local convexity is a three space property i.e. if E is an A-Banach tvs over K and F its subspace such that F and E/F are locally convex, then E is locally convex. Is the converse also true?
By L(E, F) we denote the space of all continuous linear maps between lcs E and F. A topology a on E will be called compatible with the pair (E, L(E, F)) if L({E, a), F) = L(E, F); if F =, as usual we shall say that a is compatible with the dual pair (E, E*), where E* := L(E, K).
A lcs space F will be said to have the Mackey-Arens property (MA-property) if for every lcs space E the finest topology p.(E, L(E, F)) compatible with (E, L(E, F)) exists, '7~ . As we have already mentioned Van Tiel [14] proved that if K is spherically complete, then K has the MA-property, i.e. every K-space E over spherically complete K admits the finest topology E*) compatible with the dual pair (E, E*). We have already proved the converse : If K is not spherically complete, then .~°° does not admit the Mackey topology (l~, (.2°° ) * ). Hence Corollary K is spherically complete iff it has the MA-property.
On the other hand one has the following Theorem 4 Every spherically complete normed K-space F = (F, ~.~) has the MAproperty.
We shall need the following Lemma 1 Let E, F be two vector spaces over K, where F is endowed with a norm~.~~ and p, q are seminorms on E. Let T E --~ F be a linear map such that max{p{x}, q(x)). If F is spherically complete, then there exists two linear maps T= : E ~ F, i = 1,2, such that T = Ti + T2 and P(x)~ q(x), x E E. Proof Set P(x, x) = T(x), U(x,y) = x, y E E. Then U(x,y) is a seminorm on E x E and = = U(x,x). Since F is spherically complete, then by Ingleton theorem, cf. e.g. [6], Theorem 4.18, there exists a linear map Po : E x E ~ F extending P such that ~(P0(x,y))~ U(x, y), x, y E E. To complete the proof it is enough to put Tl (x) = Po(x, 0), T2(x) = P0(0, x).
We shall also need the following lemma. Its proof uses some ideas of [1] and [4]. . Lemma 2 Let E, F be two dual-separating K-spaces over non-spherically complete K and such that F is complete and E is an infinite dimensional metrizable and complete. Then E admits two topologies T~ and T2 strictly finer than the original one of E and compatible with the pair (E, L(E, F)) and such that the topology sup{1, 2} is not compatible with {E, L{E, F)).
Proof : Observe that E contains a dense subspace G with dim(E/G)=dim(l~/c0). Let h be a non-zero linear functional on E vanishing on G. As above we construct on E two topologies r~ and r2 strictly finer than the original one r of E such that Tj ~G = r~G and (E/G, j/G) is isomorphic to the quotient space l~/c0, j = 1, 2, and h is continuous in sup{1, 2}. We show that the topologies ij, j = 1, 2, are compatible with the pair (E, L(E, F)). Fix j E {1, 2} and non-zero T E L((E, rj), F). There exists xo E E and f E F* such that 0. Suppose that T~G = {o}. Then the map q(x) --> f(Tx)) defines a non-zero continuous linear functional on (E/G, rj/G), q : E -~ E/G is the quotient map. Since (I°°~c4)* ={0}, [12], Corollary 4.3, we get a contradiction. Hence T~G is non-zero. Since G is dense in E and r and r~ coincide on G, there exists a continuous linear extension W of T to E. It is easy to see that T = W. . Hence T e L(E, F). Finally the map x ~ h(x)y, for fixed y E F, defines a T-discontinuous linear map H of E into F such that H E L((E, sup Tl, T2), F). . Proof of Theorem 4 Let E = (E, r) be a lcs and F the family of all topologies on E compatible with (E, L(E, F)). It is enough to show that the topology ~c := supF belongs to F. Let T : (E, ) -+ F be a continuous linear map. There exist seminorms pj on E, j =1, ... , n, continuous in topologies (/j E ,~'), respectively, and M > 0 such that M max pj(x) for every x E E. Using Lemma 1 one shows that T is r-continuous.
1~j~n Remarks (1) There exist complete normed K-spaces having the MA-property which are not spherically complete. In fact, assume that K is spherically complete; then .~°° is spherically complete [12] , p. 97; hence ~°° has the MA-property (by our Theorem 4). On the other hand there exists on the space ~°° another norm v which is equivalent with the usual norm, such that (~°°, v) is not spherically complete [12], p. 50 and p. 98. On the other hand the space (e°°, v) has the MA-property.
(2) Let E be an infinite dimensional normed and complete K-space. Since F := 03A0n En/~n En, where En = E for every n E IN, is spherically complete for any K [12], Theorem 4.1, then by our Theorem 4 the space F has the MA-property. For concrete spaces put E = then F = If K is not spherically complete, then by Lemma 2 the space l~ does not admit the Mackey topology but l~/c0 has the MA-property. In particular there exists on e°° the finest topology p compatible with (~°°, (3) Let E and F be A'-spaces and assume that E admits the Mackey topology p = E*). Then the finest topology on E compatible with ((E, ~z), L((E, ~), F)) exists and equals ~c.