THE NON-ARCHIMEDEAN SPACE BC ( X ) WITH THE STRICT TOPOLOG Y

Let X be a zero-dimensional, Hausdorff topological space and K a field with a non-trivial, non -archimedean valuation under whic h it is complete . Then BC(X) is the vector space of the bounde d continuous functions from X to K . We obtain necessary and sufficient conditions for BC(X ) , equipped with the strict topology , to be of countable type and to be nuclear in the non-archimedean sense .


Introductxo n
Throughout the paper K is a complete nonarchimedean valued field with a valuation • which is nat trivial, and X is a zero-dimensional Hausdorff topological space .We denote by C(X) (resp .BC(X )} th e space of the continuous (resp .bounded and continuous) functions fro m X to K .For A c X and f E BC(A) we define IlfilA = sup xEA lf(x)I .
All the specific definitions needed are included in the paper .Far mor e general facts on locally convex spaces and Banach spaces we refer to [7 ] and [81 .We consider on BC (X )the strict topology r,3 (definition below) .Many properties of the locally convex space BC(X), T,3 have already bee n investigated .See e .g .[4], [5] and [fi] .But so far no attention was pai d to the properties "being of countable type " and "being nuclear" .Thi s paper fills that gap .The curious fact is that the conditions for the stric t topology turn out to be the same as those obtained for the compact ope n topology T~, in [3] .
Denote by Bo(X ) the bounded functions ~: X ---} K which vanish at infinity.The strict topology T~on BC(X) is then defined by the family of semi -norms {p ; cp E Bfl (X) }, where The strict topology lies between the compact-open topology T( i .e .the topology of uniform convergence on compact subsets of X) an d the uniform topology Tu .Thus T~Ç TQ Ç Tu .
In [4, p . 193], Katsaras shows that a basis for the zero-neighbourhoods in BC(X), T~consists of the sets of the form : where A = (Aa) is an increasing sequence of compact subsets of X an d k = (len) is a sequence of real numbers, increasing to infinity, with k l > 1 .
The next lemma follows easily.

The strict topology TQ on BC(X) can be determined by the family o f semi-norms p A, k with A and k as aboye and where
Let k _ (kn ) be as aboye .Then, taking In = kn j-/2 , the sequence d = ( ln) gives us a continuous semi-norm pA , l on BC(X) , T,3 such that pA,a (f) ~ PA,k(f) for all f E BC(X), and limn le n • 1 n -1 = oo .This semi-norm will be used in Section 3 .

.1 . Definition . ([8, p . 661) .
A normed space E over K is said to be of countable type if it is the closed linear span of a countable set .

.. Example .
If X is compact then BC(X) = C(X), Tu is of countable type if and only if X is ultrametrizable .([S, 3 .T]) .

.4 . Proposition .
The space enEn, II • is of countable type if and only if E me, , ii is of countable type for all n .
Let E be a locally convex Hausdorff space over K, the topology o f which is determined by a family of semi -norms P (E, P in short) .For each p E P put Ep = E/ Kerp and denote by 7rp : E --~Ep the canonical surjection .The space Ep is then normed by Ii7rp(x) = p(x), x E E .

.6 . Example .
Let PA , k be one of the semi -norms determining the strict topology TQ on BC(X) .

Denot e by BC(A, k) the space BC(U n A n ), normed by PM . Then the norme d spac e B C (X ) / Ker PA , k is linearly isometri c wit h a subspace of BC(A , k) . Indeed we have a commutative diagram BC(X ) BC(X)/KerpA,k + s BC(A,k )
where ?rA , k is the canonical surjection and R is the restriction map which sends f E BC(X) onto its restriction to U n A n .Then S is the desired isometry.
For later use we prove :

.7 . Lemma . BC(A, k) is tinearZy isometric to a subspace of the space enC(An, kn) , where, for cal n, C(An, k n ) is the space C(An ), normed b y g E C(An ) .
Note that each of the spaces C(An, kn ) is a Banach space, linearly homeomorphic to C(An ) , Tu .

Broof
For f E BC(A, k) let fn stand for the restriction of f to An .
The locally convex space E, P is said to be of countable type if each of the normed spaces EP , II .II, p E P, is of countable type .

.9 . Theorem .
The following are equivalent : i) BC(X), Tp is of countable type.ii) BC(X), T~is of countable type .iii) Every compact subset of X is ultramerizable .Broof i) ~ ii} follows directly from the fact that (See 1 .1) .
We have to prove that the normed space BC(X)/ Ker p A, k is of countable type .Now every subspace of a space of countable type, is of countable type ([S, 3.16]) .Hence, by 2 .4, 2 .6, and 2 .7 it suffices to show that each of the normed spaces C(An,k n ) is of countable type .Now make use o f the remark made in 2 .7 and apply 2 .2 .■ 2 .10 .Remark .
The conditions in Theorem 2 .9 are not equivalent to "BC (X ) , Tu i s of countable type" .Indeed, take X = the natural numbers with th e discrete topology.Then BC (X ) , Tu = l °°, II .II which is not of countabl e type .Also note that the strict topology on l°°coincides with the natural topology n (100, co ) in the sense of perfect sequence spaces ([1, p .

473] ) and that the compact open topology on
is the weak topology a-(1' , Ca) .Hence the inequalities in 1 .1 may be strict .
Let E be a locally convex space over K .A subset B of E is calle d compactoid if for every zero-neighbourhood U in E there exists a finit e subset S of E such that B c U + Co(S), where Co(S) is the absolutel y convex hull of S .
A linear map T from a normed space E to a normed space F is calle d compact if if maps the unit ball of E into a compactoid subset of F .
The following is easily seen : 3 .2 .Lemma.
Let E, F and G be normed spaces over K, T : E-> F a linear map , and S : F G a linear isometry.lf SoT is compact, then so is T .

.. Definition .
Let E, P be a locally convex space over K .If p E P and q is a continuous seminorm on E with p Ç q .Then there exists a uniqu e continuous linear map ("opq : Eq -> Ep which makes the diagra m

Eq Ep
The space E, P is called nuclear if for every p E P there exists a continuous seminorm q on E with p Ç q such that the map (ppq is compact .
The following are equivalent : i) BC(X), .rp is nuclear.
ii} Every To-bounded subset of BC(X) is rQ -compactoid .iii) BC(X), rG is nuclear .iv) Every rc -bounded subset of BC(X) is Tc -compactoid.v} C(X), is nuclear .vi) Every Tc -bounded subset of C(X ) is rc -compactoid.vii) Every compact subset of X is finite .