Non-Archimedean Quantitative Grothendieck and Krein%26apos;s Theorems

作者：Kakol Jerzy^*; Kubzdela Albert

来源：Journal of Convex Analysis, 2013, 20(1): 233-242.

• Grothendieck%26apos
• s theorem
• Krein%26apos
• s theorem
• locally compact non-archimedean field
• cmpactness
• space of continuous functions

We show that the non-archimedean version of Grothendieck%26apos;s theorem about weakly compact sets for C(X, K), the space of continuous maps on X with values in a locally compact non-trivially valued non-archimedean field K, fails in general. Indeed, we prove that if X is an infinite zero-dimensional compact space, then there exists a relatively compact set H := {g(n) : n is an element of N} subset of C(X,K) in the pointwise topology tau(p) of C(X, K) which is not w-relatively compact, i.e. compact in the weak topology of C(X, K), such that all parallel to g(n)parallel to = 1 and gamma(H):= sup{vertical bar lim(m) lim(n) f(m)(x(n)) - lim(n) lim(m) f(m)(x(n))vertical bar : (f(m))(m) subset of B, (x(n))(n) subset of H} %26gt; 0, where B is the closed unit ball in the dual C(X, K)* and the involved limits exist. The latter condition gamma(H) %26gt; 0 shows in fact that a quantitative version of Grothendieck%26apos;s theorem for real spaces (due to Angosto and Cascales) fails in the non-archimedean setting. The classical Krein and Grothendieck%26apos;s theorems ensure that for any compact space X every uniformly bounded set H in a real (or complex) space C(X) is tau(p)-relatively compact if and only if the absolutely convex hull aco H of H is tau(p)-relatively compact. In contrast, we show that for an infinite zero-dimensional compact space X the absolutely convex hull aco H of a tau(p)-relatively compact and uniformly bounded set H in C(X, K) needs not be tau(p)-relatively compact for a locally compact non-archimedean K. Nevertheless, our main result states that if H subset of C(X,K) is uniformly bounded, then aco H is tau(p)-relatively compact if and only if H is w-relatively compact.

• 出版日期2013