## Non-archimedean quantitative Grothendieck and Krein's theorems

### Authors:

- Jerzy Kąkol,
- Albert Kubzdela

### Abstract

We show that the non-archimedean version of Grothendieck'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-archunedean field K, fails in general. Indeed, we prove that if is an infinite zero-dimensional compact space, then there exists a relatively compact set H := (g_{n} : n ε N) C C(X, K) hi the pointwise topology _{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 ∥g_{n}∥ = 1 and γ(H) := sup{|lim_{m} lim _{n} f_{m}(x_{n}) - lim_{n} f_{m} (x_{n})|: (f_{m})_{m}⊂ B,(x_{n})_{n} ⊂ H} > 0, where B is the closed unit ball m the dual C(X,K)* and the involved limits exist. The latter condition γ(H) > 0 shows in fact that a quantitative version of Grothendieck's theorem for real spaces (due to Angosto and Cáscales) fails in the non-archimedean setting. The classical Krein and Grothendieck's theorems ensure that for any compact space X every imiformly bounded set if in a real (or complex) space C(X) is τ_{p}-relatively compact if and only if the absolutely convex hull aco H of His τ_{p}-relatively compact. In contrast, we show that for an infinite zero-dimensional compact space X the absolutely convex hull aco II of a τ_{p}-relatively compact and tmiformly boimded set i/ in C(X, K) needs not be τ_{p}-relatively compact for a locally compact non-archimedean K. Nevertheless, our main result states that if H ⊂ C(X,K) is imiformly boimded, then acoH is τ_{p}-relatively compact if and only if if is tu-relatively compact. © Heldermann Verlag.

- Record ID
- UAMf093e4bb12b04810bb23cf08fc04c186
- Author
- Journal series
- Journal of Convex Analysis, ISSN 0944-6532
- Issue year
- 2013
- Vol
- 20
- Pages
- 233-242
- ASJC Classification
- ;
- Language
- en English
- Score (nominal)
- 30
- Score source
- journalList
- Score
- Publication indicators
- = 1.000; : 2013 = 0.900; : 2013 = 0.592 (2) - 2013=0.792 (5)

- Uniform Resource Identifier
- https://researchportal.amu.edu.pl/info/article/UAMf093e4bb12b04810bb23cf08fc04c186/

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