Abstract

Let EE be a non-Archimedean Banach space over a non-Archimedean locally compact non-trivially valued field K:=(K,∣.∣)\mathbb{K}:=(\mathbb{K},|.|). Let E′′E'' be its bidual and MM a bounded set in EE. We say that MM is ε\varepsilon-weakly relatively compact if  M‾σ(E′′,E′)⊂E+BE′′,ε\ \overline{M}^{\sigma (E'',E')}\subset E+B_{E^{\prime \prime },\varepsilon}, where BE′′,εB_{E^{\prime \prime },\varepsilon } is the closed ball in E′′E'' with the radius ε≥0\varepsilon \geq 0. In this paper we describe measures of noncompactness γ,\gamma, kk and De Blasi measure ω\omega. We show that γ(M)≤k(M)≤ω(M)=ω(acoM)≤1∣ρ∣γ(M),\gamma \left( M\right) \leq k\left( M\right) \leq \omega \left( M\right) =\omega (acoM)\leq \frac{1}{\left\vert \rho \right\vert }\gamma \left( M\right), where ρ\rho (∣ρ∣<1)\left\vert \rho \right\vert <1) is an uniformizing element in K\mathbb{K}, and ω(M)=sup⁡{lim⁡m‾   dist(xm,[x1,…,xm−1]):(xm)⊂M\omega (M)=\sup \{\overline{\lim_{m}}\,\,\,dist\left( x_{m},\left[ x_{1},\dots,x_{m-1}\right] \right):\left( x_{m}\right) \subset M }\}; the latter equality is purely non-Archimedean. In particular, assuming ∣K∣={∣∣x∣∣:x∈E},\left\vert \mathbb{K}\right\vert =\{||x||:x\in E\}, we prove that the absolutely convex hull acoMacoM of a ε−\varepsilon -weakly relatively compact subset MM in EE is ε−\varepsilon -weakly relatively compact. In fact we show that in this case for a bounded set MM in EE we have γ(M)=γ(acoM)=k(M)=k(acoM)=ω(M)\gamma \left( M\right) =\gamma \left( acoM\right) =k\left( M\right) =k(acoM)=\omega \left( M\right), Note that the above equalities fail in general for real Banach spaces by results of A. S. Granero [An extension of the Krein-Smulian theorem, Rev. Mat. Iberoam. 22 (2006) 93–100] and K. Astala and H. O. Tylli [Seminorms related to weak compactness and to Tauberian operators, Math. Proc. Cambridge Philos. Soc. 107 (1990) 367–375]. Most proofs are strictly non-Archimedean. A non-Archimedean variant of another quantitative Krein's theorem due to Fabian, Hajek, Montesinos and Zizler is also provided, see Corollary 9.

Author information

Contact details are reproduced from the original publication and may be historical.

Carlos Angosto

Depto. de Matemática Aplicada y Estadistica, Universidad Politécnica de Cartagena, 30203 Cartagena, Spain

carlos.angosto@upct.es

Jerzy Kąkol

Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Poznań, Poland

kakol@amu.edu.pl

Keywords

  • Krein's theorem
  • Compactness
  • Measures of weak noncompactness

Mathematics Subject Classification

46S10, 46A50, 54C35

References

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Suggested citation

C. Angosto, J. Kąkol, A. Kubzdela. “Measures of Weak Noncompactness in Non-Archimedean Banach Spaces.” Journal of Convex Analysis 21 (2014), No. 3, 833–849. https://doi.org/10.68381/jca21045

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.