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Journal of Convex Analysis 21 (2014), No. 3, 833--849
Copyright Heldermann Verlag 2014

Measures of Weak Noncompactness in Non-Archimedean Banach Spaces

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

Jerzy Kakol
Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Poznan, Poland

Albert Kubzdela
Institute of Civil Engineering, University of Technology, Ul. Piotrowo 5, 61-138 Poznan, Poland


Let $E$ be a non-Archimedean Banach space over a non-Archimedean locally compact non-trivially valued field $\mathbb{K}:=(\mathbb{K},|.|)$. Let $E''$ be its bidual and $M$ a bounded set in $E$. We say that $M$ is $\varepsilon$-weakly relatively compact if $\ \overline{M}^{\sigma (E'',E')}\subset E+B_{E^{\prime \prime },\varepsilon}$, where $B_{E^{\prime \prime },\varepsilon }$ is the closed ball in $E''$ with the radius $\varepsilon \geq 0$. In this paper we describe measures of noncompactness $\gamma ,$ $k$ and De Blasi measure $\omega$. We show that $\gamma \left( M\right) \leq k\left( M\right) \leq \omega \left( M\right) =\omega (acoM)\leq \frac{1}{\left\vert \rho \right\vert }\gamma \left( M\right) ,$ where $\rho $ ($\left\vert \rho \right\vert <1)$ is an uniformizing element in $\mathbb{K}$, and $\omega (M)=\sup \{\overline{\lim_{m}}\,\,\,dist\left( x_{m},\left[ x_{1},\dots ,x_{m-1}\right] \right) :\left( x_{m}\right) \subset M$ $\}$; the latter equality is purely non-Archimedean. In particular, assuming $\left\vert \mathbb{K}\right\vert =\{||x||:x\in E\},$ we prove that the absolutely convex hull $acoM$ of a $\varepsilon -$weakly relatively compact subset $M$ in $E$ is $\varepsilon -$weakly relatively compact. In fact we show that in this case for a bounded set $M$ in $E$ we have $\gamma \left( M\right) =\gamma \left( acoM\right) =k\left( M\right) =k(acoM)=\omega \left( M\right)$, Note that the above equalities fail in general for real Banach spaces by results of A. S. Granero [\textit{An extension of the Krein-Smulian theorem}, Rev. Mat. Iberoam. 22 (2006) 93--100] and K. Astala and H. O. Tylli [\textit{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.

Keywords: Krein's theorem, Compactness, Measures of weak noncompactness.

MSC: 46S10, 46A50, 54C35

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