Measures of Weak Noncompactness in Non-Archimedean Banach Spaces

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 [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.