| dc.description.abstract | The theory of X-normed spaces over non-Archimedean valued fields with valuations of higher rank was introduced by H. Ochsenius and W. H. Schikhof in [1] and further developed in [2–4, 6, 7] and [5]. In order to obtain results like the Open Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem, H. Ochsenius and W. H. Schikhof used 1st countability conditions in the value group of the based field. In this article the author develops a new tool to work with transfinite induction simplifying the techniques employed in X-normed spaces, thus accomplishing a GeneralizedBaire Category Theorem that allows the proof of an Open Mapping Theorem for X-normed spaces without restrictions on the value group of the based field. Additionally, some contributions to the theory of X-normed spaces are presented regarding quotient spaces. [1] H. Ochsenius and W. H. Schikhof, "Banach spaces over fields with an infinite rank valuation,” p-Adic Functional Analysis, Lecture Notes in Pure and Appl. Math. 207, 233–293 (Marcel Dekker,1999). [2] H. Ochsenius and W. H. Schikhof, “Lipschitz operators on Banach spaces over Krull valued fields,” Contemp. Math. 384, 203–234 (2005). [3] H. Ochsenius and W. H. Schikhof, “Norm Hilbert spaces over Krull valued fields,” Indagat. Math. 17 (1), 65–84 (2006). [4] H. Ochsenius and W. H. Schikhof, “Matrix characterizations of Lipschitz operators on Banach spaces over Krull valued fields,” Bull. Belgian Math. Soc. – Simon Stevin 14, 193–212 (2) (2007). [5] H. Ochsenius and W. H. Schikhof, “A new method for comparing two norm Hilbert spaces and their operators,” Indagat. Math. 21 (1-2), 112–126 (2011). [6] W. H. Schikhof, H. Keller and H. Ochsenius, “On the commutation relation AB-BA=I for operators on non-classical Hilbert spaces", p-Adic Functional Analysis, pp. 177–190 (2001). [7] W. H. Schikhof and H. Ochsenius, “Linear homeomorphisms of non-classical Hilbert spaces,” Indagat. Math. 10 (4), 601–613 (1999). | |
