Open Access
December 2007 The orthogonal group of a Form Hilbert space
Hans A. Keller, Herminia Ochsenius
Bull. Belg. Math. Soc. Simon Stevin 14(5): 937-946 (December 2007). DOI: 10.36045/bbms/1197908904


Form Hilbert spaces are constructed over fields that are complete in a non-archimedean valuation. They share with classical Hilbert spaces the basic property expressed by the Projection Theorem. However, there appear some remarkable geometric features which are unknown in Euclidean geometry. In fact, due to the so-called type condition there are only a few orthogonal straight lines containing vectors of the same length, so these non-archimedean spaces are utmost inhomogeneous. In the paper we consider a typical Form Hilbert space $(E, < , >)$ and we show that this geometric feature has a strong impact on the group $\mathcal{O}(E)$ of all isometries $T:E\longrightarrow E$ and on the lattice $\mathcal{L}$ of all normal subgroups of $\mathcal{O}$. In particular, we describe some remarkable sublattices of $\mathcal{L}$ which have no analogue in the classical orthogonal groups.


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Hans A. Keller. Herminia Ochsenius. "The orthogonal group of a Form Hilbert space." Bull. Belg. Math. Soc. Simon Stevin 14 (5) 937 - 946, December 2007.


Published: December 2007
First available in Project Euclid: 17 December 2007

zbMATH: 1134.46048
MathSciNet: MR2378998
Digital Object Identifier: 10.36045/bbms/1197908904

Primary: 46C99
Secondary: 51F25

Keywords: Form Hilbert spaces , isometries , Krull valuation , lattices , orthogonal group

Rights: Copyright © 2007 The Belgian Mathematical Society

Vol.14 • No. 5 • December 2007
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