Banach spaces over nonarchimedean valued fields
Banach spaces over nonarchimedean valued fields
Summary: In this survey note we present the state of the art on the theory of Banach and Hilbert spaces over complete valued scalar fields that are not isomorphic to \(\mathbb{R}\) or \(\mathbb{C}\). For convenience we treat `classical' theorems (such as Hahn-Banach Theorem, Closed Range Therem, Riesz Representation Theorem, Eberlein-Šmulian's Theorem, Krein-Milman Theorem) and discuss whether or not they remain valid in this new context, thereby stating sometimes strong negations or strong improvements. Meanwhile several concepts are being introduced (such as norm orthogonality, spherical completeness, compactoidity, modules over valuation rings) leading to results that do not have -- or at least have less important -- counterparts in the classical theory.
- Radboud University Nijmegen Netherlands
Normed linear spaces and Banach spaces; Banach lattices, Other ``topological'' linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than \(\mathbb{R}\), etc.), Banach space, non-Archimedean valued fields, Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis, Hilbert space, Research exposition (monographs, survey articles) pertaining to functional analysis, Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory, Mathematical Physics
Normed linear spaces and Banach spaces; Banach lattices, Other ``topological'' linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than \(\mathbb{R}\), etc.), Banach space, non-Archimedean valued fields, Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis, Hilbert space, Research exposition (monographs, survey articles) pertaining to functional analysis, Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory, Mathematical Physics
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