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Non-archimedean Hilbert like spaces. (English) Zbl 1134.46011

Summary: Let \(\mathbb K\) be a non-Archimedean, complete valued field. It is known that the supremum norm \(\| \cdot \| _{\infty}\) on \(c_{0}\) is induced by an inner product if and only if the residual class field of \(\mathbb K\) is formally real. One of the main problems of this inner product is that \(c_{0}\) is not orthomodular, as is any classical Hilbert space. Our goal in this work is to identify those closed subspaces of \(c_{0}\) which have a normal complement. In this study, we also involve projections, adjoint and self-adjoint operators.

MSC:

46C50 Generalizations of inner products (semi-inner products, partial inner products, etc.)
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
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Full Text: Euclid