Banach spaces over fields with an infinite rank valuation.

*(English)*Zbl 0938.46056
Kąkol, J. (ed.) et al., \(p\)-adic functional analysis. Proceedings of the 5th international conference in Poznań, Poland, June 1-5, 1998. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 207, 233-293 (1999).

Studies in functional analysis over fields with an infinite rank valuation are motivated by the fact that no analogs of a Hilbert space, where a norm is directly connected with a scalar product, are possible if the underlying field has a rank one valuation (that is a real valued one). For infinite rank valuations the situation is different, which is known since the paper by H. Keller [Math. Z. 172, 41-49 (1980; Zbl 0423.46013)], who constructed the first non-trivial example of an infinite-dimensional orthomodular space.

In the paper under review, the authors begin a systematic treatment of functional analysis over fields with an infinite rank valuation. They give the basic theory of normed spaces over such fields, and then they consider a class of “norm Hilbert spaces”, that is Banach spaces of countable type, for which closed subspaces admit projections of the norm \(\leq 1\). In such a space each closed subspace has a closed complement; on the other hand, every ball is a compactoid, a property that in the rank 1 theory is shared only by finite-dimensional spaces. In particular, a class of “form Hilbert spaces” is studied. These are norm Hilbert spaces, for which there exists a Hermitian form \((\cdot ,\cdot)\), satisfying \(|(x,x)|=\|x\|^2\) for all \(x\).

For the entire collection see [Zbl 0919.00056].

In the paper under review, the authors begin a systematic treatment of functional analysis over fields with an infinite rank valuation. They give the basic theory of normed spaces over such fields, and then they consider a class of “norm Hilbert spaces”, that is Banach spaces of countable type, for which closed subspaces admit projections of the norm \(\leq 1\). In such a space each closed subspace has a closed complement; on the other hand, every ball is a compactoid, a property that in the rank 1 theory is shared only by finite-dimensional spaces. In particular, a class of “form Hilbert spaces” is studied. These are norm Hilbert spaces, for which there exists a Hermitian form \((\cdot ,\cdot)\), satisfying \(|(x,x)|=\|x\|^2\) for all \(x\).

For the entire collection see [Zbl 0919.00056].

Reviewer: Anatoly N.Kochubei (Kiev)

##### MSC:

46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |

12J25 | Non-Archimedean valued fields |

06C15 | Complemented lattices, orthocomplemented lattices and posets |