Hilbert-like spaces over Krull valued fields

In this survey paper we give an account of the development of a theory of Hilbert-like spaces over arbitrary fields. This was first done in an algebraic setting as the study of the so-called orthomodular spaces. Then the subject was studied from the point of view of non-archimedean analysis, in the broader context of Banach spaces over Krull valued fields. The basic concepts and facts are reviewed, and it is shown that in these Hilbert-like spaces there appear strong differences with the classical ones, as far as results and methods are concerned.