Semi-indefinite inner product and generalized Minkowski spaces

In this paper we develop the theories of normed linear spaces and of linear spaces with indefinite metric, for finite dimensions both of which are also called Minkowski spaces in the literature.In the first part of this paper we collect the common properties of the semi- and indefinite inner products and define the semi-indefinite inner product as well as the corresponding semi-indefinite inner product space. We give a generalized concept of the Minkowski space embedded in a semi-indefinite inner product space using the concept of a new product, which contains the classical cases as special ones.In the second part we investigate the real, finite-dimensional generalized Minkowski space and its sphere of radius ii. We prove that it can be regarded as a so-called Minkowski–Finsler space, and if it is homogeneous with respect to linear isometries, then the Minkowski–Finsler distance of its points can be determined by the Minkowski product.