  In this paper we parallelly build up the theories of normed linear spaces and
of linear spaces with indefinite metric, called also Minkowski spaces for
finite dimensions 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 and
the corresponding structure, the semi-indefinite-inner-product space. We give a
generalized concept of Minkowski space embedded in a
semi-indefinite-inner-product space using the concept of a new product, that
contains the classical cases as special ones.
  In the second part of this paper we investigate the real, finite dimensional
generalized Minkowski space and its sphere of radius $i$. We prove that it can
be regarded as a so-called Minkowski-Finsler space and if it is homogeneous one
with respect to linear isometries, then the Minkowski-Finsler distance its
points can be determined by the Minkowski-product.
