Symmetric bilinear functions

Abstract

All the properties of an inner product space discussed in Chapter VII are based upon the bilinearity, the symmetry and the definiteness of the inner product. The question arises which of these properties do not depend on the definiteness and hence can be carried over to a real linear space with an indefinite inner product. Linear spaces of this type will be discussed in § 4. First of all, the general properties of a symmetric bilinear function will be investigated. It will be assumed throughout the chapter that all linear spaces are real.