Semi-Riemannian Geometry

Chapter 4 Scalar Product Spaces

4.1 Scalar Product Spaces

We now begin the study of a type of vector space that will occupy us, in one way or another, for the rest of the book.

Let V be a vector space, and let g : V × V → ℝ be a bilinear function. We say that g is a scalar product on V, and that the pair (V, g) is a scalar product space, if g is symmetric and nondegenerate on V. Recall from Section 3.1 that g is symmetric on V if (v, w) = (w, v) for all vectors v, w in V; and that g is nondegenerate on V if for all vectors v in V, (v, w) =0 for all vectors w in V implies v = 0.

Suppose (V, g) is in fact a scalar product space. It follows from the symmetry of g that for all bases ℋ for V, g _ℋ is a symmetric matrix. The norm corresponding to g is the function

defined by

(4.1.1)

for all vectors v in V, where q is the quadratic function corresponding to g. Taking the absolute value in (4.1.1) is necessary because a scalar product can have negative values. We refer to ||v|| as the norm of v. Observe that ||v|| =0 if and only if v = 0 or v is lightlike; and ||v|| = 1 if and only if v is a unit vector. Recall from Theorem 3.3.3 that since g is nondegenerate on V, the flat map F is a linear isomorphism, so its inverse, the sharp map S, exists and is also a linear isomorphism. ...