Scalar product Euclidean generalized

Problem Prove that the mathematical criteria (9.27a-c) for a proposed scalar product (R R7) are sufficient for the general Schwarz inequality (9.24) in the space M, thereby guaranteeing that M is Euclidean. 

Here and also in the sequel, we shall need some additional remarks concerning the generalized Euclidean scalar products and norms of vectors. See Section B.IO of Appendix B. Let S be generally a symmetric positive definite NxN matrix. Designate... 

Generalizing, in the space R of N-vectors we introduce the (canonical Euclidean) scalar product of vectors a and b... 

We generally denote scalars by lowercase Greek letters (e.g., P), column vectors by boldface lowercase Roman letters (e.g., x), and matrices by capital italic Roman letters (e.g., H). A superscriptT denotes a vector or matrix transpose. Thus xT is a row vector, xTy is an inner product, and AT is the transpose of the matrix A. Unless stated otherwise, all vectors belong to R , the u-dimen-sional vector space. Components of a vector are typically written as italic letters with subscripts (e.g., xux2,.. . , ). The standard basis vectors in R" are the n vectors ei,e2,. . . , e , where e has the entry 1 in the th component and 0 in all others. Often, the associated vector norm is the standard Euclidean norm, j 2, defined as... 

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