Inner Product Spaces

So far we have seen that the definition of an abstract vector space captures the fundamental geometric notion of dimension. There remain, however, at least two other basic geometric ideas that we have not yet addressed: length and angle. To encompass them in the abstract we need to introduce a bit m...

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Bibliographic Details
Published inLinear Algebra pp. 131 - 150
Main Author Valenza, Robert J
Format Book Chapter
LanguageEnglish
Published United States Springer 1993
Springer New York
SeriesUndergraduate Texts in Mathematics
Subjects
Canonical Basis
Nonzero Vector
Orthogonal Projection
Orthonormal Basis
Product Space
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Summary:So far we have seen that the definition of an abstract vector space captures the fundamental geometric notion of dimension. There remain, however, at least two other basic geometric ideas that we have not yet addressed: length and angle. To encompass them in the abstract we need to introduce a bit more structure, and in consequence we shall require that our ground field manifest some notion of order. Hence we no longer operate over some abstract field k but rather, for the most part, over the field R of real numbers. In the final section we shall generalize the results to the complex numbers C.
ISBN:9781461269403
1461269407
ISSN:0172-6056
DOI:10.1007/978-1-4612-0901-0_7