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CHAPTER
Inner Product Spaces
I N T R O D U C T I O N : In Section 3.3 we defined the Euclidean inner product on the spaces and . Then, in Section 4.1,
we extended that concept to and used it to define notions of length, distance, and angle in . In this section we shall
extend the concept of an inner product still further by extracting the most important properties of the Euclidean inner product on
and turning them into axioms that are applicable in general vector spaces. Thus, when these axioms are satisfied, they will
produce generalized inner products that automatically have the most important properties of Euclidean inner products. It will
then be reasonable to use these generalized inner products to define notions of length, distance, and angle in general vector
spaces.
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