Let us now get our feet back on the ground. We started in Chapter I by pointing out that we wish to generalize certain elementary properties of certain elementary spaces such as ℛ2. In our study so far we have done this, but we have entirely omitted from consideration one aspect of ℛ2. We have studied the qualitative concept of linearity; what we have entirely ignored are the usual quantitative concepts of angle and length. In the present chapter we shall fill this gap; we shall superimpose on the vector spaces to be studied certain numerical functions, corresponding to the ordinary notions of angle and length, and we shall study the new structure (vector space plus given numerical function) so obtained. For the added depth of geometric insight we gain in this way, we must sacrifice some generality; throughout the rest of this book we shall have to assume that the underlying field of scalars is either the field ℛ of real numbers or the field C of complex numbers.
KeywordsOrthonormal Basis Linear Transformation Product Space Real Vector Space Partial Isometry
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