Abstract

Our slogan is that defining length comparable regardless of direction is the same as defining angle. We temper this idealism with only one proviso, to wit the parallelogram law. An abstract vector space does not come equipped with notions of length and angle other than that one may compare vectors ‘in the same direction’: It always seems to make sense to say that the vector αv has |α | times the length of the vector v, where α is a scalar and | | is some absolute value on the field of scalars. No doubt, we also think of a non-zero vector as having nonzero length, but that plays only a concluding rôle below. Throughout, V is a vector space over a field K. We intend to restrict ourselves to subfields of R, and eventually of C. Nevertheless, to maintain the generality of our remarks, we do our best not to use properties peculiar to such fields until absolutely necessary. The field K will come accompanied with an absolute value | |, that is, a positive definite map K − → R preserving multiplication in K and obeying the triangle inequality. In mildly technical language, Definition. Length is the composition of a map V−→K respecting multiplication by scalars, and an absolute value on K. In different words, length is a map to the ordered semigroup R≥0: v ↦ → ‖v ‖ with the homogeneity property (1) ‖αv ‖ = |α|‖v ‖ , α ∈ K. Comparability of lengths arises from the ordering on R. In principle, we could assume that the absolute value | |, and hence length ‖ ‖, takes its values in some more general ordered ring.

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