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 \(\mathbb{R}^{2}\) . In our study so far we have done this, but we have entirely omitted from consideration one aspect of \(\mathbb{R}^{2}\) . 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 \(\mathbb{R}\) of real numbers or the field \(\mathbb{C}\) of complex numbers.

For a clue as to how to proceed, we first inspect \(\mathbb{R}^{2}\) . If \(x=(\xi_{1}, \xi_{2})\) and \(y=(\eta_{1}, \eta_{2})\) are any two points in \(\mathbb{R}^{2}\) , the usual formula for the distance between \(x\) and \(y\) , or the length of the segment joining \(x\) and \(y\) , is \(\sqrt{(\xi_{1}-\eta_{1})^{2}+(\xi_{2}-\eta_{2})^{2}}\) . It is convenient to introduce the notation \[\|x\|=\sqrt{\xi_{1}^{2}+\xi_{2}^{2}}\] for the distance from \(x\) to the origin \(0=(0,0)\) ; in this notation the distance between \(x\) and \(y\) becomes \(\|x-y\|\) .

So much, for the present, for lengths and distances; what about angles? It turns out that it is much more convenient to study, in the general case, not any of the usual measures of angles but rather their cosines. (Roughly speaking, the reason for this is that the angle, in the usual picture in the circle of radius one, is the length of a certain circular arc, whereas the cosine of the angle is the length of a line segment; the latter is much easier to relate to our preceding study of linear functions.) Suppose then that we let \(\alpha\) be the angle between the segment from \(0\) to \(x\) and the positive \(\xi_{1}\) axis, and let \(\beta\) be the angle between the segment from \(0\) to \(y\) and the same axis; the angle between the two vectors \(x\) and \(y\) is \(\alpha-\beta\) , so that its cosine is \[\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta=\frac{\xi_{1} \eta_{1}+\xi_{2} \eta_{2}}{\|x\| \cdot\|y\|}.\] 

Consider the expression \(\xi_{1} \eta_{1}+\xi_{2} \eta_{2}\) ; by means of it we can express both angle and length by very simple formulas. We have already seen that if we know the distance between \(0\) and \(x\) for all \(x\) , then we can compute the distance between any \(x\) and \(y\) ; we assert now that if for every pair of vectors \(x\) and \(y\) we are given the value of \(\xi_{1} \eta_{1}+\xi_{2} \eta_{2}\) , then in terms of this value we may compute all distances and all angles. Indeed, if we take \(x=y\) , then \(\xi_{1} \eta_{1}+\xi_{2} \eta_{2}\) becomes \(\xi_{1}^{2}+\xi_{2}^{2}=\|x\|^{2}\) , and this takes care of lengths; the cosine formula above gives us the angle in terms of \(\xi_{1} \eta_{1}+\xi_{2} \eta_{2}\) and the two lengths \(\|x\|\) and \(\|y\|\) . To have a concise notation, let us write, for \(x=(\xi_{1}, \xi_{2})\) and \(y=(\eta_{1}, \eta_{2})\) , \[\xi_{1} \eta_{1}+\xi_{2} \eta_{2}=(x, y);\] what we said above is summarized by the relations \begin{align} & \text{distance from $0$ to $x$ } = \|x\| = \sqrt{(x, x)},\\ & \text{distance from $x$ to $y$ } = \|x - y\|,\\ & \text{cosine of angle between $x$ and $y$ } = \frac{(x, y)}{\|x\| \cdot \|y\|}. \end{align} The important properties of \((x, y)\) , considered as a numerical function of the pair of vectors \(x\) and \(y\) , are the following: it is symmetric in \(x\) and \(y\) , it depends linearly on each of its two variables, and (unless \(x=0\) ) the value of \((x, x)\) is always strictly positive. (The notational conflict between the use of parentheses in \((x, y)\) and in \((\xi_{1}, \xi_{2})\) is only apparent. It could arise in two-dimensional spaces only, and even there confusion is easily avoided.)

Observe for a moment the much more trivial picture in \(\mathbb{R}^{1}\) . For \(x=(\xi_{1})\) and \(y=(\eta_{1})\) we should have, in this case, \((x, y)=\xi_{1} \eta_{1}\) (and it is for this reason that \((x, y)\) is known as the inner product or scalar product of \(x\) and \(y\) ). The angle between any two vectors is either \(0\) or \(\pi\) , so that its cosine is either \(+1\) or \(-1\) . This shows up the much greater sensitivity of the function given by \((x, y)\) , which takes on all possible numerical values.