The Schwarz inequality has important arithmetic, geometric, and analytic consequences.

1. In any inner product space we define the distance \(\delta(x, y)\) between two vectors \(x\) and \(y\) by \[\delta(x, y)=\|x-y\|=\sqrt{(x-y, x-y)}.\] In order for \(\delta\) to deserve to be called a distance, it should have the following three properties:
   1. \(\delta(x, y)=\delta(y, x)\) ,
   2. \(\delta(x, y) \geq 0\) ; \(\delta(x, y)=0\) if and only if \(x=y\) ,
   3. \(\delta(x, y) \leq \delta(x, z)+\delta(z, y)\) .
   4. \(\delta(x, y)=\delta(x+z, y+z)\) .)
2. In the Euclidean space \(\mathbb{R}^{n}\) , the expression \[\frac{(x, y)}{\|x\| \cdot\|y\|}\] gives the cosine of the angle between \(x\) and \(y\) . The Schwarz inequality in this case merely amounts to the statement that the cosine of a real angle is \(\leq 1\) .
3. In the unitary space \(\mathbb{C}^{n}\) , the Schwarz inequality becomes the so-called Cauchy inequality; it asserts that for any two sequences \((\xi_{1}, \ldots, \xi_{n})\) and \((\eta_{1}, \ldots, \eta_{n})\) of complex numbers, we have \[\bigg|\sum_{i=1}^{n} \xi_{i} \bar{\eta}_{i}\bigg|^{2} \leq \sum_{i=1}^{n}|\xi_{i}|^{2} \cdot \sum_{i=1}^{n}|\eta_{i}|^{2}.\] 
4. In the space \(\mathcal{P}\) , the Schwarz inequality becomes \[\bigg|\int_{0}^{1} x(t) \overline{y(t)} \,d t\bigg|^{2} \leq \int_{0}^{1}|x(t)|^{2} \,d t \cdot \int_{0}^{1}|y(t)|^{2} \,d t.\] 

It is useful to observe that the relations mentioned in (1)-(4) above are not only analogous to the general Schwarz inequality, but actually consequences or special cases of it.

1. We mention in passing that there is room between the two notions (general vector spaces and inner product spaces) for an intermediate concept of some interest. This concept is that of a normed vector space, a vector space in which there is an acceptable definition of length, but nothing is said about angles. A norm in a (real or complex) vector space is a numerically valued function \(\|x\|\) of the vectors \(x\) such that \(\|x\| \geq 0\) unless \(x=0\) , \(\|\alpha x\|=|\alpha| \cdot\|x\|\) , and \(\|x+y\| \leq\|x\|+\|y\|\) . Our discussion so far shows that an inner product space is a normed vector space; the converse is not in general true. In other words, if all we are given is a norm satisfying the three conditions just given, it may not be possible to find an inner product for which \((x, x)\) is identically equal to \(\|x\|^{2}\) . In somewhat vague but perhaps suggestive terms, we may say that the norm in an inner product space has an essentially "quadratic" character that norms in general need not possess.