Banach’s principle states that if a map T uniformly reduces the distance between points of a complete metric space, then there is a unique x such that Tx = x, called T’s fixed point. This simple statement has profound and surprising consequences, as we will see in the following chapters. For now, we will content ourselves with an example, which may appear to belong to the realm of linear algebra, but is, in fact, much easier to deal with using metric notions.