What is the present value of n periodic payments of x dollars discounted at b per period?
A special feature of a normed linear space is that its structure or geometry is uniform throughout the space. This can be seen in the special form taken by the open balls in a normed linear space. Recall that the open ball about x0 of radius r is the set

By linearity, this can be expressed as

The unit ball B is the open ball about 0 of radius 1

It is the set of all elements of norm less than 1. Any open ball can be expressed in terms of the unit ball as follows:
Br(x0) = x0 + rB
That is, any open ball in a normed linear space is simply a translation and scaling of the unit ball. Therefore many important properties of a normed linear space are related to the shape of its unit ball. Figure 1.13 illustrates the unit ball in the plane (R2) for some different norms. The uniform structure of a normed linear space enables the following refinement of exercise 1.93.

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= {x e X: ||x – xo||