ABSTRACT

In the previous chapter we introduced the notion of the scalar product of two vectors in ℝn. More generally if a scalar product is defined on some space, then this permits the definition of a norm, or length, associated with a vector, and this in turn allows us to define the distance between two vectors. A distance function or metric may be defined on a space, X, even when X admits no norm. For example let X be the surface of the earth. Clearly it is possible to say what the shortest distance, d(x,y), between two points on the earth's surface is, although it is not meaningful to talk of the “length” of a point on the surface. More general than the notion of a metric is that of a topology. Essentially a topology on a space is a mathematical notion for making more precise the idea of “nearness”. The notion of topology can then be used to define what we mean by continuity.