Definition of a normed vector space; examples

In this and the following chapters we will give an indication of the advantages to be gained by superimposing onto vector spaces the ideas we have developed for metric spaces. It is worthwhile spending a few lines now to enlarge on the reasons previously given for wanting to do this.

All the work of Chapters 2, 3 and 4 was developed from the three axioms (M1), (M2) and (M3) for a metric space. The numerous applications that we have given from many fields are a pointer to just how much can be developed in this way. In all of those applications, the metric was defined in a way suggested by our ultimate aim within the application and we then made use of the general theorems deduced earlier. Within each application our knowledge of the subject matter of that application allowed us to carry out the usual manipulations that occur in any piece of mathematics. A common operation was of course the addition of elements. The pertinent point is that this could only be done within applications because, according to the axioms of a metric space X, no meaning is attached to any form of sum of elements of X. Imagine therefore what extra general theorems could be obtained if in the axioms themselves we did incorporate such an operation.

In a vector space we may add elements together. We may also multiply them by scalars.