Examples of vector spaces

Abstract

Before continuing with our study of the elementary properties of vector spaces and their linear subspaces let us collect a list of examples of vector spaces. We have already encountered the cartesian k-space ℝ^k and so for the sake of completeness let us begin by listing this example:

Example 1. ℝ^k The first new example that we have in this chapter is primarily designed to destroy the belief that a vector is a quantity with both direction and magnitude and to give meaning to the phrase in our comments on axiomatics in Chapter 2, that “the possibility is opened of assigning to them (the axioms of a vector space) content in new and unforeseen ways.”