The Finite Dimensional Normed Linear Space Theorem

The claim that follows, which I have called the nite-dimensional normed linear space theorem, essentially says that all such spaces are topologically R with the Euclidean norm. This means that in many cases the intuition we obtain in R,R, and R by imagining intervals, circles, and spheres, respectively, will carry over into not only higher dimension R but also any vector space that has nite dimension. Throughout this discussion I assume that all scalars for forming linear combinations in this nite dimensional linear space are real numbers (I have heard of complex vector spaces, but in this mathematical foundations for economics independent study we are solely concerned with real vector spaces). By nite-dimensional normed linear space I mean a set X of vectors along with addition of vectors and scalar multiplication satisfying the usual axioms for a vector space (e.g., X is closed under arbitrary linear combinations of its elements), and a norm function || · || : X → R satisfying the usual axioms of a norm (e.g., the triangle inequality), with the induced distance function onX de ned in the usual way: ρX(x, y) = ||x−y||. Finally, by nite dimensional, I mean that there exists a nite set of vectors {x1, x2, ..., xn} ⊆ X that are linearly independent and span X (every vector in X can be written as a linear combination of vectors in {x1, x2, ..., xn}, and none of the xi are redundant ). A standard example of such a vector space is X = R with Euclidean norm and basis {e1, e2, e3} = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}. These assumptions force X to be highly structured. In particular, it can be shown that every open ball in X is simply a translation and rescaling of the unit open ball centered around the origin. So the metric on X is, in this context, determined entirely by the shape of its unit ball. For example, the Euclidean norm has this unit ball be the normal sphere; the taxicab norm leads to a diamond-shape; the sup norm has its unit ball a cube. Within this context, we have the following.