Structure of Euclidean Space
Point of Post: In this post we discuss the basic structure of Euclidean space . This will be in preparation for our multivariable analysis.
The setting for all multivariable analysis is of course Euclidean space . It makes sense that it should then be something we get our notation straight about and recall some basic theorems. Euclidean space is perhaps one of the richest mathematical structures encountered by people on a regular basis, and so consequently we will not do it any fraction of justice with our discussion here.
Properties as an Algebra/Inner Product Space
Euclidean -space, , is given the product algebra structure inherited from the -algebra itself. It is clearly -dimensional as a vector space with the canonical basis where has a in the slot and elsewhere. The canonical ordered basis for is .
One has the usual inner product ,denoted just when no confusion will arise, given by
which could equivalently be thought of as declaring the canonical basis orthonormal and extending by bilinearity. Of course we can induce the usual norm on , , given by (as expected)
From this one can prove the so-called polarization identity
Topological Properties of
There are multiple equivalent ways to define a topology on . One can give it the metric topology induced by the usual norm, in other words the metric topology induced by the metric . Or one could give it the product topology induced of thinking of as a -fold product space of (in either the order topology induced by the usual linear ordering on or the usual metric topology). The fact that these two topologies are equivalent can be cutely wrapped up in the following: “Inside every little square is a little circle, and inside every little circle is a little square.” Given this normal topology one can prove that has the Heine-Borel Property that while in general compact subspaces of metric spaces are closed and bounded the converse is true in (this can be proven first for and then extended by Tychonoff’s theorem to the general case). It is also true that is a complete metric space.
We have as always the canonical projections for which takes . Every mapping can, of course, be thought of as
where given a we denote, when no confusion arises, as and call it the coordinate function. As usual a function is continuous if and only if is continuous for each . Perhaps stronger is the fact that if one defines limits in the usual way (as per a metric space)$ then it’s true that if where that
1. Spivak, Michael. Calculus on Manifolds; a Modern Approach to Classical Theorems of Advanced Calculus. New York: W.A. Benjamin, 1965. Print.