Abstract Nonsense

Crushing one theorem at a time

Structure of Euclidean Space


Point of Post: In this post we discuss the basic structure of Euclidean space \mathbb{R}^n. This will be in preparation for our multivariable analysis.

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Motivation

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The setting for all multivariable analysis is of course Euclidean space \mathbb{R}^n. 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.

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Properties as an Algebra/Inner Product Space

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Euclidean n-space, \mathbb{R}^n, is given the product algebra structure inherited from the \mathbb{R}-algebra \mathbb{R} itself. It is clearly n-dimensional as a vector space with the canonical basis e_1,\cdots,e_n where e_k has a 1 in the k^{\text{th}} slot and 0 elsewhere. The canonical ordered basis for \mathbb{R}^n is (e_1,\cdots,e_n).

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One has the usual inner product ,denoted just \langle\cdot,\cdot\rangle when no confusion will arise, given by

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\displaystyle \left\langle (x_1\cdots,x_n),(y_1,\cdots,y_n)\right\rangle=\sum_{j=1}^{n}x_jy_j

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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 \mathbb{R}^n, \|\cdot\|, given by (as expected)

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\displaystyle \|(x_1,\cdots,x_n)\|=\sqrt{\sum_{j=1}^{n}x_j^2}=\sqrt{\langle (x_1,\cdots,x_n),(x_1,\cdots,x_n)\rangle}

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From this one can prove the so-called polarization identity 

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\displaystyle \left\langle (x_1,\cdots,x_n),(y_1,\cdots,y_n)\right\rangle=\frac{\left\|(x_1,\cdots,x_n)+(y_1,\cdots,y_n)\right\|^2-\left\|(x_1,\cdots,x_n)-(y_1,\cdots,y_n)\right\|^2}{4}

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Topological Properties of \mathbb{R}^n

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There are multiple equivalent ways to define a topology on \mathbb{R}^n. One can give it the metric topology induced by the usual norm, in other words the metric topology induced by the metric d(x,y)=\|x-y\|. Or one could give it  the product topology induced of thinking of \mathbb{R}^n as a n-fold product space of \mathbb{R} (in either the order topology induced by the usual linear ordering on \mathbb{R} 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 \mathbb{R}^n has the Heine-Borel Property that while in general compact subspaces of metric spaces are closed and bounded the converse is true in \mathbb{R}^n (this can be proven first for \mathbb{R} and then extended by Tychonoff’s theorem to the general case). It is also true that \mathbb{R}^n is a complete metric space.

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We have as always the canonical projections \pi_i:\mathbb{R}^n\to\mathbb{R} for i=1,\cdots,n which takes (x_1,\cdots,x_n)\mapsto x_i. Every mapping f:X\to\mathbb{R}^n can, of course, be thought of as

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f:x\mapsto \left(\pi_1(f(x)),\cdots,\pi_n(f(x))\right)

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where given a f we denote, when no confusion arises, \pi_i\circ f as f_i and call it the i^{\text{th}} coordinate function. As usual a function f:X\to\mathbb{R}^n is continuous if and only if f_i:X\to\mathbb{R} is continuous for each i=1,\cdots,n. 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 f:E\to\mathbb{R}^m where E\subseteq\mathbb{R}^n that

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\displaystyle \lim_{x\to y}f(x)=(a_1,\cdots,a_m)\Leftrightarrow \lim_{x\to y}f_i(x)=a_i\;\;i=1,\cdots,m

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References:

1. Spivak, Michael. Calculus on Manifolds; a Modern Approach to Classical Theorems of Advanced Calculus. New York: W.A. Benjamin, 1965. Print.

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May 22, 2011 - Posted by | Analysis | , ,

7 Comments »

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