Euclidean Space ${ℝ}^n$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 25 November 2011

Euclidean Space ${ℝ}^n$

The Euclidean space ${ℝ}^n$ is the set $${ℝ}^n=\left\{(x_1,\dots ,x_n)\right|x_1,\dots ,x_n\in ℝ\}$$ with addition operation ${ℝ}^n\times {ℝ}^n\to {ℝ}^n$ given by $$(x_1,\dots ,x_n)+(y_1,\dots ,y_n)=(x_1+y_1,\dots ,x_n+y_n),$$ and scalar multiplication $ℝ\times {ℝ}^n\to {ℝ}^n$ given by $$c(x_1,\dots ,x_n)=(c{x}_1,\dots ,c{x}_n).$$

Define the absolute value on ${ℝ}^n$ $$\begin{array}{rrcl}\left|\phantom{x}\right|:& {ℝ}^n& ⟶& {ℝ}_{\ge 0}\\ & x& ⟼& \left|x\right|\end{array}\text{by}\left|x\right|=\sqrt{x_1^2+\cdots +x_n^2}$$ if $x=(x_1,\dots ,x_n)$.

Define the inner product on ${ℝ}^n$ $$⟨,⟩:{ℝ}^n\times {ℝ}^n\to ℝ\text{by}⟨x,y⟩=x_1{y}_1+\cdots +x_n{y}_n$$ if $x=(x_1,\dots ,x_n)$ and $y=(y_1,\dots ,y_n)$.

Define the distance on ${ℝ}^n$ $$d:{ℝ}^n\times {ℝ}^n\to {ℝ}_{\ge 0}\text{by}d(x,y)=|y-x|.$$

Let $\epsilon \in {ℝ}_{>0}$. The $\epsilon$-ball at $x$ is $${ℬ}_{\epsilon }\left(x\right)=\{y\in {ℝ}^n|d(y,x)<\epsilon \}.$$

Let $E$ be a subset of ${ℝ}^n$. The set $E$ is open if

$$E\text{is a union of}\epsilon \text{-balls.}$$

(a) The set ${ℝ}^n$ with the operations of addition, scalar multiplication and open sets as in (1.1) is a topological vector space.

(b) Identifying $ℂ$ with ${ℝ}^2$, the set $ℂ$ with the operations of addition, multiplication and open sets as in (1.1) is a topological field.

Notes and References

These notes are written to highlight the analogy between the structures on $ℝ$ and the structures on ${ℝ}^n$.

References

[BouTop] N. Bourbaki, General Topology, Chapter VI, Springer-Verlag, Berlin 1989. MR?????

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