Euclidean Space ${ℝ}^{n}$

Last updates: 25 November 2011

Euclidean Space ${ℝ}^{n}$

The Euclidean space ${ℝ}^{n}$ is the set $ℝn ={ ( x1,…,xn ) | x1,…, xn ∈ℝ }$ with addition operation ${ℝ}^{n}×{ℝ}^{n}\to {ℝ}^{n}$ given by $( x1,…,xn ) + ( y1,…,yn ) = ( x1+y1, …,xn+yn ) ,$ and scalar multiplication $ℝ×{ℝ}^{n}\to {ℝ}^{n}$ given by $c ( x1,…,xn ) = ( cx1,…, cxn ).$

Define the absolute value on ${ℝ}^{n}$ $|x| : ℝn ⟶ ℝ≥0 x ⟼ |x| by |x| = x12 +⋯+ xn2$ if $x=\left({x}_{1},\dots ,{x}_{n}\right)$.

Define the inner product on ${ℝ}^{n}$ $⟨, ⟩: ℝn×ℝn →ℝ by ⟨x,y⟩ = x1y1 +⋯+xnyn$ if $x=\left({x}_{1},\dots ,{x}_{n}\right)$ and $y=\left({y}_{1},\dots ,{y}_{n}\right)$.

Define the distance on ${ℝ}^{n}$ $d:ℝn ×ℝn →ℝ≥0 by d(x,y) = |y-x| .$

Let $\epsilon \in {ℝ}_{>0}$. The $\epsilon$-ball at $x$ is $ℬε(x) = {y∈ℝn | d(y,x)<ε }.$

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

 $E is a union of ε-balls.$

(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?????