# Proposition: Definition of the Metric Space $$\mathbb R^n$$, Euclidean Norm

Let $$n$$ be a natural number. The cartesian product of the $$n$$ sets of real numbers. $\mathbb R^n:=\underbrace{\mathbb R\times \mathbb R\times\cdots\times \mathbb R}_{n\text{ times}}.$

constitutes a vector space over $$\mathbb R$$, also called the Euclidean vector space of dimension $$n$$, and denoted by $$\mathbb R^n$$. It means that:

1. $$(\mathbb R^n, + )$$ is an Abelian group.
2. For $$\alpha,\beta\in \mathbb R$$ and $$x,y\in \mathbb R^n$$ the following axioms of scalar multiplication hold:

# 1. $$1\cdot x=x$$.

With the $\text{vector addition "+"}:=\pmatrix{ x_{1}\cr x_{2}\cr \vdots \cr x_{n} \cr }+\pmatrix{ y_{1}\cr y_{2}\cr \vdots \cr y_{n} \cr }=\pmatrix{ x_{1}+y_1\cr x_{2}+y_2\cr \vdots \cr x_{n}+y_n \cr }$ $\text{and scalar multiplication "}\cdot\text{"}:=\alpha\cdot\pmatrix{ x_{1}\cr x_{2}\cr \vdots \cr x_{n} \cr }=\pmatrix{ \alpha\cdot x_{1}\cr \alpha\cdot x_{2}\cr \vdots \cr \alpha\cdot x_{n} \cr }$

We consider the dot product $$\langle x,y\rangle:=x_1y_1 + x_2y_2 + \ldots + x_ny_n$$ of two vectors $$x=(x_{1},\ldots,x_{n})^T$$ and $$y=(y_{1},\ldots,y_{n})^T$$ in $$\mathbb R^n$$. The Euclidean norm of any vector $$x\in\mathbb R^n$$ is defined as the square root of the dot product of $$x$$ with itself:

$||x||:=\sqrt{\langle x,x\rangle}=\sqrt{x_1^2+\ldots+x_n^2}.$

The Euclidean distance of any two vectors $$x,y\in\mathbb R^n$$ is the Euclidean norm of the difference of these vectors:

$d(x,y)=||x-y||:=\sqrt{\langle x-y,x-y\rangle}=\sqrt{(x_1-y_1)^2+\ldots+(x_n-y_n)^2}.$

Together with the distance defined as above, the pair $$(\mathbb R^n,d)$$ constitutes a metric space. Together with the norm defined as above, the pair $$(\mathbb R^n,||~||)$$ constitutes a normed vector space.

Proofs: 1

Definitions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Explanations: 21
Proofs: 22 23
Propositions: 24 25
Theorems: 26

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### References

#### Bibliography

1. Forster Otto: "Analysis 2, Differentialrechnung im $$\mathbb R^n$$, Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984