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. \((\alpha + \beta) x=\alpha x + \beta x\),

1. \(\alpha(x + y)=\alpha x + \alpha y\),

1. \((\alpha\beta) x=\alpha (\beta x)\),

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