Euclidean Space

Definition

A Euclidian space is a space of the form ${\mathbb{R}}^I$ for some indexing set $I$. As a metric space $\mathbb{R}$ uses the Euclidian metric:

Euclidian Metric

Idea

The Euclidian metric is a metric on the vector space ${\mathbb{R}}^n$ gives a distance measure to the distance to origin from any point in the vector space. It is the standard dinstance measure used in Eculician space and has the property of being invariant over all unital transformations.

Definition

Let $v\in {\mathbb{R}}^n$. Then

$$\parallel v{\parallel }_2:=\sqrt{\sum _{i\in [\mathrm{0..}n)}{v}_i^2}$$

Remarks

• This definition also works for any field closed that has a solution to equations of the form $x^2={\sum }_i{y}_i^2$ for fixed $y_i$. $\mathbb{C}$ is another such field, but $\mathbb{Q}$ fails since $x^2=1^2+1^2=2$ has no solution in $\mathbb{Q}$.

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