Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, let $n\in\mathbb{N}$, and let $[n]$ be as in Intervals of Natural Numbers (ii).

(i) Euclidean space $\mathbb{R}^{n}$ is the set of $n$-tuples in $\mathbb{R}$, that is, $\mathbb{R}^{[n]}$, $[n]$ being a set by Tuples in a Set: the Set of n-Tuples and Their Components (i); its elements are called points or vectors of $\mathbb{R}^{n}$.

(ii) For a set $u$: $u\in\mathbb{R}^{n}$ if and only if $u$ is a map $u:[n]\to\mathbb{R}$, by Tuples in a Set: the Set of n-Tuples and Their Components (ii). For $x\in\mathbb{R}^{n}$ and $k\in[n]$, $x_{k}=x(k)$ is the $k$-th component of $x$, and $x$ is written $(x_{1},\dots,x_{n})$.

(iii) For $x,y\in\mathbb{R}^{n}$: $x=y$ if and only if $x_{k}=y_{k}$ for every $k\in[n]$, by Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (i).

(iv) For $x,y\in\mathbb{R}^{n}$ and $t\in\mathbb{R}$, the sum $x+y$, the scalar multiple $tx$, the negative $-x$ and the difference $x-y$ are those of Pointwise Operations on Real-Valued Maps (i) with $I=[n]$, so that, for $k\in[n]$, $(x+y)_{k}=x_{k}+y_{k}$, $(tx)_{k}=t\,x_{k}$ and $(-x)_{k}=-x_{k}$, and $(x-y)_{k}=x_{k}-y_{k}$ by Rules for Pointwise Operations on Real-Valued Maps: a Commutative Ring, Scalar Multiples and Evaluation (ii).

(v) The origin of $\mathbb{R}^{n}$ is the constant map $0=(0,\dots,0)$ of Pointwise Operations on Real-Valued Maps (ii).