# Definitions > [!NOTE] Definition 1 (Euclidean space is FDVS over field with an inner product) > A Euclidean space is a [[Finite Dimensional Vector Space|FDVS]] over $\mathbb{R}$ equipped with a fixed choice of [[Inner Product|inner product]]. > [!NOTE] Definition 2 (Euclidean space is metric space isometric to $\mathbb{R}^n$ with standard Euclidean norm) > A Euclidean space is a [[Metric Space|metric space]] that is [[Isometry|isometric]] to some [[Real n-Space|real n-space]], with the standard Euclidean metric. > [!Example] > A [[Line in Real n-Space]] is a Euclidean space. # Properties The [[Euclidean Norm]] is defined by ... > [!NOTE] Definition (Length of a vector) > Let $V$ be a Euclidean space. We define the length of a vector $v\in V$ to be $||v|| = \sqrt{ \langle v, v\rangle }$which is a non-negative real number. > [!NOTE] Definition (Angle) > Let $v,w\in V\setminus \{ 0_{V}\}$ be nonzero vectors. We define the angle between $v$ and $w,$ denoted $\angle vw,$ to be the real number $\angle = \cos^{-1} \left( \frac{\langle v, w \rangle}{||v||||w||} \right)$where we take the principal preimage of $\cos,$ so that $\angle vw$ lies in the interval $[0,\pi].$ >Note that [[Upper Bound for Absolute Value of Dot Product in Real n-Space (Cauchy-Schwartz Inequality)]] guarantees that argument of $\cos^{-1}$ lies within $[-1,1].$ > [!NOTE] Theorem (Isomorphism with $\mathbb{R}^{n}$ the usual dot product) > Let $V$ be a Euclidean space and let $B$ be an [[Orthonormal Subset of Euclidean Space|orthonormal basis]]. Then the [[Existence of Coordinate Map with respect to Basis for Finite Dimensional Real Vector Space|coordinate isomorphism]] $\chi_{B}:V \to \mathbb{R}^{n}$matches the inner product of $V$ with the usual [[Dot Product in Real n-Space|dot product]] of $\mathbb{R}^{n},$ in the sense that $\langle v,w \rangle = \chi_{B}(v)\cdot \chi_{B}(w)$for all $v,w\in V.$ >*Proof*. Let $B=\{ w_{1},\dots,w_{n} \}$ so that by definition the coordinate isomorphism $\chi_{B}$ is determined by $\chi_{B}(w_{i})=\underline{e}_{i}\in\mathbb{R}^{n}.$ Observe first that therefore $\chi_{B}$ has the required property on the elements of $B$: for any $i,j\in\{ 1,\dots,n \}$ $\langle w_{i}, w_{j} \rangle= \delta_{ij} = \underline{e}_{i} \cdot \underline{e}_{j}$Now by linearity, writing any $v\in V$ with respect to $B$ as $v= \sum_{i=1}^{n} \lambda_{i}w_{i},$ we have $\begin{aligned} & \langle v,w_j\rangle=\left\langle\sum_{i=1}^n\lambda_iw_i,w_j\right\rangle && =\quad\sum_{i=1}^n\lambda_i\left<w_i,w_j\right> \\ &&&=\quad\sum_{i=1}^n\lambda_i\left(\underline{e}_i\cdot\underline{e}_j\right) \\ &&&=\quad\left(\sum_{i=1}^n\lambda_i\underline{e}_i\right)\cdot\underline{e}_j \\ &&&=\quad\chi_{\mathcal{B}}(v)\cdot\chi_{\mathcal{B}}(w_j) \end{aligned}$and similarly in the right-hand factor of $\langle \cdot,\cdot \rangle.$