Euclidean space

A Euclidean space is a real inner product space $(V,\langle\cdot,\cdot\rangle)$ with $\dim V<\infty$.

Equivalently, it is a finite-dimensional inner product space (hence a finite-dimensional vector space ) over $\mathbb{R}$. The inner product determines lengths via the Euclidean norm and the notion of perpendicularity via orthogonality .

Examples:

• $\mathbb{R}^n$ with the dot product $\langle x,y\rangle=\sum_{i=1}^n x_i y_i$.
• Any linear subspace $W\subseteq \mathbb{R}^n$ with the dot product restricted to $W$.