Euclidean spaces are generalizations of two- and three-dimensional spaces of Euclidean geometry to arbitrary finite-dimensional spaces. It is the abstraction of a geometric object into a topological and algebraic structure.
Many concepts in algebra and analysis are developed by analogy with geometry, where Euclidean space is among the first few.
Euclidean space is the space of all $n$-tuples of real numbers $(x_1, x_2, \cdots, x_n)$, with inner product $\mathbf{x}\cdot\mathbf{y} = \sum_{i=1}^n x_i y_i$. It is commonly denoted as $\mathbb{R}^n$.
Length
Angle
Orthogonal
Ortho-normal basis
Gram-Schmidt Orthonormalization, QR decomposition
Orthogonal transformation: rotation, reflection; Improper rotation
Orthogonal subspace
direct sum
orthogonal complement
Thm: Any finite-dimensional linear inner-product space over the real numbers is isometric with the Euclidean space of the same dimension.