Realn

Realn is a nonstandard term used in some mathematical and computational contexts to denote an n-dimensional real vector space, typically represented as ℝ^n or R^n. In this usage, Realn denotes the set of all ordered n-tuples (x1, x2, ..., xn) of real numbers, equipped with the structure of a real vector space over the field of real numbers. Vector operations are defined componentwise: (x1, ..., xn) + (y1, ..., yn) = (x1+y1, ..., xn+yn), and for c in ℝ, c*(x1, ..., xn) = (c x1, ..., c xn). The space is commonly endowed with a norm, most often the Euclidean norm sqrt(x1^2 + ... + xn^2), which induces the standard Euclidean topology and makes Realn a finite-dimensional inner-product space.

Usage notes: Realn is not standard mathematical notation. In rigorous texts, the same object is usually written ℝ^n. The term is more likely to appear in informal notes, introductory tutorials, or programming contexts where one emphasizes the dimension n of a real-valued vector. Variants include infinite-dimensional analogues such as ℝ^∞ or ℓ^p spaces, but these are not realn per se.

Key properties include finite dimension n, completeness, and a canonical basis e1,...,en. The dual space is isomorphic to Realn itself, and inner products induce norms and notions of angle and projection.

Examples: Real3 is often used in computer graphics to denote a three-component real vector representing coordinates or colors.

See also: ℝ^n, Euclidean space, vector space, normed space.