What is "normed vector space"

n. (context mathematics English) A vector space which is additionally equipped with a norm.

In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space R. The following properties of "vector length" are crucial.

1. The zero vector, 0, has zero length; every other vector has a positive length. ∥x∥ > 0 if x ≠ 0
2. Multiplying a vector by a positive number changes its length without changing its direction. Moreover,
   : ∥αx∥ = ∣α∣∥x∥ for any scalar α.
3. The triangle inequality holds. That is, taking norms as distances, the distance from point A through B to C is never shorter than going directly from A to C, or the shortest distance between any two points is a straight line. ∥x + y∥ ≤ ∥x∥ + ∥y∥ for any vectors x and y. (triangle inequality)

The generalization of these three properties to more abstract vector spaces leads to the notion of norm. A vector space on which a norm is defined is then called a normed space or normed vector space. Normed vector spaces are central to the study of linear algebra and functional analysis.