# Normed Vector Space

A Normed Vector Space is a vector space that is equipped with a norm function.

**Context:**- It can range from being a Normed Linear Vector Space to being a Normed Non-Linear Vector Space.

**See:**Functional Analysis, Real Vector Space, Triangle Inequality, Linear Algebra, Topological Vector Space.

## References

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Normed_vector_space Retrieved:2015-2-7.
- 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**^{n}. The following properties of "vector length" are crucial.1. The zero vector,

**0**, has zero length; every other vector has a positive length. :[math]\displaystyle{ \|x\|\gt 0 }[/math] if [math]\displaystyle{ x\ne0 }[/math]2. Multiplying a vector by a positive number changes its length without changing its direction. Moreover,

:[math]\displaystyle{ \|\alpha x\|=|\alpha| \|x\| }[/math] for any scalar [math]\displaystyle{ \alpha. }[/math]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. :[math]\displaystyle{ \|x+y\| \le \|x\|+\|y\| }[/math] 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 vector space.Normed vector spaces are central to the study of linear algebra and functional analysis.

- 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

### 1997

- (Luenberger, 1997) ⇒ David G. Luenberger. (1997). “Optimization by Vector Space Methods." Wiley Professional. ISBN:047118117X