# Normed Vector Space

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

## 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 Rn. The following properties of "vector length" are crucial.

1. The zero vector, '0, has zero length; every other vector has a positive length. :$\|x\|\gt 0$ if $x\ne0$

2. Multiplying a vector by a positive number changes its length without changing its direction. Moreover,
:$\|\alpha x\|=|\alpha| \|x\|$ for any scalar $\alpha.$

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. :$\|x+y\| \le \|x\|+\|y\|$ 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 vector space.[1]

Normed vector spaces are central to the study of linear algebra and functional analysis.

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