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 Rⁿ. 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.

1997

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

---